On Growth and Form

But it is possible that the modern engineer might be disposed to criticise the skeleton girder at two or three points; and in particular he might think the girder, as we see it for instance i Diplodocus or Stegosaurus, not deep enough for carrying the animal’s enormous weight of some twenty tons. If we adopt aq 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 be flexible as well as strong. Now this opens up a new aspect of the matter and is the beginning of a long, long story, for in every direction this double requirement of strength and flexibility imposes new conditions upon our design. To represent all the correlated quantities we should have to construct not only a diagram of moments but also a diagram of elastic deflexion and its so-called “curvature”; and the engineer would want to know something more about the material of the ligamentous tensionmember—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 @appui for the muscles concerned in locomotion. We cannot hope to arrive at a numerical or quantitative solution of this complicate problem, but we have found it possible to trace it out in part towards a qualitative solution. And speaking broadly we may certainly say that in each case the problem has been solved by nature herself, very much as she solves the difficult problems of minimal areas in a system of soap-bubbles; so that each animal is fitted with a backbone adapted to his own individual needs, or (in other words) corresponding exactly to the mean resultant of the stresses to which as a mechanical system it is exposed.

Throughout this short discussion of the principles of construction, limited to one part of the skeleton, we see the same general principles at work which we recognise in the plan and construction of an individual bone. That is to say, we see a tendency for material to be laid down just in the lines of stress, and so as to evade thereby the distortions and disruptions due to shear. In these phenomena there lies a definite law of growth, whatever its ultimate expression or explanation may come to be. Let us not press either argument or hypothesis too far: but be content to see that skeletal form, as brought about by growth, is to a very large extent determined by mechanical considerations, and tends to manifest itself as a diagram, or reflected image, of mechanical stress. If we fail, owing to the immense complexity of the case, to unravel all the mathematical principles involved in the construction of the skeleton, we yet gain something, and not a little, by applying this method to the familiar objects of our anatomical study: obvia conspicimus, nubem pellente mathesi*.

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 im strictly functional correlation. But this conception, though never denied, is sometimes apt to be forgotten in the course of that process of more and more minute analysis by which, for simplicity’s sake, we seek to unravel the intricacies of a complex organism.

We tend, as we analyse a thing into its parts or into its properties, to magnify these, to exaggerate their apparent independence, and to hide from ourselves (at least for a time) the essential integrity and individuality of the composite whole. We divide the body into its organs, the skeleton into its bones, as in very much the same fashion we make a subjective analysis of the mind, according to the teachings of psychology, into component factors: but we know very well that judgment and knowledge, courage or gentleness, love or fear, have no separate existence, but are somehow mere manifestations, or imaginary co-efficients, of a most complex integral. And likewise, as biologists, we may go so far as to say that even the bones themselves are only in a limited and even a deceptive sense, separate and individual things. -The skeleton begins as a continuum, and a continuum it remains all life long. The things that link bone with bone,

cartilage, ligaments, membranes, are fashioned out of the same primordial tissue, and come into being pari passu, with the bones themselves. The entire fabric has its soft parts and its hard, its rigid and its flexible parts; but until we disrupt and dismember its bony, gristly and fibrous parts, one from another, it exists simply as a “skeleton,” as one integral and individual whole. A bridge was once upon a time a loose heap of pillars and rods and rivets of steel. But the identity of these is lost, just as if they were fused into a solid mass, when once the bridge is built; their separate functions are only to be recognised and analysed in so far as we can analyse the stresses, the tensions and the pressures, which affect this part of the structure or that; and these forces are not themselves separate entities, but are the resultants of an analysis of the whole field of force. Moreover when the bridge is broken it is no: longer a bridge, and all its strength is gone. So is it precisely with the skeleton. In it is reflected a field of force: and keeping pace, as it were, in action and interaction with this field of force, the whole skeleton and every part thereof, down to the minute intrinsic structure of the bones themselves, is related in form and in position to the lines of force, to the resistances it has to encounter; for by one of the mysteries of biology, resistance begets resistance, and where pressure falls there growth springs up in strength to meet it. And, pursuing the same train of thought, we see that all this is true not of the skeleton alone but of the whole fabric of the body. Muscle and. bone, for instance, are inseparably associated and connected; they are moulded one with another; they come into being together, and act and react together*. 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 * John Hunter was seldom wrong; but I cannot believe that he was right when he said (Scientific Works, ed. Owen, 1, p. 371), ‘‘The bones, in a mechanical view, appear to be the first that are to be considered. We can study their shape, connexions, number, uses, etc., without considering any other part of the body.”

in this limited and subordinate sense, that they are parts of a whole which, when it loses its composite integrity, ceases to exist. The biologist, as well as the philosopher, learns to recognise that the whole is not merely the sum of its parts. It is this, and much more than this. For it is not a bundle of parts but an organisation of parts, of parts in their mutual arrangement, fitting one with another, in what Aristotle calls “a single and indivisible principle of unity”; and this is no merely metaphysical conception, but is in biology the fundamental truth which lies at the basis of Geoffroy’s (or Goethe’s) law of “compensation,” or “balancement of growth.”

Nevertheless Darwin found no difficulty in believing that ‘natural selection will tend in the long run to reduce any part of the organisation, as soon as, through changed habits, it becomes superfluous: without by any means causing some other part to be largely developed in a corresponding degree. And conversely, that natural selection may perfectly well succeed in largely developing an organ without requiring as a necessary compensation the reduction of some adjoining part*.” This view has been developed into a doctrine of the “independence of single characters” (not to be confused with the germinal “unit characters” of Mendelism), especially by the palaeontologists. Thus Osborn asserts a “principle of hereditary correlation,” combined with a “principle of hereditary separability whereby the body is a colony, a mosaic, of single individual and separable characters}.” I cannot think that there is more than a small element of truth in this doctrine. As Kant said, “die Ursache der Art der Existenz bei jedem Theile eines lebenden Korpers ist om Ganzen enthalten.” And, according to the trend or aspect of our thought, we may look upon the co-ordinated parts, now as related and fitted to the

end or function of the whole, and now as related to or resulting from the physical causes inherent in the entire system of forces to which the whole has been exposed, and under whose influence it has come into beingt. It would seem to me that the mechanical principles and phenomena which we have dealt with in this chapter are of no small impottance to the morphologist, all the more when he is inclined to direct his study of the skeleton exclusively to the problem of phylogeny; and especially when, according to the methods of modern comparative morphology, he is apt to take the skeleton to pieces, and to draw from the comparison of a series of scapulae, humeri, or individual vertebrae, conclusions as to the descent and relationship of the animals to which they belong.

It would, I dare say, be a gross exaggeration to see in every bone nothing more than a resultant of immediate and direct physical or mechanical conditions; for to do so would be te deny the existence, in this connection, of a principle of heredity. And though I have tried throughout this book to lay emphasis on the direct action of causes other than heredity, in short to circumscribe the employment of the latter as a working hypothesis in morphology, there can still be no question whatsoever but that heredity is a vastly important as well as a mysterious thing; it is one of the great factors in biology, however we may attempt to figure to ourselves, or howsoever we may fail even to imagine, its underlying physical explanation. But I maintain that it is no less an exaggeration if we tend to neglect these direct physical and mechanical modes of causation altogether, and to see in the characters of a bone merely the results of variation and of heredity, and to trust, in consequence, to those characters as a sure and certain and unquestioned guide to affinity and phylogeny. Comparative anatomy has its physiological side, which filled men’s minds in John Hunter’s day, and in Owen’s day; it has its

from the sum of its parts in the case of the organism: ‘The organism, we know, is a system the single constituents of which are inorganic in themselves; only the whole constituted by them in their typical order or arrangement owes its specificity to ‘Entelechy’” (Gifford Lectures, p. 229, 19U8): and I think it could be shewn that many other philosophers have said precisely the same thing. So far as the argument goes, I fail to see how this Entelechy is shewn to be peculiarly or specifically related to the living organism. The conception that the whole is always something very different from its parts is a very ancient doctrine. The reader will perhaps remember how, in another vein, the theme is treated by Martinus Scriblerus: ‘‘In every Jack there is a meat-roasting Quality, which neither resides in the fly, nor in the weight, nor in any particular wheel of the Jack, but is the result of the whole composition; etc., etc.”

classificatory and phylogenetic aspect, which has all but filled men’s minds during the last couple of generations; and we can lose sight of neither aspect without risk of error and misconcéption. It is certain that the question of phylogeny, always difficult, becomes especially so in cases where a great change of physical or mechanical conditions has come about, and where accordingly the physical and physiological factors in connection with change of form are bound to be large. To discuss these questions at length would be to enter on a discussion of Lamarck’s philosophy of biology, and of many other things besides. But let us take one single illustration.

The affinities of the whales constitute, as will be readily admitted, a very hard problem in phylogenetic classification. We know now that the extinct Zeuglodons are related to the old Creodont carnivores, and thereby (though distantly) to the seals; and it is supposed, but it is by no means so certain, that in turn they are to be considered as representing, or as allied to, the ancestors of the modern toothed whales*. The proof of any such a contention becomes, to my mind, extraordinarily difficult and complicated; and the arguments commonly used in such cases may be said (in Bacon’s phrase) to allure, rather than to extort assent. Though the Zeuglodonts were aquatic animals, we do not know, and we have no right to suppose or to assume, that they swam after the fashion of a whale (any more than the seal does), that they dived like a whale, and leaped like a whale. But the fact that the whale does these things, and the way in which he does them, is reflected in many parts of his skeleton—perhaps more or less in all: so much so that the lines of stress which these actions impose are the very plan and working-diagram of great part of his structure. That the Zeuglodon has a scapula like that of a whale is to my mind no necessary argument that he is akin by blood-relationship to a whale: that his dorsal vertebrae are very different from a whale’s is no conclusive argument that

* “There can be no doubt that Fraas is correct in regarding this type (Procetus) as an annectant form between the Zeuglodonts and the Creodonta, but, although the origin of the Zeuglodonts is thus made clear, it still seems to be by no means so certain as that author believes, that they may not themselves be the ancestral forms of the Odontoceti”; Andrews, Tertiary Vertebrata of the Fayum, 1906, p- 235. such blood-relationship is lacking. The former fact goes a long way to prove that he used his flippers very much as a whale does; the latter goes still farther to prove that his general movements and equilibrium in the water were totally different. The whale may be descended from the Carnivora, or might for that matter, as an older school of naturalists believed, be descended from the Ungulates; but whether or no, we need not expect to find in him the scapula, the pelvis or the vertebral column of the lion or of the cow, for it would be physically impossible that he could live the life he does with any one of them. In short, when we hope to find the missing links between a whale and his terrestrial ancestors, it must be not by means of conclusions drawn from a scapula, an axis, or even from a tooth, but by the discovery of forms so intermediate in their general structure as to indicate an organisation ’ and, zpso facto, a mode of life, intermediate between the terrestrial and the Cetacean form. There is no valid syllogism to the effect that A has a flat curved scapula like a seal’s, and B has a flat, curved scapula like a seal’s: and therefore A and B are related to the seals and to each other; it is merely a flagrant case of an “undistributed middle.” But there is validity in an argument that B shews in its general structure, extending over this bone and that bone, resemblances both to A and to the seals: and that therefore he may be presumed to be related to both, in his hereditary habits of life and in actual kinship by blood. It is cognate to this argument that (as every palaeontologist knows) we find clues to affinity more easily, that is to say with less confusion and perplexity, in certain structures than in others.

The deep-seated rhythms of growth which, as I venture to think, are the chief basis of morphological heredity, bring about similarities of form, which endure in the absence of conflicting forces; but a new system of forces, introduced by altered environment and habits, impinging on those particular parts of the fabric which le 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 modification, spread over a number of correlated parts, and at times extending over the whole, or over great portions, of the body. Whether any such general tendency to widespread and correlated transformation exists, we shall attempt to discuss in the following chapter. In the foregoing chapters of this book we have attempted to study the inter-relations of growth and form, and the part which certain of the physical forces play in this complex interaction ; and, as part of the same enquiry, we have tried in comparatively — simple cases to use mathematical methods and mathematical terminology in order to describe and define the forms of organisms. We have learned in so doing that our own study of organic form, which we call by Goethe’s name of Morphology, is but a portion of that wider Science of Form which deals with the forms assumed by matter under all aspects and conditions, and, in a still wider sense, with forms which are theoretically imaginable.

The study of form may be descriptive merely, or it may become analytical. We begin by describing the shape of an object in the simple words of common speech: we end by defining it in the precise language of mathematics; and the one method tends to follow the other in strict scientific order and historical continuity. Thus, for instance, the form of the earth, of a raindrop or a rainbow, the shape of the hanging chain, or the path of a stone thrown up into the air, may all be described, however inadequately, in common words; but when we have learned to comprehend and to define the sphere, the catenary, or the parabola, we have made a wonderful and perhaps a manifold advance. The mathematical definition of a “form” has a quality of precision which was quite lacking in our earlier stage of mere description; it is expressed in few words, or in still briefer symbols, and these

words or symbols are so pregnant with meaning that thought itself is economised; we are brought by means of it in touch with Galileo’s aphorism (as old as Plato, as old as Pythagoras, as old perhaps as the wisdom of the Egyptians), that “the Book of Nature is written in characters of Geometry.” Next, we soon reach through mathematical analysis to mathematical synthesis; we discover homologies or identities which were not obvious before, and which our descriptions obscured rather than revealed: as for instance, when we learn that, however we hold our chain, or however we fire our bullet, the contour of the one or the path of the other is always mathematically homologous. Lastly, and this is the greatest gain of all, we pass quickly and easily from the mathematical conception of form in its statical aspect to form in its dynamical relations: we pass from the conception of form to an understanding of the forces which. gave rise to it; and in the representation of form and in the comparison of kindred forms, we see in the one case a diagram of forces in equilibrium, and in the other case we discern the magnitude and the direction of the forces which have sufficed to convert the one form into the other. Here, since a change of material form is only effected by the movement of matter, we have once again the support of the schoolman’s and the philosopher’s axiom, Ignorato motu, ignoratur Natura.”

In the morphology of living things the use of mathematical methods and symbols has made slow progress; and there are various reasons for this failure to employ a method whose advantages are so obvious in the investigation of other physical forms. To begin with, there would seem to be a psychological reason lying in the fact that the student of living things is by nature and training an observer of concrete objects and phenomena, and the habit of mind which he possesses and cultivates is alien to that of the theoretical mathematician. But this is by no means the only reason; for in the kindred subject of mineralogy, for instance, crystals were still treated in the days of Linnaeus as wholly within the province of the naturalist, and were described by him after the simple methods in use for animals and plants: but as soon as Haiiy showed 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 than the features in which it agrees, and even that the peculiar aesthetic pleasure with which we regard a living thing is somehow bound up with the departure from mathematical regularity which it manifests as a peculiar attribute of life. This view seems to me to involve a misapprehension. There is no such essential difference between these phenomena of organic form and those which are manifested in portions of inanimate matter*. No chain hangs in a perfect catenary and no raindrop is a perfect sphere: and this for the simple reason that forces and resistances other than the main one are inevitably at work. The same is true of organic form, but it is for the mathematician to unravel the conflicting forces which are at work together. And this process of investigation may lead us on step by step to new phenomena, as it has done in physics, where sometimes a knowledge of form leads us to the interpretation of forces, and at other times a knowledge of the forces at work guides us towards a better insight into form. I would illustrate this by the case of the earth itself. After the fundamental advance had been made which taught us that the world was round, Newton showed that the forces at work upon it must lead to its being imperfectly spherical, and in the course of time its oblate spheroidal shape was actually verified.

But now, in turn, it has been shown that its form is still more complicated, and the next step will be to seek for the forces that have deformed the oblate spheroid. * M. Bergson repudiates, with peculiar confidence, the application of mathematics to biology. Cf. Creative Evolution, p. 21, “Calculation touches, at most, certain phenomena of organic destruction. Organic creation, on the contrary, the evolutionary phenomena which properly constitute life, we cannot in any way subject to a mathematical treatment.”

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 zdeal simplicity, so do we pass quickly to cases where the interference of numerous, though still perhaps very simple, causes leads to a resultant which lies far beyond our powers of analysis. Nor must we forget that the biologist is much more exacting in his requirements, as regards form, than the physicist; for the latter is usually content with either an ideal or a general description of form, while the student of living things must needs be specific. The physicist or mathematician can give us perfectly satisfying expressions for the form of a wave, or even of a heap of sand; but we never ask him to define the form of any particular

wave of the sea, nor the actual form of any monntain-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 until its results can be expressed mathematically. The error comes from the assumption that mathematics can express complex relations. Unfortunately mathematics have a very limited scope, and are based upon a few extremely rudimentary experiences, which we make as very little children and of which no adult has any recollection. The fact that from this basis men of genius have evolved wonderful methods of dealing with numerical relations should not blind us to another fact, namely, that the observational basis of mathematics is, psychologically speaking, very minute compared with the observational basis of even a single minor branch of biology....While therefore here and there the mathematical methods may aid us, we need a kind and degree of accuracy of which mathematics is absolutely incapable....With human minds constituted as they actually are, we cannot anticipate that there will ever be a mathematical expression for any organ or even a single cell, although formulae will continue to be useful for dealing now and then with isolated details...” (op. cit., p. 19, 1911). It were easy to discuss and criticise these sweeping assertions, which perhaps had their origin and parentage in an obiter dictum of Huxley’s, to the effect that “Mathematics is that study which knows nothing of observation, nothing of experiment,

For various reasons, then, there are a vast multitude of organic forms which we are unable to account for, or to define, in mathematical terms; and this is not seldom the case even in forms which are apparently of great simplicity and regularity. The curved outline of a leaf, for instance, is such a case; its ovate, lanceolate, or cordate shape is apparently very simple, but the difficulty of finding for it a mathematical expression is very great indeed. To define the complicated outline of a fish, for instance, or of a vertebrate skull, we never even seek for a mathematical formula.

But in a very large part of morphology, our essential task lies in the comparison of related forms rather than in the precise definition of each; and the deformation of a complicated figure may be a phenomenon easy of comprehension, though the figure itself have to be left unanalysed and undefined. This process of comparison, of recognising in one form a definite permutation or deformation of another, apart altogether from a precise and adequate understanding of the original “type” or standard of comparison, lies within the immediate province of mathematics, and finds its solution in the elementary use of a certain method of the mathematician. ‘This method is the Method of Co-ordinates, on which is based the Theory of Transformations.

I imagine that when Descartes conceived the method of co-ordinates, as a generalisation from the proportional diagrams of the artist and the architect, and long before the immense possibilities of this analysis could be foreseen, he had in mind a very simple purpose; it was perhaps no more than to find a way of translating the form of a curve into numbers and into words. This is precisely what we do, by the method of co-ordinates, every time we study a statistical curve; and conversely, we translate numbers into form whenever we “plot a curve” to illustrate a table of mortality, a rate of growth, or the daily variation of temperature or barometric pressure. In precisely the same way it is possible to inscribe in a net of rectangular co-ordinates the outline, for instance, of a fish, and so to translate

nothing of induction, nothing of causation” (cit. Cajori, Hist of Elem. Mathematics, p- 283). But Gauss called mathematics “a science of the eye’; and Sylvester assures us that “most, if not all, of the great ideas of modern mathematics have had their origin in observation” (Brit. Ass. Address, 1869, and Laws of Verse, p. 120, it into a table of numbers, from which again we may at pleasure reconstruct the curve. But it is the next step in the employment of co-ordinates which is of special interest and use to the morphologist; and this step consists in the alteration, or “transformation,” of our system of co-ordinates and in the study of the corresponding transformation of the curve or figure inscribed in the co-ordinate network. .

Let us inscribe in a system of Cartesian co-ordinates the outline of an organism, however complicated, or a part thereof: such as a fish, a crab, or a mammalian skull. We may now treat this complicated figure, in general terms, as a function of x, y. If we - submit our rectangular system to “deformation,” on simple and recognised lines, altering, for instance, the direction of the axes, the ratio of «/y, or substituting for « and y some more complicated expressions, then we shall obtain a new system of co-ordinates, whose deformation from the original type the inscribed figure will precisely follow. In other words, we obtain a new figure, which represents the old figure under strain, and is a function of the new co-ordinates in precisely the same way as the old figure was of the original co-ordinates x and y.