Unit I: Mathematical Morphology, Magnitude, and Rate of Growth¶
1917 Cambridge First Edition — Prefatory Note, Epigraphs & Chapters I–IV (pp. v–vi, 1–200)
Prefatory Note¶
1917 Cambridge First Edition (pp. v–vi)
“The reasonings about the wonderful and intricate operations of nature are so full of uncertainty, that, as the Wise-man truly observes, hardly do we guess aright at the things that are upon earth, and with labour do we find the things that are before us.” Stephen Hales, Vegetable Staticks (1727), p. 318, 1738.
This book of mine has little need of preface, for indeed it is “all preface” from beginning to end. I have written it as an easy introduction to the study of organic Form, by methods which are the common-places of physical science, which are by no means novel in their application to natural history, but which nevertheless naturalists are little accustomed to employ.
It is not the biologist with an inkling of mathematics, but the skilled and learned mathematician who must ultimately deal with such problems as are merely sketched and adumbrated here. I pretend to no mathematical skill, but I have made what use I could of what tools I had; I have dealt with simple cases, and the mathematical methods which I have introduced are of the easiest and simplest kind. Elementary as they are, my book has not been written without the help—the indispensable help—of many friends. Like Mr Pope translating Homer, when I felt myself deficient I sought assistance! And the experience which Johnson attributed to Pope has been mine also, that men of learning did not refuse to help me.
My debts are many, and I will not try to proclaim them all: but I beg to record my particular obligations to Professor Claxton Fidler, Sir George Greenhill, Sir Joseph Larmor, and Professor A. McKenzie; to a much younger but very helpful friend, Mr John Marshall, Scholar of Trinity; lastly, and (if I may say so) most of all, to my colleague Professor William Peddie, whose advice has made many useful additions to my book and whose criticism has spared me many a fault and blunder.
I am under obligations also to the authors and publishers of many books from which illustrations have been borrowed, and especially to the following:―
To the Controller of H.M. Stationery Office, for leave to reproduce a number of figures, chiefly of Foraminifera and of Radiolaria, from the Reports of the Challenger Expedition. [p. vi]
To the Council of the Royal Society of Edinburgh, and to that of the Zoological Society of London:—the former for letting me reprint from their Transactions the greater part of the text and illustrations of my concluding chapter, the latter for the use of a number of figures for my chapter on Horns.
To Professor E. B. Wilson, for his well-known and all but indispensable figures of the cell (figs. 42–51, 53); to M. A. Prenant, for other figures (41, 48) in the same chapter; to Sir Donald MacAlister and Mr Edwin Arnold for certain figures (335–7), and to Sir Edward Schäfer and Messrs Longmans for another (334), illustrating the minute trabecular structure of bone. To Mr Gerhard Heilmann, of Copenhagen, for his beautiful diagrams (figs. 388–93, 401, 402) included in my last chapter. To Professor Claxton Fidler and to Messrs Griffin, for letting me use, with more or less modification or simplification, a number of illustrations (figs. 339–346) from Professor Fidler’s Textbook of Bridge Construction. To Messrs Blackwood and Sons, for several cuts (figs. 127–9, 131, 173) from Professor Alleyne Nicholson’s Palaeontology; to Mr Heinemann, for certain figures (57, 122, 123, 205) from Dr Stéphane Leduc’s Mechanism of Life; to Mr A. M. Worthington and to Messrs Longmans, for figures (71, 75) from A Study of Splashes, and to Mr C. R. Darling and to Messrs E. and S. Spon for those (fig. 85) from Mr Darling’s Liquid Drops and Globules. To Messrs Macmillan and Co. for two figures (304, 305) from Zittel’s Palaeontology, to the Oxford University Press for a diagram (fig. 28) from Mr J. W. Jenkinson’s Experimental Embryology; and to the Cambridge University Press for a number of figures from Professor Henry Woods’s Invertebrate Palaeontology, for one (fig. 210) from Dr Willey’s Zoological Results, and for another (fig. 321) from “Thomson and Tait.”
Many more, and by much the greater part of my diagrams, I owe to the untiring help of Dr Doris L. Mackinnon, D.Sc., and of Miss Helen Ogilvie, M.A., B.Sc., of this College.
Epigraphs (Aquinas, Newton, Jellett)¶
1917 Cambridge First Edition (p. xvi)
“Cum formarum naturalium et corporalium esse non consistat nisi in unione ad materiam, ejusdem agentis esse videtur eas producere cujus est materiam transmutare. Secundo, quia cum hujusmodi formae non excedant virtutem et ordinem et facultatem principiorum agentium in natura, nulla videtur necessitas eorum originem in principia reducere altiora.” Aquinas, De Pot. Q. iii, a, 11. (Quoted in Brit. Assoc. Address, Section D, 1911.)
“…I would that all other natural phenomena might similarly be deduced from mechanical principles. For many things move me to suspect that everything depends upon certain forces, in virtue of which the particles of bodies, through forces not yet understood, are either impelled together so as to cohere in regular figures, or are repelled and recede from one another.” Newton, in Preface to the Principia. (Quoted by Mr W. Spottiswoode, Brit. Assoc. Presidential Address, 1878.)
“When Science shall have subjected all natural phenomena to the laws of Theoretical Mechanics, when she shall be able to predict the result of every combination as unerringly as Hamilton predicted conical refraction, or Adams revealed to us the existence of Neptune,—that we cannot say. That day may never come, and it is certainly far in the dim future. We may not anticipate it, we may not even call it possible. But none the less are we bound to look to that day, and to labour for it as the crowning triumph of Science:—when Theoretical Mechanics shall be recognised as the key to every physical enigma, the chart for every traveller through the dark Infinite of Nature.” J. H. Jellett, in Brit. Assoc. Address, Section A, 1874.
Chapter I: Introductory¶
1917 Cambridge First Edition (pp. 1–15)
Of the chemistry of his day and generation, Kant declared that it was “a science, but not science,”—“eine Wissenschaft, aber nicht Wissenschaft”; for that the criterion of physical science lay in its relation to mathematics. And a hundred years later Du Bois Reymond, profound student of the many sciences on which physiology is based, recalled and reiterated the old saying, declaring that chemistry would only reach the rank of science, in the high and strict sense, when it should be found possible to explain chemical reactions in the light of their causal relation to the velocities, tensions and conditions of equilibrium of the component molecules; that, in short, the chemistry of the future must deal with molecular mechanics, by the methods and in the strict language of mathematics, as the astronomy of Newton and Laplace dealt with the stars in their courses. We know how great a step has been made towards this distant and once hopeless goal, as Kant defined it, since van’t Hoff laid the firm foundations of a mathematical chemistry, and earned his proud epitaph, Physicam chemiae adiunxit [1].
We need not wait for the full realisation of Kant’s desire, in order to apply to the natural sciences the principle which he urged. Though chemistry fall short of its ultimate goal in mathematical mechanics, nevertheless physiology is vastly strengthened and enlarged by making use of the chemistry, as of the physics, of the age. Little by little it draws nearer to our conception of a true science, with each branch of physical science which it [p. 2] brings into relation with itself: with every physical law and every mathematical theorem which it learns to take into its employ. Between the physiology of Haller, fine as it was, and that of Helmholtz, Ludwig, Claude Bernard, there was all the difference in the world.
As soon as we adventure on the paths of the physicist, we learn to weigh and to measure, to deal with time and space and mass and their related concepts, and to find more and more our knowledge expressed and our needs satisfied through the concept of number, as in the dreams and visions of Plato and Pythagoras; for modern chemistry would have gladdened the hearts of those great philosophic dreamers.
But the zoologist or morphologist has been slow, where the physiologist has long been eager, to invoke the aid of the physical or mathematical sciences; and the reasons for this difference lie deep, and in part are rooted in old traditions. The zoologist has scarce begun to dream of defining, in mathematical language, even the simpler organic forms. When he finds a simple geometrical construction, for instance in the honey-comb, he would fain refer it to psychical instinct or design rather than to the operation of physical forces; when he sees in snail, or nautilus, or tiny foraminiferal or radiolarian shell, a close approach to the perfect sphere or spiral, he is prone, of old habit, to believe that it is after all something more than a spiral or a sphere, and that in this “something more” there lies what neither physics nor mathematics can explain. In short he is deeply reluctant to compare the living with the dead, or to explain by geometry or by dynamics the things which have their part in the mystery of life. Moreover he is little inclined to feel the need of such explanations or of such extension of his field of thought. He is not without some justification if he feels that in admiration of nature’s handiwork he has an horizon open before his eyes as wide as any man requires. He has the help of many fascinating theories within the bounds of his own science, which, though a little lacking in precision, serve the purpose of ordering his thoughts and of suggesting new objects of enquiry. His art of classification becomes a ceaseless and an endless search after the blood-relationships of things living, and the pedigrees of things [p. 3] dead and gone. The facts of embryology become for him, as Wolff, von Baer and Fritz Müller proclaimed, a record not only of the life-history of the individual but of the annals of its race. The facts of geographical distribution or even of the migration of birds lead on and on to speculations regarding lost continents, sunken islands, or bridges across ancient seas. Every nesting bird, every ant-hill or spider’s web displays its psychological problems of instinct or intelligence. Above all, in things both great and small, the naturalist is rightfully impressed, and finally engrossed, by the peculiar beauty which is manifested in apparent fitness or “adaptation,”—the flower for the bee, the berry for the bird.
Time out of mind, it has been by way of the “final cause,” by the teleological concept of “end,” of “purpose,” or of “design,” in one or another of its many forms (for its moods are many), that men have been chiefly wont to explain the phenomena of the living world; and it will be so while men have eyes to see and ears to hear withal. With Galen, as with Aristotle, it was the physician’s way; with John Ray, as with Aristotle, it was the naturalist’s way; with Kant, as with Aristotle, it was the philosopher’s way. It was the old Hebrew way, and has its splendid setting in the story that God made “every plant of the field before it was in the earth, and every herb of the field before it grew.” It is a common way, and a great way; for it brings with it a glimpse of a great vision, and it lies deep as the love of nature in the hearts of men.
Half overshadowing the “efficient” or physical cause, the argument of the final cause appears in eighteenth century physics, in the hands of such men as Euler [2] and Maupertuis, to whom Leibniz [3] had passed it on. Half overshadowed by the mechanical concept, it runs through Claude Bernard’s Leçons sur les [p. 4] phénomènes de la Vie [4], and abides in much of modern physiology [5]. Inherited from Hegel, it dominated Oken’s Naturphilosophie and lingered among his later disciples, who were wont to liken the course of organic evolution not to the straggling branches of a tree, but to the building of a temple, divinely planned, and the crowning of it with its polished minarets [6].
It is retained, somewhat crudely, in modern embryology, by those who see in the early processes of growth a significance “rather prospective than retrospective,” such that the embryonic phenomena must be “referred directly to their usefulness in building the body of the future animal [7]”:—which is no more, and no less, than to say, with Aristotle, that the organism is the τέλος, or final cause, of its own processes of generation and development. It is writ large in that Entelechy [8] which Driesch rediscovered, and which he made known to many who had neither learned of it from Aristotle, nor studied it with Leibniz, nor laughed at it with Voltaire. And, though it is in a very curious way, we are told that teleology was “refounded, reformed or rehabilitated [9]” by Darwin’s theory of natural selection, whereby “every variety of form and colour was urgently and absolutely called upon to produce its title to existence either as an active useful agent, or as a survival” of such active usefulness in the past. But in this last, and very important case, we have reached a “teleology” without a τέλος, [p. 5] as men like Butler and Janet have been prompt to shew: a teleology in which the final cause becomes little more, if anything, than the mere expression or resultant of a process of sifting out of the good from the bad, or of the better from the worse, in short of a process of mechanism [10]. The apparent manifestations of “purpose” or adaptation become part of a mechanical philosophy, according to which “chaque chose finit toujours par s’accommoder à son milieu [11].” In short, by a road which resembles but is not the same as Maupertuis’s road, we find our way to the very world in which we are living, and find that if it be not, it is ever tending to become, “the best of all possible worlds [12].”
But the use of the teleological principle is but one way, not the whole or the only way, by which we may seek to learn how things came to be, and to take their places in the harmonious complexity of the world. To seek not for ends but for “antecedents” is the way of the physicist, who finds “causes” in what he has learned to recognise as fundamental properties, or inseparable concomitants, or unchanging laws, of matter and of energy. In Aristotle’s parable, the house is there that men may live in it; but it is also there because the builders have laid one stone upon another: and it is as a mechanism, or a mechanical construction, that the physicist looks upon the world. Like warp and woof, mechanism and teleology are interwoven together, and we must not cleave to the one and despise the other; for their union is “rooted in the very nature of totality [13].”
Nevertheless, when philosophy bids us hearken and obey the lessons both of mechanical and of teleological interpretation, the precept is hard to follow: so that oftentimes it has come to pass, just as in Bacon’s day, that a leaning to the side of the final cause “hath intercepted the severe and diligent inquiry of all [p. 6] real and physical causes,” and has brought it about that “the search of the physical cause hath been neglected and passed in silence.” So long and so far as “fortuitous variation [14]” and the “survival of the fittest” remain engrained as fundamental and satisfactory hypotheses in the philosophy of biology, so long will these “satisfactory and specious causes” tend to stay “severe and diligent inquiry,” “to the great arrest and prejudice of future discovery.”
The difficulties which surround the concept of active or “real” causation, in Bacon’s sense of the word, difficulties of which Hume and Locke and Aristotle were little aware, need scarcely hinder us in our physical enquiry. As students of mathematical and of empirical physics, we are content to deal with those antecedents, or concomitants, of our phenomena, without which the phenomenon does not occur,—with causes, in short, which, aliae ex aliis aptae et necessitate nexae, are no more, and no less, than conditions sine quâ non. Our purpose is still adequately fulfilled: inasmuch as we are still enabled to correlate, and to equate, our particular phenomena with more and ever more of the physical phenomena around, and so to weave a web of connection and interdependence which shall serve our turn, though the metaphysician withhold from that interdependence the title of causality. We come in touch with what the schoolmen called a ratio cognoscendi, though the true ratio efficiendi is still enwrapped in many mysteries. And so handled, the quest of physical causes merges with another great Aristotelian theme,—the search for relations between things apparently disconnected, and for “similitude in things to common view unlike.” Newton did not shew the cause of the apple falling, but he shewed a similitude between the apple and the stars.
Moreover, the naturalist and the physicist will continue to speak of “causes,” just as of old, though it may be with some mental reservations: for, as a French philosopher said, in a kindred difficulty: “ce sont là des manières de s’exprimer, [p. 7] et si elles sont interdites il faut renoncer à parler de ces choses.”
The search for differences or essential contrasts between the phenomena of organic and inorganic, of animate and inanimate things has occupied many mens’ minds, while the search for community of principles, or essential similitudes, has been followed by few; and the contrasts are apt to loom too large, great as they may be. M. Dunan, discussing the “Problème de la Vie [15]” in an essay which M. Bergson greatly commends, declares: “Les lois physico-chimiques sont aveugles et brutales; là où elles règnent seules, au lieu d’un ordre et d’un concert, il ne peut y avoir qu’incohérence et chaos.” But the physicist proclaims aloud that the physical phenomena which meet us by the way have their manifestations of form, not less beautiful and scarce less varied than those which move us to admiration among living things. The waves of the sea, the little ripples on the shore, the sweeping curve of the sandy bay between its headlands, the outline of the hills, the shape of the clouds, all these are so many riddles of form, so many problems of morphology, and all of them the physicist can more or less easily read and adequately solve: solving them by reference to their antecedent phenomena, in the material system of mechanical forces to which they belong, and to which we interpret them as being due. They have also, doubtless, their immanent teleological significance; but it is on another plane of thought from the physicist’s that we contemplate their intrinsic harmony and perfection, and “see that they are good.”
Nor is it otherwise with the material forms of living things. Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed [16]. [p. 8] They are no exception to the rule that Θεὸς ἀεὶ γεωμετρεῖ. Their problems of form are in the first instance mathematical problems, and their problems of growth are essentially physical problems; and the morphologist is, ipso facto, a student of physical science.
Apart from the physico-chemical problems of modern physiology, the road of physico-mathematical or dynamical investigation in morphology has had few to follow it; but the pathway is old. The way of the old Ionian physicians, of Anaxagoras [17], of Empedocles and his disciples in the days before Aristotle, lay just by that highwayside. It was Galileo’s and Borelli’s way. It was little trodden for long afterwards, but once in a while Swammerdam and Réaumur looked that way. And of later years, Moseley and Meyer, Berthold, Errera and Roux have been among the little band of travellers. We need not wonder if the way be hard to follow, and if these wayfarers have yet gathered little. A harvest has been reaped by others, and the gleaning of the grapes is slow.
It behoves us always to remember that in physics it has taken great men to discover simple things. They are very great names indeed that we couple with the explanation of the path of a stone, the droop of a chain, the tints of a bubble, the shadows in a cup. It is but the slightest adumbration of a dynamical morphology that we can hope to have, until the physicist and the mathematician shall have made these problems of ours their own, or till a new Boscovich shall have written for the naturalist the new Theoria Philosophiae Naturalis.
How far, even then, mathematics will suffice to describe, and physics to explain, the fabric of the body no man can foresee. It may be that all the laws of energy, and all the properties of matter, and all the chemistry of all the colloids are as powerless to explain the body as they are impotent to comprehend the soul. For my part, I think it is not so. Of how it is that the soul informs the body, physical science teaches me nothing: consciousness is not explained to my comprehension by all the nerve-paths and “neurones” of the physiologist; nor do I ask of physics how goodness shines in one man’s face, and evil betrays itself in another. But of the construction and growth and working [p. 9] of the body, as of all that is of the earth earthy, physical science is, in my humble opinion, our only teacher and guide [18].
Often and often it happens that our physical knowledge is inadequate to explain the mechanical working of the organism; the phenomena are superlatively complex, the procedure is involved and entangled, and the investigation has occupied but a few short lives of men. When physical science falls short of explaining the order which reigns throughout these manifold phenomena,—an order more characteristic in its totality than any of its phenomena in themselves,—men hasten to invoke a guiding principle, an entelechy, or call it what you will. But all the while, so far as I am aware, no physical law, any more than that of gravity itself, not even among the puzzles of chemical “stereometry,” or of physiological “surface-action” or “osmosis,” is known to be transgressed by the bodily mechanism.
Some physicists declare, as Maxwell did, that atoms or molecules more complicated by far than the chemist’s hypotheses demand are requisite to explain the phenomena of life. If what is implied be an explanation of psychical phenomena, let the point be granted at once; we may go yet further, and decline, with Maxwell, to believe that anything of the nature of physical complexity, however exalted, could ever suffice. Other physicists, like Auerbach [19], or Larmor [20], or Joly [21], assure us that our laws of thermodynamics do not suffice, or are “inappropriate,” to explain the maintenance or (in Joly’s phrase) the “accelerative absorption” [p. 10] of the bodily energies, and the long battle against the cold and darkness which is death. With these weighty problems I am not for the moment concerned. My sole purpose is to correlate with mathematical statement and physical law certain of the simpler outward phenomena of organic growth and structure or form: while all the while regarding, ex hypothesi, for the purposes of this correlation, the fabric of the organism as a material and mechanical configuration.
Physical science and philosophy stand side by side, and one upholds the other. Without something of the strength of physics, philosophy would be weak; and without something of philosophy’s wealth, physical science would be poor. “Rien ne retirera du tissu de la science les fils d’or que la main du philosophe y a introduits [22].” But there are fields where each, for a while at least, must work alone; and where physical science reaches its limitations, physical science itself must help us to discover. Meanwhile the appropriate and legitimate postulate of the physicist, in approaching the physical problems of the body, is that with these physical phenomena no alien influence interferes. But the postulate, though it is certainly legitimate, and though it is the proper and necessary prelude to scientific enquiry, may some day be proven to be untrue; and its disproof will not be to the physicist’s confusion, but will come as his reward. In dealing with forms which are so concomitant with life that they are seemingly controlled by life, it is in no spirit of arrogant assertiveness that the physicist begins his argument, after the fashion of a most illustrious exemplar, with the old formulary of scholastic challenge,—An Vita sit? Dico quod non.
The terms Form and Growth, which make up the title of this little book, are to be understood, as I need hardly say, in their relation to the science of organisms. We want to see how, in some cases at least, the forms of living things, and of the parts of living things, can be explained by physical considerations, and to realise that, in general, no organic forms exist save such as are in conformity with ordinary physical laws. And while growth is a somewhat vague word for a complex matter, which may [p. 11] depend on various things, from simple imbibition of water to the complicated results of the chemistry of nutrition, it deserves to be studied in relation to form, whether it proceed by simple increase of size without obvious alteration of form, or whether it so proceed as to bring about a gradual change of form and the slow development of a more or less complicated structure.
In the Newtonian language of elementary physics, force is recognised by its action in producing or in changing motion, or in preventing change of motion or in maintaining rest. When we deal with matter in the concrete, force does not, strictly speaking, enter into the question, for force, unlike matter, has no independent objective existence. It is energy in its various forms, known or unknown, that acts upon matter. But when we abstract our thoughts from the material to its form, or from the thing moved to its motions, when we deal with the subjective conceptions of form, or movement, or the movements that change of form implies, then force is the appropriate term for our conception of the causes by which these forms and changes of form are brought about. When we use the term force, we use it, as the physicist always does, for the sake of brevity, using a symbol for the magnitude and direction of an action in reference to the symbol or diagram of a material thing. It is a term as subjective and symbolic as form itself, and so is appropriately to be used in connection therewith.
The form, then, of any portion of matter, whether it be living or dead, and the changes of form that are apparent in its movements and in its growth, may in all cases alike be described as due to the action of force. In short, the form of an object is a “diagram of forces,” in this sense, at least, that from it we can judge of or deduce the forces that are acting or have acted upon it: in this strict and particular sense, it is a diagram,—in the case of a solid, of the forces that have been impressed upon it when its conformation was produced, together with those that enable it to retain its conformation; in the case of a liquid (or of a gas) of the forces that are for the moment acting on it to restrain or balance its own inherent mobility. In an organism, great or small, it is not merely the nature of the motions of the living substance that we must interpret in terms of force (according to kinetics), but also [p. 12] the conformation of the organism itself, whose permanence or equilibrium is explained by the interaction or balance of forces, as described in statics.
If we look at the living cell of an Amoeba or a Spirogyra, we see a something which exhibits certain active movements, and a certain fluctuating, or more or less lasting, form; and its form at a given moment, just like its motions, is to be investigated by the help of physical methods, and explained by the invocation of the mathematical conception of force.
Now the state, including the shape or form, of a portion of matter, is the resultant of a number of forces, which represent or symbolise the manifestations of various kinds of energy; and it is obvious, accordingly, that a great part of physical science must be understood or taken for granted as the necessary preliminary to the discussion on which we are engaged. But we may at least try to indicate, very briefly, the nature of the principal forces and the principal properties of matter with which our subject obliges us to deal. Let us imagine, for instance, the case of a so-called “simple” organism, such as Amoeba; and if our short list of its physical properties and conditions be helpful to our further discussion, we need not consider how far it be complete or adequate from the wider physical point of view [23].
This portion of matter, then, is kept together by the intermolecular force of cohesion; in the movements of its particles relatively to one another, and in its own movements relative to adjacent matter, it meets with the opposing force of friction. It is acted on by gravity, and this force tends (though slightly, owing to the Amoeba’s small mass, and to the small difference between its density and that of the surrounding fluid), to flatten it down upon the solid substance on which it may be creeping. Our Amoeba tends, in the next place, to be deformed by any pressure from outside, even though slight, which may be applied to it, and this circumstance shews it to consist of matter in a fluid, or at least semi-fluid, state: which state is further indicated when we observe streaming or current motions in its interior. [p. 13] Like other fluid bodies, its surface, whatsoever other substance, gas, liquid or solid, it be in contact with, and in varying degree according to the nature of that adjacent substance, is the seat of molecular force exhibiting itself as a surface-tension, from the action of which many important consequences follow, which greatly affect the form of the fluid surface.
While the protoplasm of the Amoeba reacts to the slightest pressure, and tends to “flow,” and while we therefore speak of it as a fluid, it is evidently far less mobile than such a fluid, for instance, as water, but is rather like treacle in its slow creeping movements as it changes its shape in response to force. Such fluids are said to have a high viscosity, and this viscosity obviously acts in the way of retarding change of form, or in other words of retarding the effects of any disturbing action of force. When the viscous fluid is capable of being drawn out into fine threads, a property in which we know that the material of some Amoebae differs greatly from that of others, we say that the fluid is also viscid, or exhibits viscidity. Again, not by virtue of our Amoeba being liquid, but at the same time in vastly greater measure than if it were a solid (though far less rapidly than if it were a gas), a process of molecular diffusion is constantly going on within its substance, by which its particles interchange their places within the mass, while surrounding fluids, gases and solids in solution diffuse into and out of it. In so far as the outer wall of the cell is different in character from the interior, whether it be a mere pellicle as in Amoeba or a firm cell-wall as in Protococcus, the diffusion which takes place through this wall is sometimes distinguished under the term osmosis.
Within the cell, chemical forces are at work, and so also in all probability (to judge by analogy) are electrical forces; and the organism reacts also to forces from without, that have their origin in chemical, electrical and thermal influences. The processes of diffusion and of chemical activity within the cell result, by the drawing in of water, salts, and food-material with or without chemical transformation into protoplasm, in growth, and this complex phenomenon we shall usually, without discussing its nature and origin, describe and picture as a force. Indeed we shall manifestly be inclined to use the term growth in two senses, [p. 14] just indeed as we do in the case of attraction or gravitation, on the one hand as a process, and on the other hand as a force.
In the phenomena of cell-division, in the attractions or repulsions of the parts of the dividing nucleus and in the “caryokinetic” figures that appear in connection with it, we seem to see in operation forces and the effects of forces, that have, to say the least of it, a close analogy with known physical phenomena; and to this matter we shall afterwards recur. But though they resemble known physical phenomena, their nature is still the subject of much discussion, and neither the forms produced nor the forces at work can yet be satisfactorily and simply explained. We may readily admit, then, that besides phenomena which are obviously physical in their nature, there are actions visible as well as invisible taking place within living cells which our knowledge does not permit us to ascribe with certainty to any known physical force; and it may or may not be that these phenomena will yield in time to the methods of physical investigation. Whether or no, it is plain that we have no clear rule or guide as to what is “vital” and what is not; the whole assemblage of so-called vital phenomena, or properties of the organism, cannot be clearly classified into those that are physical in origin and those that are sui generis and peculiar to living things. All we can do meanwhile is to analyse, bit by bit, those parts of the whole to which the ordinary laws of the physical forces more or less obviously and clearly and indubitably apply.
Morphology then is not only a study of material things and of the forms of material things, but has its dynamical aspect, under which we deal with the interpretation, in terms of force, of the operations of Energy. And here it is well worth while to remark that, in dealing with the facts of embryology or the phenomena of inheritance, the common language of the books seems to deal too much with the material elements concerned, as the causes of development, of variation or of hereditary transmission. Matter as such produces nothing, changes nothing, does nothing; and however convenient it may afterwards be to abbreviate our nomenclature and our descriptions, we must most carefully realise in the outset that the spermatozoon, the nucleus, [p. 15] the chromosomes or the germ-plasm can never act as matter alone, but only as seats of energy and as centres of force. And this is but an adaptation (in the light, or rather in the conventional symbolism, of modern physical science) of the old saying of the philosopher: ἀρχὴ γὰρ ἡ φύσις μᾶλλον τῆς ὕλης.
Chapter II: On Magnitude¶
1917 Cambridge First Edition (pp. 16–49)
To terms of magnitude, and of direction, must we refer all our conceptions of form. For the form of an object is defined when we know its magnitude, actual or relative, in various directions; and growth involves the same conceptions of magnitude and direction, with this addition, that they are supposed to alter in time. Before we proceed to the consideration of specific form, it will be worth our while to consider, for a little while, certain phenomena of spatial magnitude, or of the extension of a body in the several dimensions of space [24].
We are taught by elementary mathematics that, in similar solid figures, the surface increases as the square, and the volume as the cube, of the linear dimensions. If we take the simple case of a sphere, with radius r, the area of its surface is equal to 4πr2, and its volume to (4/3)πr3; from which it follows that the ratio of volume to surface, or V/S, is (1/3)r. In other words, the greater the radius (or the larger the sphere) the greater will be its volume, or its mass (if it be uniformly dense throughout), in comparison with its superficial area. And, taking L to represent any linear dimension, we may write the general equations in the form
From these elementary principles a great number of consequences follow, all more or less interesting, and some of them of great importance. In the first place, though growth in length (let [p. 17] us say) and growth in volume (which is usually tantamount to mass or weight) are parts of one and the same process or phenomenon, the one attracts our attention by its increase, very much more than the other. For instance a fish, in doubling its length, multiplies its weight by no less than eight times; and it all but doubles its weight in growing from four inches long to five.
In the second place we see that a knowledge of the correlation between length and weight in any particular species of animal, in other words a determination of k in the formula W = k ⋅ L3, enables us at any time to translate the one magnitude into the other, and (so to speak) to weigh the animal with a measuring-rod; this however being always subject to the condition that the animal shall in no way have altered its form, nor its specific gravity. That its specific gravity or density should materially or rapidly alter is not very likely; but as long as growth lasts, changes of form, even though inappreciable to the eye, are likely to go on. Now weighing is a far easier and far more accurate operation than measuring; and the measurements which would reveal slight and otherwise imperceptible changes in the form of a fish—slight relative differences between length, breadth and depth, for instance,—would need to be very delicate indeed. But if we can make fairly accurate determinations of the length, which is very much the easiest dimension to measure, and then correlate it with the weight, then the value of k, according to whether it varies or remains constant, will tell us at once whether there has or has not been a tendency to gradual alteration in the general form. To this subject we shall return, when we come to consider more particularly the rate of growth.
But a much deeper interest arises out of this changing ratio of dimensions when we come to consider the inevitable changes of physical relations with which it is bound up. We are apt, and even accustomed, to think that magnitude is so purely relative that differences of magnitude make no other or more essential difference; that Lilliput and Brobdingnag are all alike, according as we look at them through one end of the glass or the other. But this is by no means so; for scale has a very marked effect upon physical phenomena, and the effect of scale constitutes what is known as the principle of similitude, or of dynamical similarity. [p. 18]
This effect of scale is simply due to the fact that, of the physical forces, some act either directly at the surface of a body, or otherwise in proportion to the area of surface; and others, such as gravity, act on all particles, internal and external alike, and exert a force which is proportional to the mass, and so usually to the volume, of the body.
The strength of an iron girder obviously varies with the cross-section of its members, and each cross-section varies as the square of a linear dimension; but the weight of the whole structure varies as the cube of its linear dimensions. And it follows at once that, if we build two bridges geometrically similar, the larger is the weaker of the two [25]. It was elementary engineering experience such as this that led Herbert Spencer [26] to apply the principle of similitude to biology.
The same principle had been admirably applied, in a few clear instances, by Lesage [27], a celebrated eighteenth century physician of Geneva, in an unfinished and unpublished work [28]. Lesage argued, for instance, that the larger ratio of surface to mass would lead in a small animal to excessive transpiration, were the skin as “porous” as our own; and that we may hence account for the hardened or thickened skins of insects and other small terrestrial animals. Again, since the weight of a fruit increases as the cube of its dimensions, while the strength of the stalk increases as the square, it follows that the stalk should grow out of apparent due proportion to the fruit; or alternatively, that tall trees should not bear large fruit on slender branches, and that melons and pumpkins must lie upon the ground. And again, that in quadrupeds a large head must be supported on a neck which is either [p. 19] excessively thick and strong, like a bull’s, or very short like the neck of an elephant.
But it was Galileo who, wellnigh 300 years ago, had first laid down this general principle which we now know by the name of the principle of similitude; and he did so with the utmost possible clearness, and with a great wealth of illustration, drawn from structures living and dead [29]. He showed that neither can man build a house nor can nature construct an animal beyond a certain size, while retaining the same proportions and employing the same materials as sufficed in the case of a smaller structure [30]. The thing will fall to pieces of its own weight unless we either change its relative proportions, which will at length cause it to become clumsy, monstrous and inefficient, or else we must find a new material, harder and stronger than was used before. Both processes are familiar to us in nature and in art, and practical applications, undreamed of by Galileo, meet us at every turn in this modern age of steel.
Again, as Galileo was also careful to explain, besides the questions of pure stress and strain, of the strength of muscles to lift an increasing weight or of bones to resist its crushing stress, we have the very important question of bending moments. This question enters, more or less, into our whole range of problems; it affects, as we shall afterwards see, or even determines the whole form of the skeleton, and is very important in such a case as that of a tall tree [31].
Here we have to determine the point at which the tree will curve under its own weight, if it be ever so little displaced from the perpendicular [32]. In such an investigation we have to make [p. 20] some assumptions,—for instance, with regard to the trunk, that it tapers uniformly, and with regard to the branches that their sectional area varies according to some definite law, or (as Ruskin assumed [33]) tends to be constant in any horizontal plane; and the mathematical treatment is apt to be somewhat difficult. But Greenhill has shewn that (on such assumptions as the above), a certain British Columbian pine-tree, which yielded the Kew flagstaff measuring 221 ft. in height with a diameter at the base of 21 inches, could not possibly, by theory, have grown to more than about 300 ft. It is very curious that Galileo suggested precisely the same height (dugento braccia alta) as the utmost limit of the growth of a tree. In general, as Greenhill shews, the diameter of a homogeneous body must increase as the power 3/2 of the height, which accounts for the slender proportions of young trees, compared with the stunted appearance of old and large ones [34]. In short, as Goethe says in Wahrheit und Dichtung, “Es ist dafür gesorgt dass die Bäume nicht in den Himmel wachsen.” But Eiffel’s great tree of steel (1000 feet high) is built to a very different plan; for here the profile of the tower follows the logarithmic curve, giving equal strength throughout, according to a principle which we shall have occasion to discuss when we come to treat of “form and mechanical efficiency” in connection with the skeletons of animals.
Among animals, we may see in a general way, without the help of mathematics or of physics, that exaggerated bulk brings with it a certain clumsiness, a certain inefficiency, a new element of risk and hazard, a vague preponderance of disadvantage. The case was well put by Owen, in a passage which has an interest of its own as a premonition (somewhat like De Candolle’s) of the “struggle for existence.” Owen wrote as follows [35]: “In proportion to the bulk of a species is the difficulty of the contest which, as a living organised whole, the individual of such species [p. 21] has to maintain against the surrounding agencies that are ever tending to dissolve the vital bond, and subjugate the living matter to the ordinary chemical and physical forces. Any changes, therefore, in such external conditions as a species may have been originally adapted to exist in, will militate against that existence in a degree proportionate, perhaps in a geometrical ratio, to the bulk of the species. If a dry season be greatly prolonged, the large mammal will suffer from the drought sooner than the small one; if any alteration of climate affect the quantity of vegetable food, the bulky Herbivore will first feel the effects of stinted nourishment.”
But the principle of Galileo carries us much further and along more certain lines.
The tensile strength of a muscle, like that of a rope or of our girder, varies with its cross-section; and the resistance of a bone to a crushing stress varies, again like our girder, with its cross-section. But in a terrestrial animal the weight which tends to crush its limbs or which its muscles have to move, varies as the cube of its linear dimensions; and so, to the possible magnitude of an animal, living under the direct action of gravity, there is a definite limit set. The elephant, in the dimensions of its limb-bones, is already shewing signs of a tendency to disproportionate thickness as compared with the smaller mammals; its movements are in many ways hampered and its agility diminished: it is already tending towards the maximal limit of size which the physical forces permit. But, as Galileo also saw, if the animal be wholly immersed in water, like the whale, (or if it be partly so, as was in all probability the case with the giant reptiles of our secondary rocks), then the weight is counterpoised to the extent of an equivalent volume of water, and is completely counterpoised if the density of the animal’s body, with the included air, be identical (as in a whale it very nearly is) with the water around. Under these circumstances there is no longer a physical barrier to the indefinite growth in magnitude of the animal [36]. Indeed, [p. 22] in the case of the aquatic animal there is, as Spencer pointed out, a distinct advantage, in that the larger it grows the greater is its velocity. For its available energy depends on the mass of its muscles; while its motion through the water is opposed, not by gravity, but by “skin-friction,” which increases only as the square of its dimensions; all other things being equal, the bigger the ship, or the bigger the fish, the faster it tends to go, but only in the ratio of the square root of the increasing length. For the mechanical work (W) of which the fish is capable being proportional to the mass of its muscles, or the cube of its linear dimensions: and again this work being wholly done in producing a velocity (V) against a resistance (R) which increases as the square of the said linear dimensions; we have at once
This is what is known as Froude’s Law of the correspondence of speeds.
But there is often another side to these questions, which makes them too complicated to answer in a word. For instance, the work (per stroke) of which two similar engines are capable should obviously vary as the cubes of their linear dimensions, for it varies on the one hand with the surface of the piston, and on the other, with the length of the stroke; so is it likewise in the animal, where the corresponding variation depends on the cross-section of the muscle, and on the space through which it contracts. But in two precisely similar engines, the actual available horse-power varies as the square of the linear dimensions, and not as the cube; and this for the obvious reason that the actual energy developed depends upon the heating-surface of the boiler [37]. So likewise must there be a similar tendency, among animals, for the rate of supply of kinetic energy to vary with the surface of the [p. 23] lung, that is to say (other things being equal) with the square of the linear dimensions of the animal. We may of course (departing from the condition of similarity) increase the heating-surface of the boiler, by means of an internal system of tubes, without increasing its outward dimensions, and in this very way nature increases the respiratory surface of a lung by a complex system of branching tubes and minute air-cells; but nevertheless in two similar and closely related animals, as also in two steam-engines of precisely the same make, the law is bound to hold that the rate of working must tend to vary with the square of the linear dimensions, according to Froude’s law of steamship comparison. In the case of a very large ship, built for speed, the difficulty is got over by increasing the size and number of the boilers, till the ratio between boiler-room and engine-room is far beyond what is required in an ordinary small vessel [38]; but though we find lung-space increased among animals where greater rate of working is required, as in general among birds, I do not know that it can be shewn to increase, as in the “over-boilered” ship, with the size of the animal, and in a ratio which outstrips that of the other bodily dimensions. If it be the case then, that the working mechanism of the muscles should be able to exert a force proportionate to the cube of the linear bodily dimensions, while the respiratory mechanism can only supply a store of energy at a rate proportional to the square of the said dimensions, the singular result ought to follow that, in swimming for instance, the larger fish ought to be able to put on a spurt of speed far in excess of the smaller one; but the distance travelled by the year’s end should be very much alike for both of them. And it should also follow that the curve of fatigue [p. 24] should be a steeper one, and the staying power should be less, in the smaller than in the larger individual. This is the case of long-distance racing, where the big winner puts on his big spurt at the end. And for an analogous reason, wise men know that in the ’Varsity boat-race it is judicious and prudent to bet on the heavier crew.
Leaving aside the question of the supply of energy, and keeping to that of the mechanical efficiency of the machine, we may find endless biological illustrations of the principle of similitude.
In the case of the flying bird (apart from the initial difficulty of raising itself into the air, which involves another problem) it may be shewn that the bigger it gets (all its proportions remaining the same) the more difficult it is for it to maintain itself aloft in flight. The argument is as follows:
In order to keep aloft, the bird must communicate to the air a downward momentum equivalent to its own weight, and therefore proportional to the cube of its own linear dimensions. But the momentum so communicated is proportional to the mass of air driven downwards, and to the rate at which it is driven: the mass being proportional to the bird’s wing-area, and also (with any given slope of wing) to the speed of the bird, and the rate being again proportional to the bird’s speed; accordingly the whole momentum varies as the wing-area, i.e. as the square of the linear dimensions, and also as the square of the speed. Therefore, in order that the bird may maintain level flight, its speed must be proportional to the square root of its linear dimensions.
Now the rate at which the bird, in steady flight, has to work in order to drive itself forward, is the rate at which it communicates energy to the air; and this is proportional to m V2, i.e. to the mass and to the square of the velocity of the air displaced. But the mass of air displaced per second is proportional to the wing-area and to the speed of the bird’s motion, and therefore to the power 2½ of the linear dimensions; and the speed at which it is displaced is proportional to the bird’s speed, and therefore to the square root of the linear dimensions. Therefore the energy communicated per second (being proportional to the mass and to the square of the speed) is jointly proportional to the power 2½ of the linear dimensions, as above, and to the first power thereof: [p. 25] that is to say, it increases in proportion to the power 3½ of the linear dimensions, and therefore faster than the weight of the bird increases.
Put in mathematical form, the equations are as follows:
(m = the mass of air thrust downwards; V its velocity, proportional to that of the bird; M its momentum; l a linear dimension of the bird; w its weight; W the work done in moving itself forward.)
The work requiring to be done, then, varies as the power 3½ of the bird’s linear dimensions, while the work of which the bird is capable depends on the mass of its muscles, and therefore varies as the cube of its linear dimensions [39]. The disproportion does not seem at first sight very great, but it is quite enough to tell. It is as much as to say that, every time we double the linear dimensions of the bird, the difficulty of flight is increased in the ratio of 23 : 23½, or 8 : 11·3, or, say, 1 : 1·4. If we take the ostrich to exceed the sparrow in linear dimensions as 25 : 1, which seems well within the mark, we have the ratio between 253½ and 253, or between 57 : 56; in other words, flight is just five times more difficult for the larger than for the smaller bird [40].
The above investigation includes, besides the final result, a number of others, explicit or implied, which are of not less importance. Of these the simplest and also the most important is [p. 26] contained in the equation V = √l, a result which happens to be identical with one we had also arrived at in the case of the fish. In the bird’s case it has a deeper significance than in the other; because it implies here not merely that the velocity will tend to increase in a certain ratio with the length, but that it must do so as an essential and primary condition of the bird’s remaining aloft. It is accordingly of great practical importance in aeronautics, for it shews how a provision of increasing speed must accompany every enlargement of our aeroplanes. If a given machine weighing, say, 500 lbs. be stable at 40 miles an hour, then one geometrically similar which weighs, say, a couple of tons must have its speed determined as follows:
That is to say, the larger machine must be capable of a speed equal to 1·414 × 40, or about 56½ miles per hour.
It is highly probable, as Lanchester [41] remarks, that Lilienthal met his untimely death not so much from any intrinsic fault in the design or construction of his machine, but simply because his engine fell somewhat short of the power required to give the speed which was necessary for stability. An arrow is a very imperfectly designed aeroplane, but nevertheless it is evidently capable, to a certain extent and at a high velocity, of acquiring “stability” and hence of actual “flight”: the duration and consequent range of its trajectory, as compared with a bullet of similar initial velocity, being correspondingly benefited. When we return to our birds, and again compare the ostrich with the sparrow, we know little or nothing about the speed in flight of the latter, but that of the swift is estimated [42] to vary from a minimum of 20 to 50 feet or more per second,—say from 14 to 35 miles per hour. Let us take the same lower limit as not far from the minimal velocity of the sparrow’s flight also; and it [p. 27] would follow that the ostrich, of 25 times the sparrow’s linear dimensions, would be compelled to fly (if it flew at all) with a minimum velocity of 5 × 14, or 70 miles an hour.
The same principle of necessary speed, or the indispensable relation between the dimensions of a flying object and the minimum velocity at which it is stable, accounts for a great number of observed phenomena. It tells us why the larger birds have a marked difficulty in rising from the ground, that is to say, in acquiring to begin with the horizontal velocity necessary for their support; and why accordingly, as Mouillard [43] and others have observed, the heavier birds, even those weighing no more than a pound or two, can be effectively “caged” in a small enclosure open to the sky. It tells us why very small birds, especially those as small as humming-birds, and à fortiori the still smaller insects, are capable of “stationary flight,” a very slight and scarcely perceptible velocity relatively to the air being sufficient for their support and stability. And again, since it is in all cases velocity relative to the air that we are speaking of, we comprehend the reason why one may always tell which way the wind blows by watching the direction in which a bird starts to fly.
It is not improbable that the ostrich has already reached a magnitude, and we may take it for certain that the moa did so, at which flight by muscular action, according to the normal anatomy of a bird, has become physiologically impossible. The same reasoning applies to the case of man. It would be very difficult, and probably absolutely impossible, for a bird to fly were it the bigness of a man. But Borelli, in discussing this question, laid even greater stress on the obvious fact that a man’s pectoral muscles are so immensely less in proportion than those of a bird, that however we may fit ourselves with wings we can never expect to move them by any power of our own relatively weaker muscles; so it is that artificial flight only became possible when an engine was devised whose efficiency was extraordinarily great in comparison with its weight and size.
Had Leonardo da Vinci known what Galileo knew, he would not have spent a great part of his life on vain efforts to make to himself wings. Borelli had learned the lesson thoroughly, and [p. 28] in one of his chapters he deals with the proposition, “Est impossible, ut homines propriis viribus artificiose volare possint [44].”
But just as it is easier to swim than to fly, so is it obvious that, in a denser atmosphere, the conditions of flight would be altered, and flight facilitated. We know that in the carboniferous epoch there lived giant dragon-flies, with wings of a span far greater than nowadays they ever attain; and the small bodies and huge extended wings of the fossil pterodactyles would seem in like manner to be quite abnormal according to our present standards, and to be beyond the limits of mechanical efficiency under present conditions. But as Harlé suggests [45], following upon a suggestion of Arrhenius, we have only to suppose that in carboniferous and jurassic days the terrestrial atmosphere was notably denser than it is at present, by reason, for instance, of its containing a much larger proportion of carbonic acid, and we have at once a means of reconciling the apparent mechanical discrepancy.
Very similar problems, involving in various ways the principle of dynamical similitude, occur all through the physiology of locomotion: as, for instance, when we see that a cockchafer can carry a plate, many times his own weight, upon his back, or that a flea can jump many inches high.
Problems of this latter class have been admirably treated both by Galileo and by Borelli, but many later writers have remained ignorant of their work. Linnaeus, for instance, remarked that, if an elephant were as strong in proportion as a stag-beetle, it would be able to pull up rocks by the root, and to level mountains. And Kirby and Spence have a well-known passage directed to shew that such powers as have been conferred upon the insect have been withheld from the higher animals, for the reason that had these latter been endued therewith they would have “caused the early desolation of the world [46].” [p. 29]
Such problems as that which is presented by the flea’s jumping powers, though essentially physiological in their nature, have their interest for us here: because a steady, progressive diminution of activity with increasing size would tend to set limits to the possible growth in magnitude of an animal just as surely as those factors which tend to break and crush the living fabric under its own weight. In the case of a leap, we have to do rather with a sudden impulse than with a continued strain, and this impulse should be measured in terms of the velocity imparted. The velocity is proportional to the impulse (x), and inversely proportional to the mass (M) moved: V = x/M. But, according to what we still speak of as “Borelli’s law,” the impulse (i.e. the work of the impulse) is proportional to the volume of the muscle by which it is produced [47], that is to say (in similarly constructed animals) to the mass of the whole body; for the impulse is proportional on the one hand to the cross-section of the muscle, and on the other to the distance through which it contracts. It follows at once from this that the velocity is constant, whatever be the size of the animals: in other words, that all animals, provided always that they are similarly fashioned, with their various levers etc., in like proportion, ought to jump, not to the same relative, but to the same actual height [48]. According to this, then, the flea is not a better, but rather a worse jumper than a horse or a man. As a matter of fact, Borelli is careful to point out that in the act of leaping the impulse is not actually instantaneous, as in the blow of a hammer, but takes some little time, during which the levers are being extended by which the centre of gravity of the animal is being propelled forwards; and this interval of time will be longer in the case of the longer levers of the larger animal. To some extent, then, this principle acts as a corrective to the more general one, [p. 30] and tends to leave a certain balance of advantage, in regard to leaping power, on the side of the larger animal [49].
But on the other hand, the question of strength of materials comes in once more, and the factors of stress and strain and bending moment make it, so to speak, more and more difficult for nature to endow the larger animal with the length of lever with which she has provided the flea or the grasshopper.
To Kirby and Spence it seemed that “This wonderful strength of insects is doubtless the result of something peculiar in the structure and arrangement of their muscles, and principally their extraordinary power of contraction.” This hypothesis, which is so easily seen, on physical grounds, to be unnecessary, has been amply disproved in a series of excellent papers by F. Plateau [50].
A somewhat simple problem is presented to us by the act of walking. It is obvious that there will be a great economy of work, if the leg swing at its normal pendulum-rate; and, though this rate is hard to calculate, owing to the shape and the jointing of the limb, we may easily convince ourselves, by counting our steps, that the leg does actually swing, or tend to swing, just as a pendulum does, at a certain definite rate [51]. When we walk quicker, we cause the leg-pendulum to describe a greater arc, but we do not appreciably cause it to swing, or vibrate, quicker, until we shorten the pendulum and begin to run. Now let two individuals, A and B, walk in a similar fashion, that is to say, with a similar angle of swing. The arc through which the leg swings, or the amplitude of each step, will therefore vary as the length of leg, or say as a/b; but the time of swing will vary as the square [p. 31] root of the pendulum-length, or √a/√b. Therefore the velocity, which is measured by amplitude/time, will also vary as the square-roots of the length of leg: that is to say, the average velocities of A and B are in the ratio of √a : √b.
The smaller man, or smaller animal, is so far at a disadvantage compared with the larger in speed, but only to the extent of the ratio between the square roots of their linear dimensions: whereas, if the rate of movement of the limb were identical, irrespective of the size of the animal,—if the limbs of the mouse for instance swung at the same rate as those of the horse,—then, as F. Plateau said, the mouse would be as slow or slower in its gait than the tortoise. M. Delisle [52] observed a “minute fly” walk three inches in half-a-second. This was good steady walking. When we walk five miles an hour we go about 88 inches in a second, or 88/6 = 14·7 times the pace of M. Delisle’s fly. We should walk at just about the fly’s pace if our stature were 1/(14·7)2, or 1/216 of our present height,—say 72/216 inches, or one-third of an inch high.
But the leg comprises a complicated system of levers, by whose various exercise we shall obtain very different results. For instance, by being careful to rise upon our instep, we considerably increase the length or amplitude of our stride, and very considerably increase our speed accordingly. On the other hand, in running, we bend and so shorten the leg, in order to accommodate it to a quicker rate of pendulum-swing [53]. In short, the jointed structure of the leg permits us to use it as the shortest possible pendulum when it is swinging, and as the longest possible lever when it is exerting its propulsive force.
Apart from such modifications as that described in the last paragraph,—apart, that is to say, from differences in mechanical construction or in the manner in which the mechanism is used,—we have now arrived at a curiously simple and uniform result. For in all the three forms of locomotion which we have attempted [p. 32] to study, alike in swimming, in flight and in walking, the general result, attained under very different conditions and arrived at by very different modes of reasoning, is in every case that the velocity tends to vary as the square root of the linear dimensions of the organism.
From all the foregoing discussion we learn that, as Crookes once upon a time remarked [54], the form as well as the actions of our bodies are entirely conditioned (save for certain exceptions in the case of aquatic animals, nicely balanced with the density of the surrounding medium) by the strength of gravity upon this globe. Were the force of gravity to be doubled, our bipedal form would be a failure, and the majority of terrestrial animals would resemble short-legged saurians, or else serpents. Birds and insects would also suffer, though there would be some compensation for them in the increased density of the air. While on the other hand if gravity were halved, we should get a lighter, more graceful, more active type, requiring less energy and less heat, less heart, less lungs, less blood.
Throughout the whole field of morphology we may find examples of a tendency (referable doubtless in each case to some definite physical cause) for surface to keep pace with volume, through some alteration of its form. The development of “villi” on the inner surface of the stomach and intestine (which enlarge its surface much as we enlarge the effective surface of a bath-towel), the various valvular folds of the intestinal lining, including the remarkable “spiral fold” of the shark’s gut, the convolutions of the brain, whose complexity is evidently correlated (in part at least) with the magnitude of the animal,—all these and many more are cases in which a more or less constant ratio tends to be maintained between mass and surface, which ratio would have been more and more departed from had it not been for the alterations of surface-form [55]. [p. 33]
In the case of very small animals, and of individual cells, the principle becomes especially important, in consequence of the molecular forces whose action is strictly limited to the superficial layer. In the cases just mentioned, action is facilitated by increase of surface: diffusion, for instance, of nutrient liquids or respiratory gases is rendered more rapid by the greater area of surface; but there are other cases in which the ratio of surface to mass may make an essential change in the whole condition of the system. We know, for instance, that iron rusts when exposed to moist air, but that it rusts ever so much faster, and is soon eaten away, if the iron be first reduced to a heap of small filings; this is a mere difference of degree. But the spherical surface of the raindrop and the spherical surface of the ocean (though both happen to be alike in mathematical form) are two totally different phenomena, the one due to surface-energy, and the other to that form of mass-energy which we ascribe to gravity. The contrast is still more clearly seen in the case of waves: for the little ripple, whose form and manner of propagation are governed by surface-tension, is found to travel with a velocity which is inversely as the square root of its length; while the ordinary big waves, controlled by gravitation, have a velocity directly proportional to the square root of their wave-length. In like manner we shall find that the form of all small organisms is largely independent of gravity, and largely if not mainly due to the force of surface-tension: either as the direct result of the continued action of surface tension on the semi-fluid body, or else as the result of its action at a prior stage of development, in bringing about a form which subsequent chemical changes have rendered rigid and lasting. In either case, we shall find a very great tendency in small organisms to assume either the spherical form or other simple forms related to ordinary inanimate surface-tension phenomena; which forms do not recur in the external morphology of large animals, or if they in part recur it is for other reasons. [p. 34]
Now this is a very important matter, and is a notable illustration of that principle of similitude which we have already discussed in regard to several of its manifestations. We are coming easily to a conclusion which will affect the whole course of our argument throughout this book, namely that there is an essential difference in kind between the phenomena of form in the larger and the smaller organisms. I have called this book a study of Growth and Form, because in the most familiar illustrations of organic form, as in our own bodies for example, these two factors are inseparably associated, and because we are here justified in thinking of form as the direct resultant and consequence of growth: of growth, whose varying rate in one direction or another has produced, by its gradual and unequal increments, the successive stages of development and the final configuration of the whole material structure. But it is by no means true that form and growth are in this direct and simple fashion correlative or complementary in the case of minute portions of living matter. For in the smaller organisms, and in the individual cells of the larger, we have reached an order of magnitude in which the intermolecular forces strive under favourable conditions with, and at length altogether outweigh, the force of gravity, and also those other forces leading to movements of convection which are the prevailing factors in the larger material aggregate.
However we shall require to deal more fully with this matter in our discussion of the rate of growth, and we may leave it meanwhile, in order to deal with other matters more or less directly concerned with the magnitude of the cell.
The living cell is a very complex field of energy, and of energy of many kinds, surface-energy included. Now the whole surface-energy of the cell is by no means restricted to its outer surface; for the cell is a very heterogeneous structure, and all its protoplasmic alveoli and other visible (as well as invisible) heterogeneities make up a great system of internal surfaces, at every part of which one “phase” comes in contact with another “phase,” and surface-energy is accordingly manifested. But still, the external surface is a definite portion of the system, with a definite “phase” of its own, and however little we may know of the distribution of the total energy of the system, it is at least plain that [p. 35] the conditions which favour equilibrium will be greatly altered by the changed ratio of external surface to mass which a change of magnitude, unaccompanied by change of form, produces in the cell. In short, however it may be brought about, the phenomenon of division of the cell will be precisely what is required to keep approximately constant the ratio between surface and mass, and to restore the balance between the surface-energy and the other energies of the system. When a germ-cell, for instance, divides or “segments” into two, it does not increase in mass; at least if there be some slight alleged tendency for the egg to increase in mass or volume during segmentation, it is very slight indeed, generally imperceptible, and wholly denied by some [56]. The development or growth of the egg from a one-celled stage to stages of two or many cells, is thus a somewhat peculiar kind of growth; it is growth which is limited to increase of surface, unaccompanied by growth in volume or in mass.
In the case of a soap-bubble, by the way, if it divide into two bubbles, the volume is actually diminished [57] while the surface-area is greatly increased. This is due to a cause which we shall have to study later, namely to the increased pressure due to the greater curvature of the smaller bubbles.
An immediate and remarkable result of the principles just described is a tendency on the part of all cells, according to their kind, to vary but little about a certain mean size, and to have, in fact, certain absolute limitations of magnitude.
Sachs [58] pointed out, in 1895, that there is a tendency for each nucleus to be only able to gather around itself a certain definite amount of protoplasm. Driesch [59], a little later, found that, by artificial subdivision of the egg, it was possible to rear dwarf sea-urchin larvae, one-half, one-quarter, or even one-eighth of their [p. 36] normal size; and that these dwarf bodies were composed of only a half, a quarter or an eighth of the normal number of cells. Similar observations have been often repeated and amply confirmed. For instance, in the development of Crepidula (a little American “slipper-limpet,” now much at home on our own oyster-beds), Conklin [60] has succeeded in rearing dwarf and giant individuals, of which the latter may be as much as twenty-five times as big as the former. But nevertheless, the individual cells, of skin, gut, liver, muscle, and of all the other tissues, are just the same size in one as in the other,—in dwarf and in giant [61]. Driesch has laid particular stress upon this principle of a “fixed cell-size.”
We get an excellent, and more familiar illustration of the same principle in comparing the large brain-cells or ganglion-cells, both of the lower and of the higher animals [62].
1917 Source Illustration
Fig. 1. Motor ganglion-cells, from the cervical spinal cord. (From Minot, after Irving Hardesty.)
In Fig. 1 we have certain identical nerve-cells taken from various mammals, from the mouse to the elephant, all represented on the same scale of magnification; and we see at once that they are all of much the same order of magnitude. The nerve-cell of the elephant is about twice that of the mouse in linear dimensions, and therefore about eight times greater in volume, or mass. But making some allowance for difference of shape, the linear dimensions of the elephant are to those of the mouse in a ratio certainly not less than one to fifty; from which it would follow that the bulk of the larger animal is something like 125,000 times that of the less. And it also follows, the size of the nerve-cells being [p. 37] about as eight to one, that, in corresponding parts of the nervous system of the two animals, there are more than 15,000 times as many individual cells in one as in the other. In short we may (with Enriques) lay it down as a general law that among animals, whether large or small, the ganglion-cells vary in size within narrow limits; and that, amidst all the great variety of structural type of ganglion observed in different classes of animals, it is always found that the smaller species have simpler ganglia than the larger, that is to say ganglia containing a smaller number of cellular elements [63]. The bearing of such simple facts as this upon the cell-theory in general is not to be disregarded; and the warning is especially clear against exaggerated attempts to correlate physiological processes with the visible mechanism of associated cells, rather than with the system of energies, or the field of force, which is associated with them. For the life of [p. 38] the body is more than the sum of the properties of the cells of which it is composed: as Goethe said, “Das Lebendige ist zwar in Elemente zerlegt, aber man kann es aus diesen nicht wieder zusammenstellen und beleben.”
Among certain lower and microscopic organisms, such for instance as the Rotifera, we are still more palpably struck by the small number of cells which go to constitute a usually complex organ, such as kidney, stomach, ovary, etc. We can sometimes number them in a few units, in place of the thousands that make up such an organ in larger, if not always higher, animals. These facts constitute one among many arguments which combine to teach us that, however important and advantageous the subdivision of organisms into cells may be from the constructional, or from the dynamical point of view, the phenomenon has less essential importance in theoretical biology than was once, and is often still, assigned to it.
Again, just as Sachs shewed that there was a limit to the amount of cytoplasm which could gather round a single nucleus, so Boveri has demonstrated that the nucleus itself has definite limitations of size, and that, in cell-division after fertilisation, each new nucleus has the same size as its parent-nucleus [64].
In all these cases, then, there are reasons, partly no doubt physiological, but in very large part purely physical, which set limits to the normal magnitude of the organism or of the cell. But as we have already discussed the existence of absolute and definite limitations, of a physical kind, to the possible increase in magnitude of an organism, let us now enquire whether there be not also a lower limit, below which the very existence of an organism is impossible, or at least where, under changed conditions, its very nature must be profoundly modified.
Among the smallest of known organisms we have, for instance, Micromonas mesnili, Bonel, a flagellate infusorian, which measures about ·34 µ, or ·00034 mm., by ·00025 mm.; smaller even than this we have a pathogenic micrococcus of the rabbit, M. progrediens, Schröter, the diameter of which is said to be only ·00015 mm. or ·15 µ, or 1·5 × 10−5 cm.,—about equal to the thickness of [p. 39] the thinnest gold-leaf; and as small if not smaller still are a few bacteria and their spores. But here we have reached, or all but reached the utmost limits of ordinary microscopic vision; and there remain still smaller organisms, the so-called “filter-passers,” which the ultra-microscope reveals, but which are mainly brought within our ken only by the maladies, such as hydrophobia, foot-and-mouth disease, or the “mosaic” disease of the tobacco-plant, to which these invisible micro-organisms give rise [65]. Accordingly, since it is only by the diseases which they occasion that these tiny bodies are made known to us, we might be tempted to suppose that innumerable other invisible organisms, smaller and yet smaller, exist unseen and unrecognised by man.
1917 Source Illustration
Fig. 2. Relative magnitudes of: A, human blood-corpuscle (7·5 µ in diameter); B, Bacillus anthracis (4 – 15 µ × 1 µ); C, various Micrococci (diam. 0·5 – 1 µ, rarely 2 µ); D, Micromonas progrediens, Schröter (diam. 0·15 µ).
To illustrate some of these small magnitudes I have adapted the preceding diagram from one given by Zsigmondy [66]. Upon the [p. 40] same scale the minute ultramicroscopic particles of colloid gold would be represented by the finest dots which we could make visible to the naked eye upon the paper.
A bacillus of ordinary, typical size is, say, 1 µ in length. The length (or height) of a man is about a million and three-quarter times as great, i.e. 1·75 metres, or 1·75 × 106 µ; and the mass of the man is in the neighbourhood of five million, million, million (5 × 1018) times greater than that of the bacillus. If we ask whether there may not exist organisms as much less than the bacillus as the bacillus is less than the dimensions of a man, it is very easy to see that this is quite impossible, for we are rapidly approaching a point where the question of molecular dimensions, and of the ultimate divisibility of matter, begins to call for our attention, and to obtrude itself as a crucial factor in the case.
Clerk Maxwell dealt with this matter in his article “Atom [67],” and, in somewhat greater detail, Errera discusses the question on the following lines [68]. The weight of a hydrogen molecule is, according to the physical chemists, somewhere about 8·6 × 2 × 10−22 milligrammes; and that of any other element, whose molecular weight is M, is given by the equation
Accordingly, the weight of the atom of sulphur may be taken as
The analysis of ordinary bacteria shews them to consist [69] of about 85% of water, and 15% of solids; while the solid residue of vegetable protoplasm contains about one part in a thousand of sulphur. We may assume, therefore, that the living protoplasm contains about
parts of sulphur, taking the total weight as = 1.
But our little micrococcus, of 0·15 µ in diameter, would, if it were spherical, have a volume of
and therefore (taking its density as equal to that of water), a weight of
But of this total weight, the sulphur represents only
And if we divide this by the weight of an atom of sulphur, we have
According to this estimate, then, our little Micrococcus progrediens should contain only about 10,000 atoms of sulphur, an element indispensable to its protoplasmic constitution; and it follows that an organism of one-tenth the diameter of our micrococcus would only contain 10 sulphur-atoms, and therefore only ten chemical “molecules” or units of protoplasm!
It may be open to doubt whether the presence of sulphur be really essential to the constitution of the proteid or “protoplasmic” molecule; but Errera gives us yet another illustration of a similar kind, which is free from this objection or dubiety. The molecule of albumin, as is generally agreed, can scarcely be less than a thousand times the size of that of such an element as sulphur: according to one particular determination [70], serum albumin has a constitution corresponding to a molecular weight of 10,166, and even this may be far short of the true complexity of a typical albuminoid molecule. The weight of such a molecule is
Now the bacteria contain about 14% of albuminoids, these constituting by far the greater part of the dry residue; and therefore (from equation (5)), the weight of albumin in our micrococcus is about
If we divide this weight by that which we have arrived at as the weight of an albumin molecule, we have
in other words, our micrococcus apparently contains something less than 30,000 molecules of albumin. [p. 42]
According to the most recent estimates, the weight of the hydrogen molecule is somewhat less than that on which Errera based his calculations, namely about 16 × 10−22 mgms. and according to this value, our micrococcus would contain just about 27,000 albumin molecules. In other words, whichever determination we accept, we see that an organism one-tenth as large as our micrococcus, in linear dimensions, would only contain some thirty molecules of albumin; or, in other words, our micrococcus is only about thirty times as large, in linear dimensions, as a single albumin molecule [71].
We must doubtless make large allowances for uncertainty in the assumptions and estimates upon which these calculations are based; and we must also remember that the data with which the physicist provides us in regard to molecular magnitudes are, to a very great extent, maximal values, above which the molecular magnitude (or rather the sphere of the molecule’s range of motion) is not likely to lie: but below which there is a greater element of uncertainty as to its possibly greater minuteness. But nevertheless, when we shall have made all reasonable allowances for uncertainty upon the physical side, it will still be clear that the smallest known bodies which are described as organisms draw nigh towards molecular magnitudes, and we must recognise that the subdivision of the organism cannot proceed to an indefinite extent, and in all probability cannot go very much further than it appears to have done in these already discovered forms. For, even, after giving all due regard to the complexity of our unit (that is to say the albumin-molecule), with all the increased possibilities of interrelation with its neighbours which this complexity implies, we cannot but see that physiologically, and comparatively speaking, we have come down to a very simple thing.
While such considerations as these, based on the chemical composition of the organism, teach us that there must be a definite lower limit to its magnitude, other considerations of a purely physical kind lead us to the same conclusion. For our discussion of the principle of similitude has already taught us that, long before we reach these almost infinitesimal magnitudes, the [p. 43] diminishing organism will have greatly changed in all its physical relations, and must at length arrive under conditions which must surely be incompatible with anything such as we understand by life, at least in its full and ordinary development and manifestation.
We are told, for instance, that the powerful force of surface-tension, or capillarity, begins to act within a range of about 1/500,000 of an inch, or say 0·05 µ. A soap-film, or a film of oil upon water, may be attenuated to far less magnitudes than this; the black spots upon a soap-bubble are known, by various concordant methods of measurement, to be only about 6 × 10−7 cm., or about ·006 µ thick, and Lord Rayleigh and M. Devaux [72] have obtained films of oil of ·002 µ, or even ·001 µ in thickness.
But while it is possible for a fluid film to exist in these almost molecular dimensions, it is certain that, long before we reach them, there must arise new conditions of which we have little knowledge and which it is not easy even to imagine.
It would seem that, in an organism of ·1 µ in diameter, or even rather more, there can be no essential distinction between the interior and the surface layers. No hollow vesicle, I take it, can exist of these dimensions, or at least, if it be possible for it to do so, the contained gas or fluid must be under pressures of a formidable kind [73], and of which we have no knowledge or experience. Nor, I imagine, can there be any real complexity, or heterogeneity, of its fluid or semi-fluid contents; there can be no vacuoles within such a cell, nor any layers defined within its fluid substance, for something of the nature of a boundary-film is the necessary condition of the existence of such layers. Moreover, the whole organism, provided that it be fluid or semi-fluid, can only be spherical in form. What, then, can we attribute, in the way of properties, to an organism of a size as small as, or smaller than, say ·05 µ? It must, in all probability, be a homogeneous, structureless sphere, composed of a very small number of albuminoid or other molecules. Its vital properties and functions must be extraordinarily limited; its specific outward characters, even if we could see it, must be nil; and its specific properties must be little more than those of an ion-laden corpuscle, enabling it to perform [p. 44] this or that chemical reaction, or to produce this or that pathogenic effect. Even among inorganic, non-living bodies, there must be a certain grade of minuteness at which the ordinary properties become modified. For instance, while under ordinary circumstances crystallisation starts in a solution about a minute solid fragment or crystal of the salt, Ostwald has shewn that we may have particles so minute that they fail to serve as a nucleus for crystallisation,—which is as much as to say that they are too minute to have the form and properties of a “crystal”; and again, in his thin oil-films, Lord Rayleigh has noted the striking change of physical properties which ensues when the film becomes attenuated to something less than one close-packed layer of molecules [74].
Thus, as Clerk Maxwell put it, “molecular science sets us face to face with physiological theories. It forbids the physiologist from imagining that structural details of infinitely small dimensions [such as Leibniz assumed, one within another, ad infinitum] can furnish an explanation of the infinite variety which exists in the properties and functions of the most minute organisms.” And for this reason he reprobates, with not undue severity, those advocates of pangenesis and similar theories of heredity, who would place “a whole world of wonders within a body so small and so devoid of visible structure as a germ.” But indeed it scarcely needed Maxwell’s criticism to shew forth the immense physical difficulties of Darwin’s theory of Pangenesis: which, after all, is as old as Democritus, and is no other than that Promethean particulam undique desectam of which we have read, and at which we have smiled, in our Horace.
There are many other ways in which, when we “make a long excursion into space,” we find our ordinary rules of physical behaviour entirely upset. A very familiar case, analysed by Stokes, is that the viscosity of the surrounding medium has a relatively powerful effect upon bodies below a certain size. A droplet of water, a thousandth of an inch (25 µ) in diameter, cannot fall in still air quicker than about an inch and a half per second; and as its size decreases, its resistance varies as the diameter, and not (as with larger bodies) as the surface of the [p. 45] drop. Thus a drop one-tenth of that size (2·5 µ), the size, apparently, of the drops of water in a light cloud, will fall a hundred times slower, or say an inch a minute; and one again a tenth of this diameter (say ·25 µ, or about twice as big, in linear dimensions, as our micrococcus), will scarcely fall an inch in two hours. By reason of this principle, not only do the smaller bacteria fall very slowly through the air, but all minute bodies meet with great proportionate resistance to their movements in a fluid. Even such comparatively large organisms as the diatoms and the foraminifera, laden though they are with a heavy shell of flint or lime, seem to be poised in the water of the ocean, and fall in it with exceeding slowness.
The Brownian movement has also to be reckoned with,—that remarkable phenomenon studied nearly a century ago (1827) by Robert Brown, facile princeps botanicorum. It is one more of those fundamental physical phenomena which the biologists have contributed, or helped to contribute, to the science of physics.
The quivering motion, accompanied by rotation, and even by translation, manifested by the fine granular particles issuing from a crushed pollen-grain, and which Robert Brown proved to have no vital significance but to be manifested also by all minute particles whatsoever, organic and inorganic, was for many years unexplained. Nearly fifty years after Brown wrote, it was said to be “due, either directly to some calorical changes continually taking place in the fluid, or to some obscure chemical action between the solid particles and the fluid which is indirectly promoted by heat [75].” Very shortly after these last words were written, it was ascribed by Wiener to molecular action, and we now know that it is indeed due to the impact or bombardment of molecules upon a body so small that these impacts do not for the moment, as it were, “average out” to approximate equality on all sides. The movement becomes manifest with particles of somewhere about 20 µ in diameter, it is admirably displayed by particles of about 12 µ in diameter, and becomes more marked the smaller the particles are. The bombardment causes our particles to behave just like molecules of uncommon size, and this [p. 46] behaviour is manifested in several ways [76]. Firstly, we have the quivering movement of the particles; secondly, their movement backwards and forwards, in short, straight, disjointed paths; thirdly, the particles rotate, and do so the more rapidly the smaller they are, and by theory, confirmed by observation, it is found that particles of 1 µ in diameter rotate on an average through 100° per second, while particles of 13 µ in diameter turn through only 14° per minute. Lastly, the very curious result appears, that in a layer of fluid the particles are not equally distributed, nor do they all ever fall, under the influence of gravity, to the bottom. But just as the molecules of the atmosphere are so distributed, under the influence of gravity, that the density (and therefore the number of molecules per unit volume) falls off in geometrical progression as we ascend to higher and higher layers, so is it with our particles, even within the narrow limits of the little portion of fluid under our microscope. It is only in regard to particles of the simplest form that these phenomena have been theoretically investigated [77], and we may take it as certain that more complex particles, such as the twisted body of a Spirillum, would show other and still more complicated manifestations. It is at least clear that, just as the early microscopists in the days before Robert Brown never doubted but that these phenomena were purely vital, so we also may still be apt to confuse, in certain cases, the one phenomenon with the other. We cannot, indeed, without the most careful scrutiny, decide whether the movements of our minutest organisms are intrinsically “vital” (in the sense of being beyond a physical mechanism, or working model) or not. For example, Schaudinn has suggested that the undulating movements of Spirochaete pallida must be due to the presence of a minute, unseen, “undulating membrane”; and Doflein says of the same species that “sie verharrt oft mit eigenthümlich zitternden Bewegungen zu einem Orte.” Both movements, the trembling or quivering [p. 47] movement described by Doflein, and the undulating or rotating movement described by Schaudinn, are just such as may be easily and naturally interpreted as part and parcel of the Brownian phenomenon.
While the Brownian movement may thus simulate in a deceptive way the active movements of an organism, the reverse statement also to a certain extent holds good. One sometimes lies awake of a summer’s morning watching the flies as they dance under the ceiling. It is a very remarkable dance. The dancers do not whirl or gyrate, either in company or alone; but they advance and retire; they seem to jostle and rebound; between the rebounds they dart hither or thither in short straight snatches of hurried flight; and turn again sharply in a new rebound at the end of each little rush. Their motions are wholly “erratic,” independent of one another, and devoid of common purpose. This is nothing else than a vastly magnified picture, or simulacrum, of the Brownian movement; the parallel between the two cases lies in their complete irregularity, but this in itself implies a close resemblance. One might see the same thing in a crowded market-place, always provided that the bustling crowd had no business whatsoever. In like manner Lucretius, and Epicurus before him, watched the dust-motes quivering in the beam, and saw in them a mimic representation, rei simulacrum et imago, of the eternal motions of the atoms. Again the same phenomenon may be witnessed under the microscope, in a drop of water swarming with Paramoecia or suchlike Infusoria; and here the analogy has been put to a numerical test. Following with a pencil the track of each little swimmer, and dotting its place every few seconds (to the beat of a metronome), Karl Przibram found that the mean successive distances from a common base-line obeyed with great exactitude the “Einstein formula,” that is to say the particular form of the “law of chance” which is applicable to the case of the Brownian movement [78]. The phenomenon is (of course) merely analogous, and by no means identical with the Brownian movement; for the range of motion of the little active organisms, whether they be gnats or infusoria, is vastly greater than that of the minute particles which are [p. 48] passive under bombardment; but nevertheless Przibram is inclined to think that even his comparatively large infusoria are small enough for the molecular bombardment to be a stimulus, though not the actual cause, of their irregular and interrupted movements.
There is yet another very remarkable phenomenon which may come into play in the case of the minutest of organisms; and this is their relation to the rays of light, as Arrhenius has told us. On the waves of a beam of light, a very minute particle (in vacuo) should be actually caught up, and carried along with an immense velocity; and this “radiant pressure” exercises its most powerful influence on bodies which (if they be of spherical form) are just about ·00016 mm., or ·16 µ in diameter. This is just about the size, as we have seen, of some of our smallest known protozoa and bacteria, while we have some reason to believe that others yet unseen, and perhaps the spores of many, are smaller still. Now we have seen that such minute particles fall with extreme slowness in air, even at ordinary atmospheric pressures: our organism measuring ·16 µ would fall but 83 metres in a year, which is as much as to say that its weight offers practically no impediment to its transference, by the slightest current, to the very highest regions of the atmosphere. Beyond the atmosphere, however, it cannot go, until some new force enable it to resist the attraction of terrestrial gravity, which the viscosity of an atmosphere is no longer at hand to oppose. But it is conceivable that our particle may go yet farther, and actually break loose from the bonds of earth. For in the upper regions of the atmosphere, say fifty miles high, it will come in contact with the rays and flashes of the Northern Lights, which consist (as Arrhenius maintains) of a fine dust, or cloud of vapour-drops, laden with a charge of negative electricity, and projected outwards from the sun. As soon as our particle acquires a charge of negative electricity it will begin to be repelled by the similarly laden auroral particles, and the amount of charge necessary to enable a particle of given size (such as our little monad of ·16 µ) to resist the attraction of gravity may be calculated, and is found to be such as the actual conditions can easily supply. Finally, when once set free from the entanglement of the earth’s [p. 49] atmosphere, the particle may be propelled by the “radiant pressure” of light, with a velocity which will carry it.—like Uriel gliding on a sunbeam,—as far as the orbit of Mars in twenty days, of Jupiter in eighty days, and as far as the nearest fixed star in three thousand years! This, and much more, is Arrhenius’s contribution towards the acceptance of Lord Kelvin’s hypothesis that life may be, and may have been, disseminated across the bounds of space, throughout the solar system and the whole universe!
It may well be that we need attach no great practical importance to this bold conception; for even though stellar space be shewn to be mare liberum to minute material travellers, we may be sure that those which reach a stellar or even a planetary bourne are infinitely, or all but infinitely, few. But whether or no, the remote possibilities of the case serve to illustrate in a very vivid way the profound differences of physical property and potentiality which are associated in the scale of magnitude with simple differences of degree.
Chapter III: The Rate of Growth¶
1917 Cambridge First Edition (pp. 50–155)
When we study magnitude by itself, apart, that is to say, from the gradual changes to which it may be subject, we are dealing with a something which may be adequately represented by a number, or by means of a line of definite length; it is what mathematicians call a scalar phenomenon. When we introduce the conception of change of magnitude, of magnitude which varies as we pass from one direction to another in space, or from one instant to another in time, our phenomenon becomes capable of representation by means of a line of which we define both the length and the direction; it is (in this particular aspect) what is called a vector phenomenon.
When we deal with magnitude in relation to the dimensions of space, the vector diagram which we draw plots magnitude in one direction against magnitude in another,—length against height, for instance, or against breadth; and the result is simply what we call a picture or drawing of an object, or (more correctly) a “plane projection” of the object. In other words, what we call Form is a ratio of magnitudes, referred to direction in space.
When in dealing with magnitude we refer its variations to successive intervals of time (or when, as it is said, we equate it with time), we are then dealing with the phenomenon of growth; and it is evident, therefore, that this term growth has wide meanings. For growth may obviously be positive or negative; that is to say, a thing may grow larger or smaller, greater or less; and by extension of the primitive concrete signification of the word, we easily and legitimately apply it to non-material things, such as temperature, and say, for instance, that a body “grows” hot or cold. When in a two-dimensional diagram, we represent a magnitude (for instance length) in relation to time (or “plot” [p. 51] length against time, as the phrase is), we get that kind of vector diagram which is commonly known as a “curve of growth.” We perceive, accordingly, that the phenomenon which we are now studying is a velocity (whose “dimensions” are Space/Time or L/T); and this phenomenon we shall speak of, simply, as a rate of growth.
In various conventional ways we can convert a two-dimensional into a three-dimensional diagram. We do so, for example, by means of the geometrical method of “perspective” when we represent upon a sheet of paper the length, breadth and depth of an object in three-dimensional space; but we do it more simply, as a rule, by means of “contour-lines,” and always when time is one of the dimensions to be represented. If we superimpose upon one another (or even set side by side) pictures, or plane projections, of an organism, drawn at successive intervals of time, we have such a three-dimensional diagram, which is a partial representation (limited to two dimensions of space) of the organism’s gradual change of form, or course of development; and in such a case our contour-lines may, for the purposes of the embryologist, be separated by intervals representing a few hours or days, or, for the purposes of the palaeontologist, by interspaces of unnumbered and innumerable years [79].
Such a diagram represents in two of its three dimensions form, and in two, or three, of its dimensions growth; and so we see how intimately the two conceptions are correlated or inter-related to one another. In short, it is obvious that the form of an animal is determined by its specific rate of growth in various directions; accordingly, the phenomenon of rate of growth deserves to be studied as a necessary preliminary to the theoretical study of form, and, mathematically speaking, organic form itself appears to us as a function of time [80]. [p. 52]
At the same time, we need only consider this part of our subject somewhat briefly. Though it has an essential bearing on the problems of morphology, it is in greater degree involved with physiological problems; and furthermore, the statistical or numerical aspect of the question is peculiarly adapted for the mathematical study of variation and correlation. On these important subjects we shall scarcely touch; for our main purpose will be sufficiently served if we consider the characteristics of a rate of growth in a few illustrative cases, and recognise that this rate of growth is a very important specific property, with its own characteristic value in this organism or that, in this or that part of each organism, and in this or that phase of its existence.
The statement which we have just made that “the form of an organism is determined by its rate of growth in various directions,” is one which calls (as we have partly seen in the foregoing chapter) for further explanation and for some measure of qualification. Among organic forms we shall have frequent occasion to see that form is in many cases due to the immediate or direct action of certain molecular forces, of which surface-tension is that which plays the greatest part. Now when surface-tension (for instance) causes a minute semi-fluid organism to assume a spherical form, or gives the form of a catenary or an elastic curve to a film of protoplasm in contact with some solid skeletal rod, or when it acts in various other ways which are productive of definite contours, this is a process of conformation that, both in appearance and reality, is very different from the process by which an ordinary plant or animal grows into its specific form. In both cases, change of form is brought about by the movement of portions of matter, and in both cases it is ultimately due to the action of molecular forces; but in the one case the movements of the particles of matter lie for the most part within molecular range, while in the other we have to deal chiefly with the transference of portions of matter into the system from without, and from one widely distant part of the organism to another. It is to this latter class of phenomena that we usually restrict the term growth; and it is in regard to them that we are in a position to study the rate of action in different directions, and to see that it is merely on a difference of velocities that the modification of form essentially depends. [p. 53] The difference between the two classes of phenomena is somewhat akin to the difference between the forces which determine the form of a rain-drop and those which, by the flowing of the waters and the sculpturing of the solid earth, have brought about the complex configuration of a river; molecular forces are paramount in the conformation of the one, and molar forces are dominant in the other.
At the same time it is perfectly true that all changes of form, inasmuch as they necessarily involve changes of actual and relative magnitude, may, in a sense, be properly looked upon as phenomena of growth; and it is also true, since the movement of matter must always involve an element of time [81], that in all cases the rate of growth is a phenomenon to be considered. Even though the molecular forces which play their part in modifying the form of an organism exert an action which is, theoretically, all but instantaneous, that action is apt to be dragged out to an appreciable interval of time by reason of viscosity or some other form of resistance in the material. From the physical or physiological point of view the rate of action even in such cases may be well worth studying; for example, a study of the rate of cell-division in a segmenting egg may teach us something about the work done, and about the various energies concerned. But in such cases the action is, as a rule, so homogeneous, and the form finally attained is so definite and so little dependent on the time taken to effect it, that the specific rate of change, or rate of growth, does not enter into the morphological problem.
To sum up, we may lay down the following general statements. The form of organisms is a phenomenon to be referred in part to the direct action of molecular forces, in part to a more complex and slower process, indirectly resulting from chemical, osmotic and other forces, by which material is introduced into the organism and transferred from one part of it to another. It is this latter complex phenomenon which we usually speak of as “growth.” [p. 54]
Every growing organism, and every part of such a growing organism, has its own specific rate of growth, referred to a particular direction. It is the ratio between the rates of growth in various directions by which we must account for the external forms of all, save certain very minute, organisms. This ratio between rates of growth in various directions may sometimes be of a simple kind, as when it results in the mathematically definable outline of a shell, or in the smooth curve of the margin of a leaf. It may sometimes be a very constant one, in which case the organism, while growing in bulk, suffers little or no perceptible change in form; but such equilibrium seldom endures for more than a season, and when the ratio tends to alter, then we have the phenomenon of morphological “development,” or steady and persistent change of form.
This elementary concept of Form, as determined by varying rates of Growth, was clearly apprehended by the mathematical mind of Haller,—who had learned his mathematics of the great John Bernoulli, as the latter in turn had learned his physiology from the writings of Borelli. Indeed it was this very point, the apparently unlimited extent to which, in the development of the chick, inequalities of growth could and did produce changes of form and changes of anatomical “structure,” that led Haller to surmise that the process was actually without limits, and that all development was but an unfolding, or “evolutio,” in which no part came into being which had not essentially existed before [82]. In short the celebrated doctrine of “preformation” implied on the one hand a clear recognition of what, throughout the later stages of development, growth can do, by hastening the increase in size of one part, hindering that of another, changing their relative magnitudes and positions, and altering their forms; while on the other hand it betrayed a failure (inevitable in those days) to recognise the essential difference between these movements of masses and the molecular processes which precede and accompany [p. 55] them, and which are characteristic of another order of magnitude.
By other writers besides Haller the very general, though not strictly universal connection between form and rate of growth has been clearly recognised. Such a connection is implicit in those “proportional diagrams” by which Dürer and some of his brother artists were wont to illustrate the successive changes of form, or of relative dimensions, which attend the growth of the child, to boyhood and to manhood. The same connection was recognised, more explicitly, by some of the older embryologists, for instance by Pander [83], and appears, as a survival of the doctrine of preformation, in his study of the development of the chick. And long afterwards, the embryological aspect of the case was emphasised by His, who pointed out, for instance, that the various foldings of the blastoderm, by which the neural and amniotic folds were brought into being, were essentially and obviously the resultant of unequal rates of growth,—of local accelerations or retardations of growth,—in what to begin with was an even and uniform layer of embryonic tissue. If we imagine a flat sheet of paper, parts of which are caused (as by moisture or evaporation) to expand or to contract, the plane surface is at once dimpled, or “buckled,” or folded, by the resultant forces of expansion or contraction: and the various distortions to which the plane surface of the “germinal disc” is subject, as His shewed once and for all, are precisely analogous. An experimental demonstration still more closely comparable to the actual case of the blastoderm, is obtained by making an “artificial blastoderm,” of little pills or pellets of dough, which are caused to grow, with varying velocities, by the addition of varying quantities of yeast. Here, as Roux is careful to point out [84], we observe that it is not only the growth of the individual cells, but the traction exercised through their mutual interconnections, which brings about the foldings and other distortions of the entire structure. [p. 56]
But this again was clearly present to Haller’s mind, and formed an essential part of his embryological doctrine. For he has no sooner treated of incrementum, or celeritas incrementi, than he proceeds to deal with the contributory and complementary phenomena of expansion, traction (adtractio) [85], and pressure, and the more subtle influences which he denominates vis derivationis et revulsionis [86]: these latter being the secondary and correlated effects on growth in one part, brought about, through such changes as are produced (for instance) in the circulation, by the growth of another.
Let us admit that, on the physiological side, Haller’s or His’s methods of explanation carry us back but a little way; yet even this little way is something gained. Nevertheless, I can well remember the harsh criticism, and even contempt, which His’s doctrine met with, not merely on the ground that it was inadequate, but because such an explanation was deemed wholly inappropriate, and was utterly disavowed [87]. Hertwig, for instance, asserted that, in embryology, when we found one embryonic stage preceding another, the existence of the former was, for the embryologist, an all-sufficient “causal explanation” of the latter. “We consider (he says), that we are studying and explaining a causal relation when we have demonstrated that the gastrula arises by invagination of a blastosphere, or the neural canal by the infolding of a cell plate so as to constitute a tube [88].” For Hertwig, therefore, as [p. 57] Roux remarks, the task of investigating a physical mechanism in embryology,—“der Ziel das Wirken zu erforschen,”—has no existence at all. For Balfour also, as for Hertwig, the mechanical or physical aspect of organic development had little or no attraction. In one notable instance, Balfour himself adduced a physical, or quasi-physical, explanation of an organic process, when he referred the various modes of segmentation of an ovum, complete or partial, equal or unequal and so forth, to the varying amount or the varying distribution of food yolk in association with the germinal protoplasm of the egg [89]. But in the main, Balfour, like all the other embryologists of his day, was engrossed by the problems of phylogeny, and he expressly defined the aims of comparative embryology (as exemplified in his own textbook) as being “twofold: (1) to form a basis for Phylogeny. and (2) to form a basis for Organogeny or the origin and evolution of organs [90].”
It has been the great service of Roux and his fellow-workers of the school of “Entwickelungsmechanik,” and of many other students to whose work we shall refer, to try, as His tried [91] to import into embryology, wherever possible, the simpler concepts of physics, to introduce along with them the method of experiment, and to refuse to be bound by the narrow limitations which such teaching as that of Hertwig would of necessity impose on the work and the thought and on the whole philosophy of the biologist.
Before we pass from this general discussion to study some of the particular phenomena of growth, let me give a single illustration, from Darwin, of a point of view which is in marked contrast to Haller’s simple but essentially mathematical conception of Form.
There is a curious passage in the Origin of Species [92], where Darwin is discussing the leading facts of embryology, and in particular Von Baer’s “law of embryonic resemblance.” Here Darwin says “We are so much accustomed to see a difference in [p. 58] structure between the embryo and the adult, that we are tempted to look at this difference as in some necessary manner contingent on growth. But there is no reason why, for instance, the wing of a bat, or the fin of a porpoise, should not have been sketched out with all their parts in proper proportion, as soon as any part became visible.” After pointing out with his habitual care various exceptions, Darwin proceeds to lay down two general principles, viz. “that slight variations generally appear at a not very early period of life,” and secondly, that “at whatever age a variation first appears in the parent, it tends to reappear at a corresponding age in the offspring.” He then argues that it is with nature as with the fancier, who does not care what his pigeons look like in the embryo, so long as the full-grown bird possesses the desired qualities; and that the process of selection takes place when the birds or other animals are nearly grown up,—at least on the part of the breeder, and presumably in nature as a general rule. The illustration of these principles is set forth as follows; “Let us take a group of birds, descended from some ancient form and modified through natural selection for different habits. Then, from the many successive variations having supervened in the several species at a not very early age, and having been inherited at a corresponding age, the young will still resemble each other much more closely than do the adults,—just as we have seen with the breeds of the pigeon …. Whatever influence long-continued use or disuse may have had in modifying the limbs or other parts of any species, this will chiefly or solely have affected it when nearly mature, when it was compelled to use its full powers to gain its own living; and the effects thus produced will have been transmitted to the offspring at a corresponding nearly mature age. Thus the young will not be modified, or will be modified only in a slight degree, through the effects of the increased use or disuse of parts.” This whole argument is remarkable, in more ways than we need try to deal with here; but it is especially remarkable that Darwin should begin by casting doubt upon the broad fact that a “difference in structure between the embryo and the adult” is “in some necessary manner contingent on growth”; and that he should see no reason why complicated structures of the adult “should not have been sketched out [p. 59] with all their parts in proper proportion, as soon as any part became visible.” It would seem to me that even the most elementary attention to form in its relation to growth would have removed most of Darwin’s difficulties in regard to the particular phenomena which he is here considering. For these phenomena are phenomena of form, and therefore of relative magnitude; and the magnitudes in question are attained by growth, proceeding with certain specific velocities, and lasting for certain long periods of time. And it is accordingly obvious that in any two related individuals (whether specifically identical or not) the differences between them must manifest themselves gradually, and be but little apparent in the young. It is for the same simple reason that animals which are of very different sizes when adult, differ less and less in size (as well as in form) as we trace them backwards through the foetal stages.
Though we study the visible effects of varying rates of growth throughout wellnigh all the problems of morphology, it is not very often that we can directly measure the velocities concerned. But owing to the obvious underlying importance which the phenomenon has to the morphologist we must make shift to study it where we can, even though our illustrative cases may seem to have little immediate bearing on the morphological problem [93].
In a very simple organism, of spherical symmetry, such as the single spherical cell of Protococcus or of Orbulina, growth is reduced to its simplest terms, and indeed it becomes so simple in its outward manifestations that it is no longer of special interest to the morphologist. The rate of growth is measured by the rate of change in length of a radius, i.e. V = (R′ − R)/T, and from this we may calculate, as already indicated, the rate of growth in terms of surface and of volume. The growing body remains of constant form, owing to the symmetry of the system; because, that is to say, on the one hand the pressure exerted by the growing protoplasm is exerted equally in all directions, after the manner of a hydrostatic pressure, which indeed it actually is: while on the other hand, the “skin” or surface layer of the cell is sufficiently [p. 60] homogeneous to exert at every point an approximately uniform resistance. Under these conditions then, the rate of growth is uniform in all directions, and does not affect the form of the organism.
But in a larger or a more complex organism the study of growth, and of the rate of growth, presents us with a variety of problems, and the whole phenomenon becomes a factor of great morphological importance. We no longer find that it tends to be uniform in all directions, nor have we any right to expect that it should. The resistances which it meets with will no longer be uniform. In one direction but not in others it will be opposed by the important resistance of gravity; and within the growing system itself all manner of structural differences will come into play, setting up unequal resistances to growth by the varying rigidity or viscosity of the material substance in one direction or another. At the same time, the actual sources of growth, the chemical and osmotic forces which lead to the intussusception of new matter, are not uniformly distributed; one tissue or one organ may well manifest a tendency to increase while another does not; a series of bones, their intervening cartilages, and their surrounding muscles, may all be capable of very different rates of increment. The differences of form which are the resultants of these differences in rate of growth are especially manifested during that part of life when growth itself is rapid: when the organism, as we say, is undergoing its development. When growth in general has become slow, the relative differences in rate between different parts of the organism may still exist, and may be made manifest by careful observation, but in many, or perhaps in most cases, the resultant change of form does not strike the eye. Great as are the differences between the rates of growth in different parts of an organism, the marvel is that the ratios between them are so nicely balanced as they actually are, and so capable, accordingly, of keeping for long periods of time the form of the growing organism all but unchanged. There is the nicest possible balance of forces and resistances in every part of the complex body; and when this normal equilibrium is disturbed, then we get abnormal growth, in the shape of tumours, exostoses, and malformations of every kind. [p. 61]
The rate of growth in Man.
Man will serve us as well as another organism for our first illustrations of rate of growth; and we cannot do better than go for our first data concerning him to Quetelet’s Anthropométrie [94], an epoch-making book for the biologist. For not only is it packed with information, some of it still unsurpassed, in regard to human growth and form, but it also merits our highest admiration as the first great essay in scientific statistics, and the first work in which organic variation was discussed from the point of view of the mathematical theory of probabilities.
1917 Source Illustration
Fig. 3. Curve of Growth in Man, from birth to 20 yrs (♂); from Quetelet’s Belgian data. The upper curve of stature from Bowditch’s Boston data.
If the child be some 20 inches, or say 50 cm. tall at birth, and the man some six feet high, or say 180 cm., at twenty, we may say that his average rate of growth has been (180 − 50)/20 cm., or 6·5 centimetres per annum. But we know very well that this is [p. 62] but a very rough preliminary statement, and that the boy grew quickly during some, and slowly during other, of his twenty years. It becomes necessary therefore to study the phenomenon of growth in successive small portions; to study, that is to say, the successive lengths, or the successive small differences, or increments, of length (or of weight, etc.), attained in successive short increments of time. This we do in the first instance in the usual way, by the “graphic method” of plotting length against time, and so constructing our “curve of growth.” Our curve of growth, whether of weight or length (Fig. 3), has always a certain characteristic form, or characteristic curvature. This is our immediate proof of the fact that the rate of growth changes as time goes on; for had it not been so, had an equal increment of length been added in each equal interval of time, our “curve” would have appeared as a straight line. Such as it is, it tells us not only that the rate of growth tends to alter, but that it alters in a definite and orderly way; for, subject to various minor interruptions, due to secondary causes, our curves of growth are, on the whole, “smooth” curves.
The curve of growth for length or stature in man indicates a rapid increase at the outset, that is to say during the quick growth of babyhood; a long period of slower, but still rapid and almost steady growth in early boyhood; as a rule a marked quickening soon after the boy is in his teens, when he comes to “the growing age”; and finally a gradual arrest of growth as the boy “comes to his full height,” and reaches manhood.
If we carried the curve further, we should see a very curious thing. We should see that a man’s full stature endures but for a spell; long before fifty [95] it has begun to abate, by sixty it is notably lessened, in extreme old age the old man’s frame is shrunken and it is but a memory that “he once was tall.” We have already seen, and here we see again, that growth may have a “negative value.” The phenomenon of negative growth in old age extends to weight also, and is evidently largely chemical in origin: the organism can no longer add new material to its fabric fast enough to keep pace with the wastage of time. Our curve [p. 63] of growth is in fact a diagram of activity, or “time-energy” diagram [96]. As the organism grows it is absorbing energy beyond its daily needs, and accumulating it at a rate depicted in our
Stature in metres Weight in kgm. Span of % ratio
arms, of stature
Age Male Female % F/M Male Female % F/M male to span
0 0·500 0·494 98·8 3·2 2·9 90·7 0·496 100·8
1 0·698 0·690 98·8 9·4 8·8 93·6 0·695 100·4
2 0·791 0·781 98·7 11·3 10·7 94·7 0·789 100·3
3 0·864 0·854 98·8 12·4 11·8 95·2 0·863 100·1
4 0·927 0·915 98·7 14·2 13·0 91·5 0·927 100·0
5 0·987 0·974 98·7 15·8 14·4 91·1 0·988 99·9
6 1·046 1·031 98·5 17·2 16·0 93·0 1·048 99·8
7 1·104 1·087 98·4 19·1 17·5 91·6 1·107 99·7
8 1·162 1·142 98·2 20·8 19·1 91·8 1·166 99·6
9 1·218 1·196 98·2 22·6 21·4 94·7 1·224 99·5
10 1·273 1·249 98·1 24·5 23·5 95·9 1·281 99·4
11 1·325 1·301 98·2 27·1 25·6 94·5 1·335 99·2
12 1·375 1·352 98·3 29·8 29·8 100·0 1·388 99·1
13 1·423 1·400 98·4 34·4 32·9 95·6 1·438 98·9
14 1·469 1·446 98·4 38·8 36·7 94·6 1·489 98·7
15 1·513 1·488 98·3 43·6 40·4 92·7 1·538 99·4
16 1·554 1·521 97·8 49·7 43·6 87·7 1·584 98·1
17 1·594 1·546 97·0 52·8 47·3 89·6 1·630 97·9
18 1·630 1·563 95·9 57·8 49·0 84·8 1·670 97·6
19 1·655 1·570 94·9 58·0 51·6 89·0 1·705 97·1
20 1·669 1·574 94·3 60·1 52·3 87·0 1·728 96·6
25 1·682 1·578 93·8 62·9 53·3 84·7 1·731 97·2
30 1·686 1·580 93·7 63·7 54·3 85·3 1·766 95·5
40 1·686 1·580 93·7 63·7 55·2 86·7 1·766 95·5
50 1·686 1·580 93·7 63·5 56·2 88·4 — —
60 1·676 1·571 93·7 61·9 54·3 87·7 — —
70 1·660 1·556 93·7 59·5 51·5 86·5 — —
80 1·636 1·534 93·8 57·8 49·4 85·5 — —
90 1·610 1·510 93·8 57·8 49·3 85·3 — —
curve; but the time comes when it accumulates no longer, and at last it is constrained to draw upon its dwindling store. But in part, the slow decline in stature is an expression of an unequal contest between our bodily powers and the unchanging force of gravity, [p. 64] which draws us down when we would fain rise up [97]. For against gravity we fight all our days, in every movement of our limbs, in every beat of our hearts; it is the indomitable force that defeats us in the end, that lays us on our deathbed, that lowers us to the grave [98].
Side by side with the curve which represents growth in length, or stature, our diagram shows the curve of weight [99]. That this curve is of a very different shape from the former one, is accounted for in the main (though not wholly) by the fact which we have already dealt with, that, whatever be the law of increment in a linear dimension, the law of increase in volume, and therefore in weight, will be that these latter magnitudes tend to vary as the cubes of the linear dimensions. This however does not account for the change of direction, or “point of inflection” which we observe in the curve of weight at about one or two years old, nor for certain other differences between our two curves which the scale of our diagram does not yet make clear. These differences are due to the fact that the form of the child is altering with growth, that other linear dimensions are varying somewhat differently from length or stature, and that consequently the growth in bulk or weight is following a more complicated law.
Our curve of growth, whether for weight or length, is a direct picture of velocity, for it represents, as a connected series, the successive epochs of time at which successive weights or lengths are attained. But, as we have already in part seen, a great part of the interest of our curve lies in the fact that we can see from it, not only that length (or some other magnitude) is changing, but that the rate of change of magnitude, or rate of growth, is itself changing. We have, in short, to study the phenomenon of acceleration: we have begun by studying a velocity, or rate of [p. 65] change of magnitude; we must now study an acceleration, or rate of change of velocity. The rate, or velocity, of growth is measured by the slope of the curve; where the curve is steep, it means that growth is rapid, and when growth ceases the curve appears as a horizontal line. If we can find a means, then, of representing at successive epochs the corresponding slope, or steepness, of the curve, we shall have obtained a picture of the rate of change of velocity, or the acceleration of growth. The measure of the steepness of a curve is given by the tangent to the curve, or we may estimate it by taking for equal intervals of time (strictly speaking, for each infinitesimal interval of time) the actual increment added during that interval of time: and in practice this simply amounts to taking the successive differences between the values of length (or of weight) for the successive ages which we have begun by studying. If we then plot these successive differences against time, we obtain a curve each point upon which represents a velocity, and the whole curve indicates the rate of change of velocity, and we call it an acceleration-curve. It contains, in truth, nothing whatsoever that was not implicit in our former curve; but it makes clear to our eye, and brings within the reach of further investigation, phenomena that were hard to see in the other mode of representation.
The acceleration-curve of height, which we here illustrate, in Fig. 4, is very different in form from the curve of growth which we have just been looking at; and it happens that, in this case, there is a very marked difference between the curve which we obtain from Quetelet’s data of growth in height and that which we may draw from any other series of observations known to me from British, French, American or German writers. It begins (as will be seen from our next table) at a very high level, such as it never afterwards attains; and still stands too high, during the first three or four years of life, to be represented on the scale of the accompanying diagram. From these high velocities it falls away, on the whole, until the age when growth itself ceases, and when the rate of growth, accordingly, has, for some years together, the constant value of nil; but the rate of fall, or rate of change of velocity, is subject to several changes or interruptions. During the first three or four years of life the fall is continuous and rapid, [p. 66] but it is somewhat arrested for a while in childhood, from about five years old to eight. According to Quetelet’s data, there is another slight interruption in the falling rate between the ages of about fourteen and sixteen; but in place of this almost insignificant interruption, the English and other statistics indicate a sudden
1917 Source Illustration
Fig. 4. Mean annual increments of stature (♂), Belgian and American.
and very marked acceleration of growth beginning at about twelve years of age, and lasting for three or four years; when this period of acceleration is over, the rate begins to fall again, and does so with great rapidity. We do not know how far the absence of this striking feature in the Belgian curve is due to the imperfections of Quetelet’s data, or whether it is a real and significant feature in the small-statured race which he investigated.
────[D]─── ───────[E]─────── ──────[F]────── ────────[G]───────
Height Height [I] Height [V] [H] [H]
Age (Boys) [H] [J] [K] [J] [K] (Boys) (6) [H] ([J]) [V] ([K]) [V]
0 50·0 — — — — — — — — — — — —
1 69·8 19·8 74·2 73·6 — — — — — — — — —
2 79·1 9·3 82·7 81·8 8·5 8·2 — — — — — — —
3 86·4 7·3 89·1 88·4 6·4 6·6 — — — — — — —
4 92·7 6·3 96·8 95·8 7·7 7·4 — — — — — — —
5 98·7 6·0 103·3 101·9 6·5 6·1 105·90 4·40 — — — — —
6 104·0 5·9 109·9 108·9 6·6 7·0 111·58 4·62 5·68 6·55 1·57 5·75 0·88
7 110·4 5·8 114·4 113·8 4·5 4·9 116·83 4·93 5·25 5·70 0·68 5·90 0·98
8 116·2 5·8 119·7 119·5 5·3 5·7 122·04 5·34 5·21 5·37 0·86 5·70 1·10
9 121·8 5·6 125·0 124·7 5·3 4·8 126·91 5·49 4·87 4·89 0·96 5·50 0·97
10 127·3 5·5 130·3 129·5 5·3 5·2 131·78 5·75 4·87 5·10 1·03 5·97 1·23
11 132·5 5·2 133·6 134·4 3·3 4·9 136·20 6·19 4·42 5·02 0·88 6·17 1·85
12 137·5 5·0 137·6 141·5 4·0 7·1 140·74 6·66 4·54 4·99 1·26 6·98 1·89
13 142·3 4·8 145·1 148·6 7·5 7·1 146·00 7·54 5·26 5·91 1·86 6·71 2·06
14 146·9 4·6 153·8 152·9 8·7 4·3 152·39 8·49 6·39 7·88 2·39 5·44 2·89
15 151·3 4·4 159·6 154·2 5·8 1·3 159·72 8·78 7·33 6·23 2·91 5·34 2·71
16 155·4 4·1 — — — — 164·90 7·73 5·18 5·64 3·46 — —
17 159·4 4·0 — — — — 168·91 7·22 4·01 — — — —
18 163·0 3·6 — — — — 171·07 6·74 2·16 — — — —
19 165·5 2·5 — — — — — — — — — — —
20 167·0 1·5 — — — — — — — — — — —
‡ Direct observations on actual, or individualised, increase of stature from year to year: between the ages of 5–6, 6–7, etc.
Even apart from these data of Quetelet’s (which seem to constitute an extreme case), it is evident that there are very [p. 68] marked differences between different races, as we shall presently see there are between the two sexes, in regard to the epochs of acceleration of growth, in other words, in the “phase” of the curve.
It is evident that, if we pleased, we might represent the rate of change of acceleration on yet another curve, by constructing a table of “second differences”; this would bring out certain very interesting phenomena, which here however we must not stay to discuss.
Increment
Age Male Female
0–1 5·9 5·6
1–2 2·0 2·4
2–3 1·5 1·4
3–4 1·5 1·5
4–5 1·9 1·4
5–6 1·9 1·4
6–7 1·9 1·1
7–8 1·9 1·2
8–9 1·9 2·0
9–10 1·7 2·1
10–11 1·8 2·4
11–12 2·0 3·5
12–13 4·1 3·5
13–14 4·0 3·8
14–15 4·1 3·7
15–16 4·2 3·5
16–17 4·3 3·3
17–18 4·2 3·0
18–19 3·7 2·3
19–20 1·9 1·1
20–21 1·7 1·1
21–22 1·7 0·5
22–23 1·6 0·4
23–24 0·9 −0·2
24–25 0·8 −0·2
* The values given in this table are not in precise accord with those of the Table on p. 63. The latter represent Quetelet’s results arrived at in 1835; the former are the means of his determinations in 1835–40.
The acceleration-curve for man’s weight (Fig. 5), whether we draw it from Quetelet’s data, or from the British, American and other statistics of later writers, is on the whole similar to that which we deduce from the statistics of these latter writers in regard to height or stature; that is to say, it is not a curve which continually descends, but it indicates a rate of growth which is subject to important fluctuations at certain epochs of life. We see that it begins at a high level, and falls continuously and rapidly [100] [p. 69] during the first two or three years of life. After a slight recovery, it runs nearly level during boyhood from about five to twelve years old; it then rapidly rises, in the “growing period” of the early teens, and slowly and steadily falls from about the age of sixteen onwards. It does not reach the base-line till the man is about seven or eight and twenty, for normal increase of weight continues during the years when the man is “filling out,” long after growth in height has ceased; but at last, somewhere about thirty, the velocity reaches zero, and even falls below it, for then the man usually begins to lose weight a little. The subsequent slow changes in this acceleration-curve we need not stop to deal with.
1917 Source Illustration
Fig. 5. Mean annual increments of weight, in man and woman; from Quetelet’s data.
In the same diagram (Fig. 5) I have set forth the acceleration-curves in respect of increment of weight for both man and woman, according to Quetelet. That growth in boyhood and growth in girlhood follow a very different course is a matter of common knowledge; but if we simply plot the ordinary curve of growth, or velocity-curve, the difference, on the small scale of our diagrams, [p. 70] is not very apparent. It is admirably brought out, however, in the acceleration-curves. Here we see that, after infancy, say from three years old to eight, the velocity in the girl is steady, just as in the boy, but it stands on a lower level in her case than in his: the little maid at this age is growing slower than the boy. But very soon, and while his acceleration-curve is still represented by a straight line, hers has begun to ascend, and until the girl is about thirteen or fourteen it continues to ascend rapidly. After that age, as after sixteen or seventeen in the boy’s case, it begins to descend. In short, throughout all this period, it is a very similar curve in the two sexes; but it has its notable differences, in amplitude and especially in phase. Last of all, we may notice that while the acceleration-curve falls to a negative value in the male about or even a little before the age of thirty years, this does not happen among women. They continue to grow in weight, though slowly, till very much later in life; until there comes a final period, in both sexes alike, during which weight, and height and strength all alike diminish.
From certain corrected, or “typical” values, given for American children by Boas and Wissler (l.c. p. 42), we obtain the following still clearer comparison of the annual increments of stature in boys and girls: the typical stature at the commencement of the period, i.e. at the age of eleven, being 135·1 cm. and 136·9 cm. for the boys and girls respectively, and the annual increments being as follows:
Age 12 13 14 15 16 17 18 19 20
Boys (cm.) 4·1 6·3 8·7 7·9 5·2 3·2 1·9 0·9 0·3
Girls (cm.) 7·5 7·0 4·6 2·1 0·9 0·4 0·1 0·0 0·0
Difference −3·4 −0·7 4·1 5·8 4·3 2·8 1·8 0·9 0·3
The result of these differences (which are essentially phase-differences) between the two sexes in regard to the velocity of growth and to the rate of change of that velocity, is to cause the ratio between the weights of the two sexes to fluctuate in a somewhat complicated manner. At birth the baby-girl weighs on the average nearly 10 per cent. less than the boy. Till about two years old she tends to gain upon him, but she then loses again until the age of about five; from five she gains for a few years somewhat rapidly, and the girl of ten to twelve is only some 3 per cent. less in weight than the boy. The boy in his teens gains [p. 71] steadily, and the young woman of twenty is nearly 15 per cent. lighter than the man. This ratio of difference again slowly diminishes, and between fifty and sixty stands at about 12 per cent., or not far from the mean for all ages; but once more as old age advances, the difference tends, though very slowly, to increase (Fig. 6).
1917 Source Illustration
Fig. 6. Percentage ratio, throughout life, of female weight to male; from Quetelet’s data.
While careful observations on the rate of growth in other animals are somewhat scanty, they tend to show so far as they go that the general features of the phenomenon are always much the same. Whether the animal be long-lived, as man or the elephant, or short-lived, like horse or dog, it passes through the same phases of growth [101]. In all cases growth begins slowly; it attains a maximum velocity early in its course, and afterwards slows down (subject to temporary accelerations) towards a point where growth ceases altogether. But especially in the cold-blooded animals, such as fishes, the slowing-down period is very greatly protracted, and the size of the creature would seem never actually to reach, but only to approach asymptotically, to a maximal limit.
The size ultimately attained is a resultant of the rate, and of [p. 72] the duration, of growth. It is in the main true, as Minot has said, that the rabbit is bigger than the guinea-pig because he grows the faster; but that man is bigger than the rabbit because he goes on growing for a longer time.
In ordinary physical investigations dealing with velocities, as for instance with the course of a projectile, we pass at once from the study of acceleration to that of momentum and so to that of force; for change of momentum, which is proportional to force, is the product of the mass of a body into its acceleration or change of velocity. But we can take no such easy road of kinematical investigation in this case. The “velocity” of growth is a very different thing from the “velocity” of the projectile. The forces at work in our case are not susceptible of direct and easy treatment; they are too varied in their nature and too indirect in their action for us to be justified in equating them directly with the mass of the growing structure.
It was apparently from a feeling that the velocity of growth ought in some way to be equated with the mass of the growing structure that Minot [102] introduced a curious, and (as it seems to me) an unhappy method of representing growth, in the form of what he called “percentage-curves”; a method which has been followed by a number of other writers and experimenters. Minot’s method was to deal, not with the actual increments added in successive periods, such as years or days, but with these increments represented as percentages of the amount which had been reached at the end of the former period. For instance, taking Quetelet’s values for the height in centimetres of a male infant from birth to four years old, as follows:
Years 0 1 2 3 4
cm. 50·0 69·8 79·1 86·4 92·7
Now when we plot actual length against time, we have a perfectly definite thing. When we differentiate this L/T, we have dL/dT, which is (of course) velocity; and from this, by a second differentiation, we obtain d2 L/dT2, that is to say, the acceleration. [p. 73]
that is to say, you are multiplying the thing you wish to represent by another quantity which is itself continually varying; and the result is that you are dealing with something very much less easily grasped by the mind than the original factors. Professor Minot is, of course, dealing with a perfectly legitimate function of x and y; and his method is practically tantamount to plotting log y against x, that is to say, the logarithm of the increment against the time. This could only be defended and justified if it led to some simple result, for instance if it gave us a straight line, or some other simpler curve than our usual curves of growth. As a matter of fact, it is manifest that it does nothing of the kind.
Pre-natal and post-natal growth.
In the acceleration-curves which we have shown above (Figs. 2, 3), it will be seen that the curve starts at a considerable interval from the actual date of birth; for the first two increments which we can as yet compare with one another are those attained during the first and second complete years of life. Now we can in many cases “interpolate” with safety between known points upon a curve, but it is very much less safe, and is not very often justifiable (at least until we understand the physical principle involved, and its mathematical expression), to “extrapolate” beyond the limits of our observations. In short, we do not yet know whether our curve continued to ascend as we go backwards to the date of birth, or whether it may not have changed its direction, and descended, perhaps, to zero-value. In regard to length, or stature, however, we can obtain the requisite information from certain tables of Rüssow’s [103], who gives the stature of the infant month by month during the first year of its life, as follows:
Age in months 0 1 2 3 4 5 6 7 8 9 10 11 12
Length in cm. (50) 54 58 60 62 64 65 66 67·5 68 69 70·5 72
[Differences (in cm.) 4 4 2 2 2 1 1 1·5 ·5 1 1·5 1·5]
If we multiply these monthly differences, or mean monthly velocities, by 12, to bring them into a form comparable with the [p. 74] annual velocities already represented on our acceleration-curves, we shall see that the one series of observations joins on very well with the other; and in short we see at once that our acceleration-curve rises steadily and rapidly as we pass back towards the date of birth.
1917 Source Illustration
Fig. 7. Curve of growth (in length or stature) of child, before and after birth. (From His and Rüssow’s data.)
But birth itself, in the case of a viviparous animal, is but an unimportant epoch in the history of growth. It is an epoch whose relative date varies according to the particular animal: the foal and the lamb are born relatively later, that is to say when development has advanced much farther, than in the case of man; the kitten and the puppy are born earlier and therefore more helpless than we are; and the mouse comes into the world still earlier and more inchoate, so much so that even the little marsupial is scarcely more unformed and embryonic. In all these cases alike, we must, in order to study the curve of growth in its entirety, take full account of prenatal or intra-uterine growth. [p. 75]
According to His [104], the following are the mean lengths of the unborn human embryo, from month to month.
Months 0 1 2 3 4 5 6 7 8 9 10
(Birth)
Length in mm. 0 7·5 40 84 162 275 352 402 443 472 490–500
Increment per
month in mm. — 7·5 32·5 44 78 113 77 50 41 29 18–28
1917 Source Illustration
Fig. 8. Mean monthly increments of length or stature of child (in cms.).
These data link on very well to those of Rüssow, which we have just considered, and (though His’s measurements for the pre-natal months are more detailed than are those of Rüssow for the first year of post-natal life) we may draw a continuous curve of growth (Fig. 7) and curve of acceleration of growth (Fig. 8) for the combined periods. It will at once be seen that there is a “point of inflection” somewhere about the fifth month of intra-uterine life [105]: up to that date growth proceeds with a continually increasing [p. 76] velocity; but after that date, though growth is still rapid, its velocity tends to fall away. There is a slight break between our two separate sets of statistics at the date of birth, while this is the very epoch regarding which we should particularly like to have precise and continuous information. Undoubtedly there is a certain slight arrest of growth, or diminution of the rate of growth, about the epoch of birth: the sudden change in the [p. 77] method of nutrition has its inevitable effect; but this slight temporary set-back is immediately followed by a secondary, and temporary, acceleration.
1917 Source Illustration
Fig. 9. Curve of pre-natal growth (length or stature) of child; and corresponding curve of mean monthly increments (mm.).
1917 Source Illustration
Fig. 10. Curve of growth of bamboo (from Ostwald, after Kraus).
It is worth our while to draw a separate curve to illustrate on a larger scale His’s careful data for the ten months of pre-natal life (Fig. 9). We see that this curve of growth is a beautifully regular one, and is nearly symmetrical on either side of that point of inflection of which we have already spoken; it is a curve for which we might well hope to find a simple mathematical expression. The acceleration-curve shown in Fig. 9 together with the pre-natal curve of growth, is not taken directly from His’s recorded data, but is derived from the tangents drawn to a smoothed curve, corresponding as nearly as possible to the actual curve of growth: the rise to a maximal velocity about the fifth month and the subsequent gradual fall are now demonstrated even more clearly than before. In Fig. 10, which is a curve of growth of the bamboo [106], we see (so far as it goes) the same essential features, [p. 78] the slow beginning, the rapid increase of velocity, the point of inflection, and the subsequent slow negative acceleration [107].
Variability and Correlation of Growth.
The magnitudes and velocities which we are here dealing with are, of course, mean values derived from a certain number, sometimes a large number, of individual cases. But no statistical account of mean values is complete unless we also take account of the amount of variability among the individual cases from which the mean value is drawn. To do this throughout would lead us into detailed investigations which lie far beyond the scope of this elementary book; but we may very briefly illustrate the nature of the process, in connection with the phenomena of growth which we have just been studying.
It was in connection with these phenomena, in the case of man, that Quetelet first conceived the statistical study of variation, on lines which were afterwards expounded and developed by Galton, and which have grown, in the hands of Karl Pearson and others, into the modern science of Biometrics.
When Quetelet tells us, for instance, that the mean stature of the ten-year old boy is 1·273 metres, this implies, according to the law of error, or law of probabilities, that all the individual measurements of ten-year-old boys group themselves in an orderly way, that is to say according to a certain definite law, about this mean value of 1·273. When these individual measurements are grouped and plotted as a curve, so as to show the number of individual cases at each individual length, we obtain a characteristic curve of error or curve of frequency; and the “spread” of this curve is a measure of the amount of variability in this particular case. A certain mathematical measure of this “spread,” as described in works upon statistics, is called the Index of Variability, or Standard Deviation, and is usually denominated by the letter σ. It is practically equivalent to a determination of the point upon the frequency curve where it changes its curvature on either side of the mean, and where, from being concave towards the middle line, it spreads out to be convex thereto. When we divide this [p. 79] value by the mean, we get a figure which is independent of any particular units, and which is called the Coefficient of Variability. (It is usually multiplied by 100, to make it of a more convenient amount; and we may then define this coefficient, C, as = (σ/M) × 100.)
In regard to the growth of man, Pearson has determined this coefficient of variability as follows: in male new-born infants, the coefficient in regard to weight is 15·66, and in regard to stature, 6·50; in male adults, for weight 10·83, and for stature, 3·66. The amount of variability tends, therefore, to decrease with growth or age.
Similar determinations have been elaborated by Bowditch, by Boas and Wissler, and by other writers for intermediate ages, especially from about five years old to eighteen, so covering a great part of the whole period of growth in man [108].
Age 5 6 7 8 9
Stature (Bowditch) 4·76 4·60 4·42 4·49 4·40
Stature (Boas and Wissler) 4·15 4·14 4·22 4·37 4·33
Weight (Bowditch) 11·56 10·28 11·08 9·92 11·04
Age 10 11 12 13 14
Stature (Bowditch) 4·55 4·70 4·90 5·47 5·79
Stature (Boas and Wissler) 4·36 4·54 4·73 5·16 5·57
Weight (Bowditch) 11·60 11·76 13·72 13·60 16·80
Age 15 16 17 18
Stature (Bowditch) 5·57 4·50 4·55 3·69
Stature (Boas and Wissler) 5·50 4·69 4·27 3·94
Weight (Bowditch) 15·32 13·28 12·96 10·40
The result is very curious indeed. We see, from Fig. 11, that the curve of variability is very similar to what we have called the acceleration-curve (Fig. 4): that is to say, it descends when the rate of growth diminishes, and rises very markedly again when, in late boyhood, the rate of growth is temporarily accelerated. We [p. 80] see, in short, that the amount of variability in stature or in weight is a function of the rate of growth in these magnitudes, though we are not yet in a position to equate the terms precisely, one with another.
1917 Source Illustration
Fig. 11. Coefficients of variability of stature in Man (♂). from Boas and Wissler’s data.
If we take not merely the variability of stature or weight at a given age, but the variability of the actual successive increments in each yearly period, we see that this latter coefficient of variability tends to increase steadily, and more and more rapidly, within the limits of age for which we have information; and this phenomenon is, in the main, easy of explanation. For a great part of the difference, in regard to rate of growth, between one individual and another is a difference of phase,—a difference in the epochs of acceleration and retardation, and finally in the epoch when growth comes to an end. And it follows that the variability of rate will be more and more marked, as we approach and reach the period when some individuals still continue, and others have already ceased, to grow. In the following epitomised table, [p. 81] I have taken Boas’s determinations of variability (σ) (op. cit. p. 1548), converted them into the corresponding coefficients of variability ((σ/M) × 100), and then smoothed the resulting numbers.
Age 7 8 9 10 11 12 13 14 15
Boys 17·3 15·8 18·6 19·1 21·0 24·7 29·0 36·2 46·1
Girls 17·1 17·8 19·2 22·7 25·9 29·3 37·0 44·8 —
The greater variability of annual increment in the girls, as compared with the boys, is very marked, and is easily explained by the more rapid rate at which the girls run through the several phases of the phenomenon.
Just as there is a marked difference in “phase” between the growth-curves of the two sexes, that is to say a difference in the periods when growth is rapid or the reverse, so also, within each sex, will there be room for similar, but individual phase-differences. Thus we may have children of accelerated development, who at a given epoch after birth are both rapidly growing and already “big for their age”; and others of retarded development who are comparatively small and have not reached the period of acceleration which, in greater or less degree, will come to them in turn. In other words, there must under such circumstances be a strong positive “coefficient of correlation” between stature and rate of growth, and also between the rate of growth in one year and the next. But it does not by any means follow that a child who is precociously big will continue to grow rapidly, and become a man or woman of exceptional stature. On the contrary, when in the case of the precocious or “accelerated” children growth has begun to slow down, the backward ones may still be growing rapidly, and so making up (more or less completely) to the others. In other words, the period of high positive correlation between stature and increment will tend to be followed by one of negative correlation. This interesting and important point, due to Boas and Wissler [109], is confirmed by the following table:―
Age 6 7 8 9 10 11 12 13 14 15
Stature (B) 112·7 115·5 123·2 127·4 133·2 136·8 142·7 147·3 155·9 162·2
(G) 111·4 117·7 121·4 127·9 131·8 136·7 144·6 149·7 153·8 157·2
Increment (B) 5·7 5·3 4·9 5·1 5·0 4·7 5·9 7·5 6·2 5·2
(G) 5·9 5·5 5·5 5·9 6·2 7·2 6·5 5·4 3·3 1·7
Correlation (B) ·25 ·11 ·08 ·25 ·18 ·18 ·48 ·29 −·42 −·44
(G) ·44 ·14 ·24 ·47 ·18 −·18 −·42 −·39 −·63 ·11
A minor, but very curious point brought out by the same investigators is that, if instead of stature we deal with height in the sitting posture (or, practically speaking, with length of trunk or back), then the correlations between this height and its annual increment are throughout negative. In other words, there would seem to be a general tendency for the long trunks to grow slowly throughout the whole period under investigation. It is a well-known anatomical fact that tallness is in the main due not to length of body but to length of limb.
The whole phenomenon of variability in regard to magnitude and to rate of increment is in the highest degree suggestive: inasmuch as it helps further to remind and to impress upon us that specific rate of growth is the real physiological factor which we want to get at, of which specific magnitude, dimensions and form, and all the variations of these, are merely the concrete and visible resultant. But the problems of variability, though they are intimately related to the general problem of growth, carry us very soon beyond our present limitations.
Rate of growth in other organisms. [110]
Just as the human curve of growth has its slight but well-marked interruptions, or variations in rate, coinciding with such epochs as birth and puberty, so is it with other animals, and this phenomenon is particularly striking in the case of animals which undergo a regular metamorphosis.
In the accompanying curve of growth in weight of the mouse (Fig. 12), based on W. Ostwald’s observations [111], we see a distinct slackening of the rate when the mouse is about a fortnight old, at which period it opens its eyes and very soon afterwards is weaned. At about six weeks old there is another well-marked retardation of growth, following on a very rapid period, and coinciding with the epoch of puberty. [p. 83]
Fig. 13 shews the curve of growth of the silkworm [112], during its whole larval life, up to the time of its entering the chrysalis stage.
The silkworm moults four times, at intervals of about a week, the first moult being on the sixth or seventh day after hatching. A distinct retardation of growth is exhibited on our curve in the case of the third and fourth moults; while a similar retardation accompanies the first and second moults also, but the scale of our diagram does not render it visible. When the worm is about seven weeks old, a remarkable process of “purgation” takes place, as a preliminary to entering on the pupal, or chrysalis, stage; and the great and sudden loss of weight which accompanies this process is the most marked feature of our curve.
1917 Source Illustration
Fig. 12. Growth in weight of Mouse. (After W. Ostwald.)
The rate of growth in the tadpole [113] (Fig. 14) is likewise marked by epochs of retardation, and finally by a sudden and drastic change. There is a slight diminution in weight immediately after [p. 84] the little larva frees itself from the egg; there is a retardation of growth about ten days later, when the external gills disappear; and finally, the complete metamorphosis, with the loss of the tail, the growth of the legs and the cessation of branchial respiration, is accompanied by a loss of weight amounting to wellnigh half the weight of the full-grown larva. [p. 85]
1917 Source Illustration
Fig. 13. Growth in weight of Silkworm. (From Ostwald, after Luciani and Lo Monaco.)
While as a general rule, the better the animals be fed the quicker they grow and the sooner they metamorphose, Barfürth has pointed out the curious fact that a short spell of starvation, just before metamorphosis is due, appears to hasten the change.
1917 Source Illustration
Fig. 14. Growth in weight of Tadpole. (From Ostwald, after Schaper.)
The negative growth, or actual loss of bulk and weight which often, and perhaps always, accompanies metamorphosis, is well shewn in the case of the eel [114]. The contrast of size is great between [p. 87] the flattened, lancet-shaped Leptocephalus larva and the little black cylindrical, almost thread-like elver, whose magnitude is less than that of the Leptocephalus in every dimension, even, at first, in length (Fig. 15).
1917 Source Illustration
Fig. 15. Development of Eel; from Leptocephalus larvae to young Elver. (From Ostwald after Joh. Schmidt.)
1917 Source Illustration
Fig. 16. Growth in length of Spirogyra. (From Ostwald, after Hofmeister.)
From the higher study of the physiology of growth we learn that such fluctuations as we have described are but special interruptions in a process which is never actually continuous, but is perpetually interrupted in a rhythmic manner [115]. Hofmeister shewed, for instance, that the growth of Spirogyra proceeds by fits and starts, by periods of activity and rest, which alternate with one another at intervals of so many minutes (Fig. 16). And Bose, by very refined methods of experiment, has shewn that plant-growth really proceeds by tiny and perfectly rhythmical pulsations recurring at regular intervals of a few seconds of time. Fig. 17 shews, according to Bose’s observations [116], the growth of a crocus, under a very high magnification. The stalk grows by little jerks, each with an amplitude of about ·002 mm., every [p. 88] twenty seconds or so, and after each little increment there is a partial recoil.
1917 Source Illustration
Fig. 17. Pulsations of growth in Crocus, in micro-millimetres. (After Bose.)
The rate of growth of various parts or organs. [117]
The differences in regard to rate of growth between various parts or organs of the body, internal and external, can be amply illustrated in the case of man, and also, but chiefly in regard to external form, in some few other creatures [118]. It is obvious that there lies herein an endless field for the mathematical study of correlation and of variability, but with this aspect of the case we cannot deal.
In the accompanying table, I shew, from some of Vierordt’s data, the relative weights, at various ages, compared with the weight at birth, of the entire body, of the brain, heart and liver; [p. 89] and also the percentage relation which each of these organs bears, at the several ages, to the weight of the whole body.
Percentage weights compared
Weight Relative weights of with total body-weights
of body† ───────────────────────── ───────────────────────────
Age in kg. Body Brain Heart Liver Body Brain Heart Liver
0 3·1 1 1 1 1 100 12·29 0·76 4·57
1 9·0 2·90 2·48 1·75 2·35 100 10·50 0·46 3·70
2 11·0 3·55 2·69 2·20 3·02 100 9·32 0·47 3·89
3 12·5 4·03 2·91 2·75 3·42 100 8·86 0·52 3·88
4 14·0 4·52 3·49 3·14 4·15 100 9·50 0·53 4·20
5 15·9 5·13 3·32 3·43 3·80 100 7·94 0·51 3·39
6 17·8 5·74 3·57 3·60 4·34 100 7·63 0·48 3·45
7 19·7 6·35 3·54 3·95 4·86 100 6·84 0·47 3·49
8 21·6 6·97 3·62 4·02 4·59 100 6·38 0·44 3·01
9 23·5 7·58 3·74 4·59 4·95 100 6·06 0·46 2·99
10 25·2 8·13 3·70 5·41 5·90 100 5·59 0·51 3·32
11 27·0 8·71 3·57 5·97 6·14 100 5·04 0·52 3·22
12 29·0 9·35 3·78 (4·13) 6·21 100 4·88 (0·34) 3·03
13 33·1 10·68 3·90 6·95 7·31 100 4·49 0·50 3·13
14 37·1 11·97 3·38 9·16 8·39 100 3·47 0·58 3·20
15 41·2 13·29 3·91 8·45 9·22 100 3·62 0·48 3·17
16 45·9 14·81 3·77 9·76 9·45 100 3·16 0·51 2·95
17 49·7 16·03 3·70 10·63 10·46 100 2·84 0·51 2·98
18 53·9 17·39 3·73 10·33 10·65 100 2·64 0·46 2·80
19 57·6 18·58 3·67 11·42 11·61 100 2·43 0·51 2·86
20 59·5 19·19 3·79 12·94 11·01 100 2·43 0·51 2·62
21 61·2 19·74 3·71 12·59 11·48 100 2·31 0·49 2·66
22 62·9 20·29 3·54 13·24 11·82 100 2·14 0·50 2·66
23 64·5 20·81 3·66 12·42 10·79 100 2·16 0·46 2·37
24 — — 3·74 13·09 13·04 100 — — —
25 66·2 21·36 3·76 12·74 12·84 100 2·16 0·46 2·75
From the first portion of the table, it will be seen that none of these organs by any means keep pace with the body as a whole in regard to growth in weight; in other words, there must be some other part of the fabric, doubtless the muscles and the bones, which increase more rapidly than the average increase of the body. Heart and liver both grow nearly at the same rate, and by the [p. 90] age of twenty-five they have multiplied their weight at birth by about thirteen times, while the weight of the entire body has been multiplied by about twenty-one; but the weight of the brain has meanwhile been multiplied only about three and a quarter times. In the next place, we see the very remarkable phenomenon that the brain, growing rapidly till the child is about four years old, then grows more much slowly till about eight or nine years old, and after that time there is scarcely any further perceptible increase. These phenomena are diagrammatically illustrated in Fig. 18.
1917 Source Illustration
Fig. 18. Relative growth in weight (in Man) of Brain, Heart, and whole Body.
Many statistics indicate a decrease of brain-weight during adult life. Boas [119] was inclined to attribute this apparent phenomenon to our statistical methods, and to hold that it could “hardly be explained in any other way than by assuming an increased death-rate among men with very large brains, at an age of about twenty years.” But Raymond Pearl has shewn that there is evidence of a steady and very gradual decline in the weight of the brain with advancing age, beginning at or before the twentieth year, and continuing throughout adult life [120]. [p. 91]
The second part of the table shews the steadily decreasing weights of the organs in question as compared with the body; the brain falling from over 12 per cent. at birth to little over 2 per cent. at five and twenty; the heart from ·75 to ·46 per cent.; and the liver from 4·57 to 2·75 per cent. of the whole bodily weight.
It is plain, then, that there is no simple and direct relation, holding good throughout life, between the size of the body as a whole and that of the organs we have just discussed; and the changing ratio of magnitude is especially marked in the case of the brain, which, as we have just seen, constitutes about one-eighth of the whole bodily weight at birth, and but one-fiftieth at five and twenty. The same change of ratio is observed in other animals, in equal or even greater degree. For instance, Max Weber [121] tells us that in the lion, at five weeks, four months, eleven months, and lastly when full-grown, the brain-weight represents the following fractions of the weight of the whole body, viz. 1/18, 1/80, 1/184, and 1/546. And Kellicott has, in like manner, shewn that in the dogfish, while some organs (e.g. rectal gland, pancreas, etc.) increase steadily and very nearly proportionately to the body as a whole, the brain, and some other organs also, grow in a diminishing ratio, which is capable of representation, approximately, by a logarithmic curve [122].
But if we confine ourselves to the adult, then, as Raymond Pearl has shewn in the case of man, the relation of brain-weight to age, to stature, or to weight, becomes a comparatively simple one, and may be sensibly expressed by a straight line, or simple equation.
Thus, if W be the brain-weight (in grammes), and A be the age, or S the stature, of the individual, then (in the case of Swedish males) the following simple equations suffice to give the required ratios:
These equations are applicable to ages between fifteen and eighty; if we take narrower limits, say between fifteen and fifty, we can get a closer agreement by using somewhat altered constants. In the two sexes, and in different races, these empirical constants will be greatly changed [123]. Donaldson has further shewn that the correlation between brain-weight and body-weight is very much closer in the rat than in man [124].
The falling ratio of weight of brain to body with increase of size or age finds its parallel in comparative anatomy, in the general law that the larger the animal the less is the relative weight of the brain.
Weight of Weight of
entire animal brain
gms. gms. Ratio
Marmoset 335 12·5 1 : 26
Spider monkey 1845 126 1 : 15
Felis minuta 1234 23·6 1 : 56
F. domestica 3300 31 1 : 107
Leopard 27,700 164 1 : 168
Lion 119,500 219 1 : 546
Elephant 3,048,000 5430 1 : 560
Whale (Globiocephalus) 1,000,000 2511 1 : 400
For much information on this subject, see Dubois, “Abhängigkeit des Hirngewichtes von der Körpergrösse bei den Säugethieren,” Arch. f. Anthropol. XXV, 1897. Dubois has attempted, but I think with very doubtful success, to equate the weight of the brain with that of the animal. We may do this, in a very simple way, by representing the weight of the body as a power of that of the brain; thus, in the above table of the weights of brain and body in four species of cat, if we call W the weight of the body (in grammes), and w the weight of the brain, then if in all four cases we express the ratio by W = wn, we find that n is almost constant, and differs little from 2·24 in all four species: the values being respectively, in the order of the table 2·36, 2·24, 2·18, and 2·17. But this evidently amounts to no more than an empirical rule; for we can easily see that it depends on the particular scale which we have used, and that if the weights had been taken, for instance, in kilogrammes or in milligrammes, the agreement or coincidence would not have occurred [125]. [p. 93]
Men Women
────────────────────────── ──────────────────────
Age Total height Head Ratio Height Head† Ratio
m. m. m. m.
Birth 0·500 0·111 4·50 0·494 0·111 4·45
1 year 0·698 0·154 4·53 0·690 0·154 4·48
2 years 0·791 0·173 4·57 0·781 0·172 4·54
3 years 0·864 0·182 4·74 0·854 0·180 4·74
5 years 0·987 0·192 5·14 0·974 0·188 5·18
10 years 1·273 0·205 6·21 1·249 0·201 6·21
15 years 1·513 0·215 7·04 1·488 0·213 6·99
20 years 1·669 0·227 7·35 1·574 0·220 7·15
30 years 1·686 0·228 7·39 1·580 0·221 7·15
40 years 1·686 0·228 7·39 1·580 0·221 7·15
† A smooth curve, very similar to this, for the growth in “auricular height” of the girl’s head, is given by Pearson, in Biometrika, III, p. 141. 1904.
As regards external form, very similar differences exist, which however we must express in terms not of weight but of length. Thus the annexed table shews the changing ratios of the vertical length of the head to the entire stature; and while this ratio constantly diminishes, it will be seen that the rate of change is greatest (or the coefficient of acceleration highest) between the ages of about two and five years.
In one of Quetelet’s tables (supra, p. 63), he gives measurements of the total span of the outstretched arms in man, from year to year, compared with the vertical stature. The two measurements are so nearly identical in actual magnitude that a direct comparison by means of curves becomes unsatisfactory; but I have reduced Quetelet’s data to percentages, and it will be seen from Fig. 19 that the percentage proportion of span to height undergoes a remarkable and steady change from birth to the age of twenty years; the man grows more rapidly in stretch of arms than he does in height, and the span which was less than [p. 94] the stature at birth by about 1 per cent. exceeds it at the age of twenty by about 4 per cent. After the age of twenty, Quetelet’s data are few and irregular, but it is clear that the span goes on for a long while increasing in proportion to the stature. How far the phenomenon is due to actual growth of the arms and how far to the increasing breadth of the chest is not yet ascertained.
[Illustration: Fig. 19. Ratio of stature in Man, to span of outstretched arms.
(From Quetelet’s data.)]
The differences of rate of growth in different parts of the body are very simply brought out by the following table, which shews the relative growth of certain parts and organs of a young trout, at intervals of a few days during the period of most rapid development. It would not be difficult, from a picture of the little trout at any one of these stages, to draw its approximate form at any other, by the help of the numerical data here set forth [126]. [p. 95]
Days Total 1st Ventral 2nd Breadth
old length Eye Head dorsal fin dorsal Tail-fin of tail
49 100 100 100 100 100 100 100 100
63 129·9 129·4 148·3 148·6 148·5 108·4 173·8 155·9
77 154·9 147·3 189·2 (203·6) (193·6) 139·2 257·9 220·4
92 173·4 179·4 220·0 (193·2) (182·1) 154·5 307·6 272·2
106 194·6 192·5 242·5 173·2 165·3 173·4 337·3 287·7
While it is inequality of growth in different directions that we can most easily comprehend as a phenomenon leading to gradual change of outward form, we shall see in another chapter [127] that differences of rate at different parts of a longitudinal system, though always in the same direction, also lead to very notable and regular transformations. Of this phenomenon, the difference in rate of longitudinal growth between head and body is a simple case, and the difference which accompanies and results from it in the bodily form of the child and the man is easy to see. A like phenomenon has been studied in much greater detail in the case of plants, by Sachs and certain other botanists, after a method in use by Stephen Hales a hundred and fifty years before [128].
On the growing root of a bean, ten narrow zones were marked off, starting from the apex, each zone a millimetre in breadth. After twenty-four hours’ growth, at a certain constant temperature, the whole marked portion had grown from 10 mm. to 33 mm. in length; but the individual zones had grown at very unequal rates, as shewn in the annexed table [129].
Zone Increment
mm.
Apex 1·5
2nd 5·8
3rd 8·2
4th 3·5
5th 1·6
6th 1·3
7th 0·5
8th 0·3
9th 0·2
10th 0·1
1917 Source Illustration
Fig. 20. Rate of growth in successive zones near the tip of the bean-root.
The several values in this table lie very nearly (as we see by Fig. 20) in a smooth curve; in other words a definite law, or principle of continuity, connects the rates of growth at successive points along the growing axis of the root. Moreover this curve, in its general features, is singularly like those acceleration-curves which we have already studied, in which we plotted the rate of growth against successive intervals of time, as here we have plotted it against successive spatial intervals of an actual growing structure. If we suppose for a moment that the velocities of growth had been transverse to the axis, instead of, as in this case, longitudinal and parallel with it, it is obvious that these same velocities would have given us a leaf-shaped structure, of which our curve in Fig. 20 (if drawn to a suitable scale) would represent the actual outline on either side of the median axis; or, again, if growth had been not confined to one plane but symmetrical about the axis, we should have had a sort of turnip-shaped root, [p. 97] having the form of a surface of revolution generated by the same curve. This then is a simple and not unimportant illustration of the direct and easy passage from velocity to form.
A kindred problem occurs when, instead of “zones” artificially marked out in a stem, we deal with the rates of growth in successive actual “internodes”; and an interesting variation of this problem occurs when we consider, not the actual growth of the internodes, but the varying number of leaves which they successively produce. Where we have whorls of leaves at each node, as in Equisetum and in many water-weeds, then the problem presents itself in a simple form, and in one such case, namely in Ceratophyllum, it has been carefully investigated by Mr Raymond Pearl [130].
It is found that the mean number of leaves per whorl increases with each successive whorl; but that the rate of increment diminishes from whorl to whorl, as we ascend the axis. In other words, the increase in the number of leaves per whorl follows a logarithmic ratio; and if y be the mean number of leaves per whorl, and x the successional number of the whorl from the root or main stem upwards, then
where A, C, and a are certain specific constants, varying with the part of the plant which we happen to be considering. On the main stem, the rate of change in the number of leaves per whorl is very slow; when we come to the small twigs, or “tertiary branches,” it has become rapid, as we see from the following abbreviated table:
Position of whorl 1 2 3 4 5 6
Mean number of leaves 6·55 8·07 9·00 9·20 9·75 10·00
Increment — 1·52 ·93 ·20 (·55) (·25)
We have seen that a slow but definite change of form is a common accompaniment of increasing age, and is brought about as the simple and natural result of an altered ratio between the rates of growth in different dimensions: or rather by the progressive change necessarily brought about by the difference in their accelerations. There are many cases however in which the change is all but imperceptible to ordinary measurement, and many others in which some one dimension is easily measured, but others are hard to measure with corresponding accuracy. [p. 98] For instance, in any ordinary fish, such as a plaice or a haddock, the length is not difficult to measure, but measurements of breadth or depth are very much more uncertain. In cases such as these, while it remains difficult to define the precise nature of the change of form, it is easy to shew that such a change is taking place if we make use of that ratio of length to weight which we have spoken of in the preceding chapter. Assuming, as we may fairly do, that weight is directly proportional to bulk or volume, we may express this relation in the form W/L3 = k, where k is a constant, to be determined for each particular case. (W and L are expressed in grammes and centimetres, and it is usual to multiply the result by some figure, such as 1000, so as to give the constant k a value near to unity.)
Plaice caught in a certain area, March, 1907. Variation of k (the weight-length coefficient) with size. (Data taken from the Department of Agriculture and Fisheries’ Plaice-Report, vol. I, p. 107, 1908.)
Size in cm. Weight in gm. W/L^3 × 10,000 W/L^3 (smoothed)
23 113 92·8 —
24 128 92·6 94·3
25 152 97·3 96·1
26 173 98·4 97·9
27 193 98·1 99·0
28 221 100·6 100·4
29 250 102·5 101·2
30 271 100·4 101·2
31 300 100·7 100·4
32 328 100·1 99·8
33 354 98·5 98·8
34 384 97·7 98·0
35 419 97·7 97·6
36 454 97·3 96·7
37 492 95·2 96·3
38 529 96·4 95·6
39 564 95·1 95·0
40 614 95·9 95·0
41 647 93·9 93·8
42 679 91·6 92·5
43 732 92·1 92·5
44 800 93·9 94·0
45 875 96·0 —
Now while this k may be spoken of as a “constant,” having a certain mean value specific to each species of organism, and depending on the form of the organism, any change to which it may be subject will be a very delicate index of progressive changes of form; for we know that our measurements of length are, on the average, very accurate, and weighing is a still more delicate method of comparison than any linear measurement.
1917 Source Illustration
Fig. 21. Changes in the weight-length ratio of Plaice, with increasing size.
Thus, in the case of plaice, when we deal with the mean values for a large number of specimens, and when we are careful to deal only with such as are caught in a particular locality and at a particular time, we see that k is by no means constant, but steadily increases to a maximum, and afterwards slowly declines with the increasing size of the fish (Fig. 21). To begin with, therefore, the weight is increasing more rapidly than the cube of the length, and it follows that the length itself is increasing less rapidly than some other linear dimension; while in later life this condition is reversed. The maximum is reached when the length of the fish is somewhere near to 30 cm., and it is tempting to suppose that with this “point of inflection” there is associated some well-marked epoch in the fish’s life. As a matter of fact, the size of 30 cm. is approximately that at which sexual maturity may be said to begin, or is at least near enough to suggest a close connection between the two phenomena. The first step towards further investigation of the [p. 100] apparent coincidence would be to determine the coefficient k of the two sexes separately, and to discover whether or not the point of inflection is reached (or sexual maturity is reached) at a smaller size in the male than in the female plaice; but the material for this investigation is at present scanty.
1917 Source Illustration
Fig. 22. Periodic annual change in the weight-length ratio of Plaice.
A still more curious and more unexpected result appears when we compare the values of k for the same fish at different seasons of the year [131]. When for simplicity’s sake (as in the accompanying table and Fig. 22) we restrict ourselves to fish of one particular size, it is not necessary to determine the value of k, because a change in the ratio of length to weight is obvious enough; but when we have small numbers, and various sizes, to deal with, the determination of k may help us very much. It will be seen, then, that in the case of plaice the ratio of weight to length exhibits a regular periodic variation with the course of the seasons. [p. 101]
Relation of Weight to Length in Plaice of 55 cm. long, from Month to Month. (Data taken from the Department of Agriculture and Fisheries Plaice-Report, vol. II, p. 92, 1909.)
Average weight
in grammes W/L^3 × 100 W/L^3 (smoothed)
Jan. 2039 1·226 1·157
Feb. 1735 1·043 1·080
March 1616 0·971 0·989
April 1585 0·953 0·967
May 1624 0·976 0·985
June 1707 1·026 1·005
July 1686 1·013 1·037
August 1783 1·072 1·042
Sept. 1733 1·042 1·111
Oct. 2029 1·220 1·160
Nov. 2026 1·218 1·213
Dec. 1998 1·201 1·215
With unchanging length, the weight and therefore the bulk of the fish falls off from about November to March or April, and again between May or June and November the bulk and weight are gradually restored. The explanation is simple, and depends wholly on the process of spawning, and on the subsequent building up again of the tissues and the reproductive organs. It follows that, by this method, without ever seeing a fish spawn, and without ever dissecting one to see the state of its reproductive system, we can ascertain its spawning season, and determine the beginning and end thereof, with great accuracy.
As a final illustration of the rate of growth, and of unequal growth in various directions, I give the following table of data regarding the ox, extending over the first three years, or nearly so, of the animal’s life. The observed data are (1) the weight of the animal, month by month, (2) the length of the back, from the occiput to the root of the tail, and (3) the height to the withers. To these data I have added (1) the ratio of length to height, (2) the coefficient (k) expressing the ratio of weight to the cube of the length, and (3) a similar coefficient (k′) for the height of the animal. It will be seen that, while all these ratios tend to alter continuously, shewing that the animal’s form is steadily altering as it approaches maturity, the ratio between length and weight [p. 102] changes comparatively little. The simple ratio between length and height increases considerably, as indeed we should expect; for we know that in all Ungulate animals the legs are remarkably
k = k′ =
Age in W, wt. L, length H, W/L^3 W/H^3
months in kg. of back height L/H × 10 × 10
0 37 ·78 ·70 1·114 ·779 1·079
1 55·3 ·94 ·77 1·221 ·665 1·210
2 86·3 1·09 ·85 1·282 ·666 1·406
3 121·3 1·207 ·94 1·284 ·690 1·460
4 150·3 1·314 ·95 1·383 ·662 1·754
5 179·3 1·404 1·040 1·350 ·649 1·600
6 210·3 1·484 1·087 1·365 ·644 1·638
7 247·3 1·524 1·122 1·358 ·699 1·751
8 267·3 1·581 1·147 1·378 ·677 1·791
9 282·8 1·621 1·162 1·395 ·664 1·802
10 303·7 1·651 1·192 1·385 ·675 1·793
11 327·7 1·694 1·215 1·394 ·674 1·794
12 350·7 1·740 1·238 1·405 ·666 1·849
13 374·7 1·765 1·254 1·407 ·682 1·900
14 391·3 1·785 1·264 1·412 ·688 1·938
15 405·9 1·804 1·270 1·420 ·692 1·982
16 417·9 1·814 1·280 1·417 ·700 2·092
17 423·9 1·832 1·290 1·420 ·689 1·974
18 423·9 1·859 1·297 1·433 ·660 1·943
19 427·9 1·875 1·307 1·435 ·649 1·916
20 437·9 1·884 1·311 1·437 ·655 1·944
21 447·9 1·893 1·321 1·433 ·661 1·943
22 464·4 1·901 1·333 1·426 ·676 1·960
23 480·9 1·909 1·345 1·419 ·691 1·977
24 500·9 1·914 1·352 1·416 ·714 2·027
25 520·9 1·919 1·359 1·412 ·737 2·075
26 534·1 1·924 1·361 1·414 ·750 2·119
27 547·3 1·929 1·363 1·415 ·762 2·162
28 554·5 1·929 1·363 1·415 ·772 2·190
29 561·7 1·929 1·363 1·415 ·782 2·218
30 586·2 1·949 1·383 1·409 ·792 2·216
31 610·7 1·969 1·403 1·403 ·800 2·211
32 625·7 1·983 1·420 1·396 ·803 2·186
33 640·7 1·997 1·437 1·390 ·805 2·159
34 655·7 2·011 1·454 1·383 ·806 2·133
long at birth in comparison with other dimensions of the body. It is somewhat curious, however, that this ratio seems to fall off a little in the third year of growth, the animal continuing to grow in height to a marked degree after growth in length has become very slow. The ratio between height and weight is by much the most variable of our three ratios; the coefficient W/H3 steadily increases, and is more than twice as great at three years old as it was at birth. This illustrates the important, but obvious fact, that the coefficient k is most variable in the case of that dimension which grows most uniformly, that is to say most nearly in proportion to the general bulk of the animal. In short, the successive values of k, as determined (at successive epochs) for one dimension, are a measure of the variability of the others.
From the whole of the foregoing discussion we see that a certain definite rate of growth is a characteristic or specific phenomenon, deep-seated in the physiology of the organism; and that a very large part of the specific morphology of the organism depends upon the fact that there is not only an average, or aggregate, rate of growth common to the whole, but also a variation of rate in different parts of the organism, tending towards a specific rate characteristic of each different part or organ. The smallest change in the relative magnitudes of these partial or localised velocities of growth will be soon manifested in more and more striking differences of form. This is as much as to say that the time-element, which is implicit in the idea of growth, can never (or very seldom) be wholly neglected in our consideration of form [132]. It is scarcely necessary to enlarge here upon our statement, for not only is the truth of it self-evident, but it will find illustration again and again throughout this book. Nevertheless, let us go out of our way for a moment to consider it in reference to a particular case, and to enquire whether it helps to remove any of the difficulties which that case appears to present. [p. 104]
1917 Source Illustration
Fig. 23. Variability of length of tail-forceps in a sample of Earwigs. (After Bateson, P. Z. S. 1892, p. 588.)
In a very well-known paper, Bateson shewed that, among a large number of earwigs, collected in a particular locality, the males fell into two groups, characterised by large or by small tail-forceps, with very few instances of intermediate magnitude. This distribution into two groups, according to magnitude, is illustrated in the accompanying diagram (Fig. 23); and the phenomenon was described, and has been often quoted, as one of dimorphism, or discontinuous variation. In this diagram the time-element does not appear; but it is certain, and evident, that it lies close behind. Suppose we take some organism which is born not at all times of the year (as man is) but at some one particular season (for instance a fish), then any random sample will consist of individuals whose ages, and therefore whose magnitudes, will form a discontinuous series; and by plotting these magnitudes on a curve in relation to the number of individuals of each particular magnitude, we obtain a curve such as that shewn in Fig. 24, the first practical use of which is to enable us to analyse our sample into its constituent “age-groups,” or in other words to determine approximately the age, or ages of the fish. And if, instead of measuring the whole length of our fish, we had confined ourselves to particular parts, such as head, or [p. 105] tail or fin, we should have obtained discontinuous curves of distribution, precisely analogous to those for the entire animal. Now we know that the differences with which Bateson was dealing were entirely a question of magnitude, and we cannot help seeing that the discontinuous distributions of magnitude represented by his earwigs’ tails are just such as are illustrated by the magnitudes of the older and younger fish; we may indeed go so far as to say that the curves are precisely comparable, for in both cases we see a characteristic feature of detail, namely that the “spread” of the curve is greater in the second wave than in the first, that is to say (in the case of the fish) in the older as well as larger series. Over the reason for this phenomenon, which is simple and all but obvious, we need not pause.
1917 Source Illustration
Fig. 24. Variability of length of body in a sample of Plaice.
It is evident, then, that in this case of “dimorphism,” the tails of the one group of earwigs (which Bateson calls the “high males”) have either grown faster, or have been growing for a longer period of time, than those of the “low males.” If we could be certain that the whole random sample of earwigs were of one and the same age, then we should have to refer the phenomenon of dimorphism to a physiological phenomenon, simple in kind (however remarkable and unexpected); viz. that there were two alternative [p. 106] values, very different from one another, for the mean velocity of growth, and that the individual earwigs varied around one or other of these mean values, in each case according to the law of probabilities. But on the other hand, if we could believe that the two groups of earwigs were of different ages, then the phenomenon would be simplicity itself, and there would be no more to be said about it [133].
Before we pass from the subject of the relative rate of growth of different parts or organs, we may take brief note of the fact that various experiments have been made to determine whether the normal ratios are maintained under altered circumstances of nutrition, and especially in the case of partial starvation. For instance, it has been found possible to keep young rats alive for many weeks on a diet such as is just sufficient to maintain life without permitting any increase of weight. The rat of three weeks old weighs about 25 gms., and under a normal diet should weigh at ten weeks old about 150 gms., in the male, or 115 gms. in the female; but the underfed rat is still kept at ten weeks old to the weight of 25 gms. Under normal diet the proportions of the body change very considerably between the ages of three and ten weeks. For instance the tail gets relatively longer; and even when the total growth of the rat is prevented by underfeeding, the form continues to alter so that this increasing length of the tail is still manifest [134]. [p. 107]
Full-fed Rats.
Age in Length of Length of Total
weeks body (mm.) tail (mm.) length % of tail
0 48·7 16·9 65·6 25·8
1 64·5 29·4 93·9 31·3
3 90·4 59·1 149·5 39·5
6 128·0 110·0 238·0 46·2
10 173·0 150·0 323·0 46·4
6 98·0 72·3 170·3 42·5
10 99·6 83·9 183·5 45·7
Again as physiologists have long been aware, there is a marked difference in the variation of weight of the different organs, according to whether the animal’s total weight remain constant, or be caused to diminish by actual starvation; and further striking differences appear when the diet is not only scanty, but ill-balanced. But these phenomena of abnormal growth, however interesting from the physiological view, are of little practical importance to the morphologist.
The effect of temperature. [135]
The rates of growth which we have hitherto dealt with are based on special investigations, conducted under particular local conditions. For instance, Quetelet’s data, so far as we have used them to illustrate the rate of growth in man, are drawn from his study of the population of Belgium. But apart from that “fortuitous” individual variation which we have already considered, it is obvious that the normal rate of growth will be found to vary, in man and in other animals, just as the average stature varies, in different localities, and in different “races.” This phenomenon is a very complex one, and is doubtless a resultant of many undefined contributory causes; but we at least gain something in regard to it, when we discover that the rate of growth is directly affected by temperature, and probably by other physical [p. 108] conditions. Réaumur was the first to shew, and the observation was repeated by Bonnet [136], that the rate of growth or development of the chick was dependent on temperature, being retarded at temperatures below and somewhat accelerated at temperatures above the normal temperature of incubation, that is to say the temperature of the sitting hen. In the case of plants the fact that growth is greatly affected by temperature is a matter of familiar knowledge; the subject was first carefully studied by Alphonse De Candolle, and his results and those of his followers are discussed in the textbooks of Botany [137].
That variation of temperature constitutes only one factor in determining the rate of growth is admirably illustrated in the case of the Bamboo. It has been stated (by Lock) that in Ceylon the rate of growth of the Bamboo is directly proportional to the humidity of the atmosphere: and again (by Shibata) that in Japan it is directly proportional to the temperature. The two statements have been ingeniously and satisfactorily reconciled by Blackman [138], who suggests that in Ceylon the temperature-conditions are all that can be desired, but moisture is apt to be deficient: while in Japan there is rain in abundance but the average temperature is somewhat too low. So that in the one country it is the one factor, and in the other country it is the other, which is essentially variable.
The annexed diagram (Fig. 25), shewing the growth in length of the roots of some common plants during an identical period of forty-eight hours, at temperatures varying from about 14° to 37° C., is a sufficient illustration of the phenomenon. We see that in all cases there is a certain optimum temperature at which the rate of growth is a maximum, and we can also see that on either side of this optimum temperature the acceleration of growth, positive or negative, with increase of temperature is rapid, while at a distance from the optimum it is very slow. From the data given by Sachs and others, we see further that this optimum temperature is very much the same for all the common plants of our own climate which have as yet been studied; in them it is [p. 109] somewhere about 26° C. (or say 77° F.), or about the temperature of a warm summer’s day; while it is found, very naturally, to be considerably higher in the case of plants such as the melon or the maize, which are at home in warmer regions that our own.
1917 Source Illustration
Fig. 25. Relation of rate of growth to temperature in certain plants. (From Sachs’s data.)
In a large number of physical phenomena, and in a very marked degree in all chemical reactions, it is found that rate of action is affected, and for the most part accelerated, by rise of temperature; and this effect of temperature tends to follow a definite “exponential” law, which holds good within a considerable range of temperature, but is altered or departed from when we pass beyond certain normal limits. The law, as laid down by van’t Hoff for chemical reactions, is, that for an interval of n degrees the velocity varies as xn, x being called the “temperature coefficient” [139] for the reaction in question. [p. 110]
Van’t Hoff’s law, which has become a fundamental principle of chemical mechanics, is likewise applicable (with certain qualifications) to the phenomena of vital chemistry; and it follows that, on very much the same lines, we may speak of the “temperature coefficient” of growth. At the same time we must remember that there is a very important difference (though we can scarcely call it a fundamental one) between the purely physical and the physiological phenomenon, in that in the former we study (or seek and profess to study) one thing at a time, while in the latter we have always to do with various factors which intersect and interfere; increase in the one case (or change of any kind) tends to be continuous, in the other case it tends to be brought to arrest. This is the simple meaning of that Law of Optimum, laid down by Errera and by Sachs as a general principle of physiology: namely that every physiological process which varies (like growth itself) with the amount or intensity of some external influence, does so according to a law in which progressive increase is followed by progressive decrease; in other words the function has its optimum condition, and its curve shews a definite maximum. In the case of temperature, as Jost puts it, it has on the one hand its accelerating effect which tends to follow van’t Hoff’s law. But it has also another and a cumulative effect upon the organism: “Sie schädigt oder sie ermüdet ihn, und je höher sie steigt, desto rascher macht sie die Schädigung geltend und desto schneller schreitet sie voran.” It would seem to be this double effect of temperature in the case of the organism which gives us our “optimum” curves, which are the expression, accordingly, not of a primary phenomenon, but of a more or less complex resultant. Moreover, as Blackman and others have pointed out, our “optimum” temperature is very ill-defined until we take account also of the duration of our experiment; for obviously, a high temperature may lead to a short, but exhausting, spell of rapid growth, while the slower rate manifested at a lower temperature may be the best in the end. [p. 111] The mile and the hundred yards are won by different runners; and maximum rate of working, and maximum amount of work done, are two very different things [140].
In the case of maize, a certain series of experiments shewed that the growth in length of the roots varied with the temperature as follows [141]:
Temperature Growth in 48 hours
°C. mm.
18·0 1·1
23·5 10·8
26·6 29·6
28·5 26·5
30·2 64·6
33·5 69·5
36·5 20·7
Let us write our formula in the form
Then choosing two values out of the above experimental series (say the second and the second-last), we have t = 23·5, n = 10, and V, V′ = 10·8 and 69·5 respectively.
Accordingly 69·5/10·8 = 6·4 = x10.
Therefore (log 6·4)/10, or ·0806 = log x.
And, x = 1·204 (for an interval of 1° C.).
This first approximation might be considerably improved by taking account of all the experimental values, two only of which we have as yet made use of; but even as it is, we see by Fig. 26 that it is in very fair accordance with the actual results of observation, within those particular limits of temperature to which the experiment is confined. [p. 112]
For an experiment on Lupinus albus, quoted by Asa Gray [142], I have worked out the corresponding coefficient, but a little more carefully. Its value I find to be 1·16, or very nearly identical with that we have just found for the maize; and the correspondence between the calculated curve and the actual observations is now a close one.
1917 Source Illustration
Fig. 26. Relation of rate of growth to temperature in Maize. Observed values (after Köppen), and calculated curve.
Since the above paragraphs were written, new data have come to hand. Miss I. Leitch has made careful observations of the rate of growth of rootlets of the Pea; and I have attempted a further analysis of her principal results [143]. In Fig. 27 are shewn the mean rates of growth (based on about a hundred experiments) at some thirty-four different temperatures between 0·8° and 29·3°, each experiment lasting rather less than twenty-four hours. Working out the mean temperature coefficient for a great many combinations of these values, I obtain a value of 1·092 per C.°, or 2·41 for an interval of 10°, and a mean value for the whole series showing a rate of growth of just about 1 mm. per hour at a temperature of 20°. My curve in Fig. 27 is drawn from these determinations; and it will be seen that, while it is by no means exact at the lower temperatures, and will of course fail us altogether at very high [p. 113] temperatures, yet it serves as a very satisfactory guide to the relations between rate and temperature within the ordinary limits of healthy growth. Miss Leitch holds that the curve is not a van’t Hoff curve; and this, in strict accuracy, we need not dispute. But the phenomenon seems to me to be one into which the van’t Hoff ratio enters largely, though doubtless combined with other factors which we cannot at present determine or eliminate.
1917 Source Illustration
Fig. 27. Relation of rate of growth to temperature in rootlets of Pea. (From Miss I. Leitch’s data.)
While the above results conform fairly well to the law of the temperature coefficient, it is evident that the imbibition of water plays so large a part in the process of elongation of the root or stem that the phenomenon is rather a physical than a chemical one: and on this account, as Blackman has remarked, the data commonly given for the rate of growth in plants are apt to be [p. 114] irregular, and sometimes (we might even say) misleading [144]. The fact also, which we have already learned, that the elongation of a shoot tends to proceed by jerks, rather than smoothly, is another indication that the phenomenon is not purely and simply a chemical one. We have abundant illustrations, however, among animals, in which we may study the temperature coefficient under circumstances where, though the phenomenon is always complicated by osmotic factors, true metabolic growth or chemical combination plays a larger role. Thus Mlle. Maltaux and Professor Massart [145] have studied the rate of division in a certain flagellate, Chilomonas paramoecium, and found the process to take 29 minutes at 15° C., 12 at 25°, and only 5 minutes at 35° C. These velocities are in the ratio of 1 : 2·4 : 5·76, which ratio corresponds precisely to a temperature coefficient of 2·4 for each rise of 10°, or about 1·092 for each degree centigrade.
By means of this principle we may throw light on the apparently complicated results of many experiments. For instance, Fig. 28 is an illustration, which has been often copied, of O. Hertwig’s work on the effect of temperature on the rate of development of the tadpole [146].
From inspection of this diagram, we see that the time taken to attain certain stages of development (denoted by the numbers III–VII) was as follows, at 20° and at 10° C., respectively.
At 20° At 10°
Stage III 2·0 6·5 days
Stage IV 2·7 8·1 days
Stage V 3·0 10·7 days
Stage VI 4·0 13·5 days
Stage VII 5·0 16·8 days
Total 16·7 55·6 days
That is to say, the time taken to produce a given result at [p. 115] 10° was (on the average) somewhere about 55·6/16·7, or 3·33, times as long as was required at 20°.
1917 Source Illustration
Fig. 28. Diagram shewing time taken (in days), at various temperatures (°C.), to reach certain stages of development in the Frog: viz. I, gastrula; II, medullary plate; III, closure of medullary folds; IV, tail-bud; V, tail and gills; VI, tail-fin; VII, operculum beginning; VIII, do. closing; IX, first appearance of hind-legs. (From Jenkinson, after O. Hertwig, 1898.)
We may then put our equation again in the simple form, [p. 116]
That is to say, between the intervals of 10° and 20° C., if it take m days, at a certain given temperature, for a certain stage of development to be attained, it will take m × 1·128n days, when the temperature is n degrees less, for the same stage to be arrived at.
1917 Source Illustration
Fig. 29. Calculated values, corresponding to preceding figure.
Fig. 29 is calculated throughout from this value; and it will be seen that it is extremely concordant with the original diagram, as regards all the stages of development and the whole range of temperatures shewn: in spite of the fact that the coefficient on which it is based was derived by an easy method from a very few points in the original curves. [p. 117]
Karl Peter [147], experimenting chiefly on echinoderm eggs, and also making use of Hertwig’s experiments on young tadpoles, gives the normal temperature coefficients for intervals of 10° C. (commonly written Q10) as follows.
Sphaerechinus 2·15,
Echinus 2·13,
Rana 2·86.
These values are not only concordant, but are evidently of the same order of magnitude as the temperature-coefficient in ordinary chemical reactions. Peter has also discovered the very interesting fact that the temperature-coefficient alters with age, usually but not always becoming smaller as age increases.
Sphaerechinus; Segmentation Q^{10} = 2·29,
Later stages Q^{10} = 2·03.
Echinus; Segmentation Q^{10} = 2·30,
Later stages Q^{10} = 2·08.
Rana; Segmentation Q^{10} = 2·23,
Later stages Q^{10} = 3·34.
Furthermore, the temperature coefficient varies with the temperature, diminishing as the temperature rises,—a rule which van’t Hoff has shewn to hold in ordinary chemical operations. Thus, in Rana the temperature coefficient at low temperatures may be as high as 5·6: which is just another way of saying that at low temperatures development is exceptionally retarded.
In certain fish, such as plaice and haddock, I and others have found clear evidence that the ascending curve of growth is subject to seasonal interruptions, the rate during the winter months being always slower than in the months of summer: it is as though we superimposed a periodic, annual, sine-curve upon the continuous curve of growth. And further, as growth itself grows less and less from year to year, so will the difference between the winter and the summer rate also grow less and less. The fluctuation in rate [p. 118] will represent a vibration which is gradually dying out; the amplitude of the sine-curve will gradually diminish till it disappears; in short, our phenomenon is simply expressed by what is known as a “damped sine-curve.” Exactly the same thing occurs in man, though neither in his case nor in that of the fish have we sufficient data for its complete illustration.
We can demonstrate the fact, however, in the case of man by the help of certain very interesting measurements which have been recorded by Daffner [148], of the height of German cadets, measured at half-yearly intervals.
Height in cent. Increment in cm.
Number ─────────────────────── Winter Summer
observed Age October April October ½-year ½-year Year
12 11–12 139·4 141·0 143·3 1·6 2·3 3·9
80 12–13 143·0 144·5 147·4 1·5 2·9 4·4
146 13–14 147·5 149·5 152·5 2·0 3·0 5·0
162 14–15 152·2 155·0 158·5 2·5 3·5 6·0
162 15–16 158·5 160·8 163·8 2·3 3·0 5·3
150 16–17 163·5 165·4 167·7 1·9 2·3 4·2
82 17–18 167·7 168·9 170·4 1·2 1·5 2·7
22 18–19 169·8 170·6 171·5 0·8 0·9 1·7
6 19–20 170·7 171·1 171·5 0·4 0·4 0·8
In the accompanying diagram (Fig. 30) the half-yearly increments are set forth, from the above table, and it will be seen that they form two even and entirely separate series. The curve joining up each series of points is an acceleration-curve; and the comparison of the two curves gives a clear view of the relative rates of growth during winter and summer, and the fluctuation which these velocities undergo during the years in question. The dotted line represents, approximately, the acceleration-curve in its continuous fluctuation of alternate seasonal decrease and increase.
In the case of trees, the seasonal fluctuations of growth [149] admit [p. 119] of easy determination, and it is a point of considerable interest to compare the phenomenon in evergreen and in deciduous trees. I happen to have no measurements at hand with which to make this comparison in the case of our native trees, but from a paper by Mr Charles E. Hall [150] I have compiled certain mean values for growth in the climate of Uruguay.
1917 Source Illustration
Fig. 30. Half-yearly increments of growth, in cadets of various ages. (From Daffner’s data.)
Mean monthly increase in Girth of Evergreen and Deciduous Trees, at San Jorge, Uruguay. (After C. E. Hall.) Values expressed as percentages of total annual increase.
Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.
Evergreens 9·1 8·8 8·6 8·9 7·7 5·4 4·3 6·0 9·1 11·1 10·8 10·2
Deciduous
trees 20·3 14·6 9·0 2·3 0·8 0·3 0·7 1·3 3·5 9·9 16·7 21·0
The measurements taken were those of the girth of the tree, in mm., at three feet from the ground. The evergreens included species of Pinus, Eucalyptus and Acacia; the deciduous trees included Quercus, Populus, Robinia and Melia. I have merely taken mean values for these two groups, and expressed the monthly values as percentages of the mean annual increase. The result (as shewn by Fig. 31) is very much what we might have expected. The growth of the deciduous trees is completely arrested in winter-time, and the arrest is all but complete over [p. 120] a considerable period of time; moreover, during the warm season, the monthly values are regularly graded (approximately in a sine-curve) with a clear maximum (in the southern hemisphere) about the month of December. In the evergreen trees, on the other hand, the amplitude of the periodic wave is very much less; there is a notable amount of growth all the year round, and, while there is a marked diminution in rate during the coldest months, there is a tendency towards equality over a considerable part of the warmer season. It is probable that some of the species examined, and especially the pines, were definitely retarded in growth, either by a temperature above their optimum, or by deficiency of moisture, during the hottest period of the year; with the result that the seasonal curve in our diagram has (as it were) its region of maximum cut off.
1917 Source Illustration
Fig. 31. Periodic annual fluctuation in rate of growth of trees (in the southern hemisphere).
In the case of trees, the seasonal periodicity of growth is so well marked that we are entitled to make use of the phenomenon in a converse way, and to draw deductions as to variations in [p. 121] climate during past years from the record of varying rates of growth which the tree, by the thickness of its annual rings, has preserved for us. Mr. A. E. Douglass, of the University of Arizona, has made a careful study of this question [151], and I have received (through Professor H. H. Turner of Oxford) some measurements of the average width of the successive annual rings in “yellow pine,” 500 years old, from Arizona, in which trees the annual rings are very clearly distinguished. From the year 1391 to 1518, the mean of two trees was used; from 1519 to 1912, the mean of five; and the means of these, and sometimes of larger numbers, were found to be very concordant. A correction was applied by drawing a long, nearly straight line through the curve for the whole period, which line was assumed to represent the slowly diminishing mean width of ring accompanying the increase of size, or age, of the tree; and the actual growth as measured was equated with this diminishing mean. The figures used give, accordingly, the ratio of the actual growth in each year to the mean growth corresponding to the age or magnitude of the tree at that epoch.
It was at once manifest that the rate of growth so determined shewed a tendency to fluctuate in a long period of between 100 and 200 years. I then smoothed in groups of 100 (according to Gauss’s method) the yearly values, so that each number thus found represented the mean annual increase during a century: that is to say, the value ascribed to the year 1500 represented the average annual growth during the whole period between 1450 and 1550, and so on. These values give us a curve of beautiful and surprising smoothness, from which we seem compelled to draw the direct conclusion that the climate of Arizona, during the last 500 years, has fluctuated with a regular periodicity of almost precisely 150 years. Here again we should be left in doubt (so far as these [p. 123] observations go) whether the essential factor be a fluctuation of temperature or an alternation of moisture and aridity; but the character of the Arizona climate, and the known facts of recent years, encourage the belief that the latter is the more direct and more important factor.
1917 Source Illustration
Fig. 32. Long-period fluctuation in rate of growth of Arizona trees (smoothed in 100-year periods), from A.D. 1390–1490 to A.D. 1810–1910.
It has been often remarked that our common European trees, such for instance as the elm or the cherry, tend to have larger leaves the further north we go; but in this case the phenomenon is to be ascribed rather to the longer hours of daylight than to any difference of temperature [152]. The point is a physiological one, and consequently of little importance to us here [153]; the main point for the morphologist is the very simple one that physical or climatic conditions have greatly influenced the rate of growth. The case is analogous to the direct influence of temperature in modifying the colouration of organisms, such as certain butterflies. Now if temperature affects the rate of growth in strict uniformity, alike in all directions and in all parts or organs, its direct effect must be limited to the production of local races or varieties differing from one another in actual magnitude, as the Siberian goldfinch or bullfinch, for instance, differ from our own. But if there be even ever so little of a discriminating action in the enhancement of growth by temperature, such that it accelerates the growth of one tissue or one organ more than another, then it is evident that it must at once lead to an actual difference of racial, or even “specific” form.
It is not to be doubted that the various factors of climate have some such discriminating influence. The leaves of our northern trees may themselves be an instance of it; and we have, [p. 124] probably, a still better instance of it in the case of Alpine plants [154], whose general habit is dwarfed, though their floral organs suffer little or no reduction. The subject, however, has been little investigated, and great as its theoretic importance would be to us, we must meanwhile leave it alone.
Osmotic factors in growth.
The curves of growth which we have now been studying represent phenomena which have at least a two-fold interest, morphological and physiological. To the morphologist, who recognises that form is a “function” of growth, the important facts are mainly these: (1) that the rate of growth is an orderly phenomenon, with general features common to very various organisms, while each particular organism has its own characteristic phenomena, or “specific constants”; (2) that rate of growth varies with temperature, that is to say with season and with climate, and with various other physical factors, external and internal; (3) that it varies in different parts of the body, and according to various directions or axes; such variations being definitely correlated with one another, and thus giving rise to the characteristic proportions, or form, of the organism, and to the changes in form which it undergoes in the course of its development. But to the physiologist, the phenomenon suggests many other important considerations, and throws much light on the very nature of growth itself, as a manifestation of chemical and physical energies.
To be content to shew that a certain rate of growth occurs in a certain organism under certain conditions, or to speak of the phenomenon as a “reaction” of the living organism to its environment or to certain stimuli, would be but an example of that “lack of particularity [155]” in regard to the actual mechanism of physical cause and effect with which we are apt in biology to be too easily satisfied. But in the case of rate of growth we pass somewhat [p. 125] beyond these limitations; for the affinity with certain types of chemical reaction is plain, and has been recognised by a great number of physiologists.
A large part of the phenomenon of growth, both in animals and still more conspicuously in plants, is associated with “turgor,” that is to say, is dependent on osmotic conditions; in other words, the velocity of growth depends in great measure (as we have already seen, p. 113) on the amount of water taken up into the living cells, as well as on the actual amount of chemical metabolism performed by them [156]. Of the chemical phenomena which result in the actual increase of protoplasm we shall speak presently, but the rôle of water in growth deserves also a passing word, even in our morphological enquiry.
It has been shewn by Loeb that in Cerianthus or Tubularia, for instance, the cells in order to grow must be turgescent; and this turgescence is only possible so long as the salt water in which the cells lie does not overstep a certain limit of concentration. The limit, in the case of Tubularia, is passed when the salt amounts to about 5·4 per cent. Sea-water contains some 3·0 to 3·5 p.c. of salts; but it is when the salinity falls much below this normal, to about 2·2 p.c., that Tubularia exhibits its maximal turgescence, and maximal growth. A further dilution is said to act as a poison to the animal. Loeb has also shewn [157] that in certain eggs (e.g. those of the little fish Fundulus) an increasing concentration of the sea-water (leading to a diminishing “water-content” of the egg) retards the rate of segmentation and at length renders segmentation impossible; though nuclear division, by the way, goes on for some time longer.
Among many other observations of the same kind, those of Bialaszewicz [158], on the early growth of the frog, are notable. He shews that the growth of the embryo while still within the [p. 126] vitelline membrane depends wholly on the absorption of water; that whether rate of growth be fast or slow (in accordance with temperature) the quantity of water absorbed is constant; and that successive changes of form correspond to definite quantities of water absorbed. The solid residue, as Davenport has also shewn, may actually and notably diminish, while the embryo organism is increasing rapidly in bulk and weight.
On the other hand, in later stages and especially in the higher animals, the percentage of water tends to diminish. This has been shewn by Davenport in the frog, by Potts in the chick, and particularly by Fehling in the case of man [159]. Fehling’s results are epitomised as follows:
Age in weeks 6 17 22 24 26 30 35 39
Percentage of water 97·5 91·8 92·0 89·9 86·4 83·7 82·9 74·2
Age in weeks 1 2 5 7 9 14 41 84
Percentage of water 56·3 58·5 76·7 89·3 93·1 95·0 90·2 87·5
To such phenomena of osmotic balance as the above, or in other words to the dependence of growth on the uptake of water, Höber [160] and also Loeb are inclined to refer the modifications of form which certain phyllopod crustacea undergo, when the highly saline waters which they inhabit are further concentrated, or are abnormally diluted. Their growth, according to Schmankewitsch, is retarded by increase of concentration, so that the individuals from the more saline waters appear stunted and dwarfish; and they become altered or transformed in other ways, which for the most part suggest “degeneration,” or a failure to attain full and perfect development [161]. Important physiological changes also ensue. The rate of multiplication is increased, and parthenogenetic reproduction is encouraged. Male individuals become plentiful in the less saline waters, and here the females bring forth [p. 127] their young alive; males disappear altogether in the more concentrated brines, and then the females lay eggs, which, however, only begin to develop when the salinity is somewhat reduced.
The best-known case is the little “brine-shrimp,” Artemia salina, found, in one form or another, all the world over, and first discovered more than a century and a half ago in the salt-pans at Lymington. Among many allied forms, one, A. milhausenii, inhabits the natron-lakes of Egypt and Arabia, where, under the name of “loul,” or “Fezzan-worm,” it is eaten by the Arabs [162]. This fact is interesting, because it indicates (and investigation has apparently confirmed) that the tissues of the creature are not impregnated with salt, as is the medium in which it lives. The fluids of the body, the milieu interne (as Claude Bernard called them [163]), are no more salt than are those of any ordinary crustacean or other animal, but contain only some 0·8 per cent. of NaCl [164], while the milieu externe may contain 10, 20, or more per cent. of this and other salts; which is as much as to say that the skin, or body-wall, of the creature acts as a “semi-permeable membrane,” through which the dissolved salts are not permitted to diffuse, though water passes through freely: until a statical equilibrium (doubtless of a complex kind) is at length attained.
Among the structural changes which result from increased concentration of the brine (partly during the life-time of the individual, but more markedly during the short season which suffices for the development of three or four, or perhaps more, successive generations), it is found that the tail comes to bear fewer and fewer bristles, and the tail-fins themselves tend at last to disappear; these changes corresponding to what have been [p. 128] described as the specific characters of A. milhausenii, and of a still more extreme form, A. köppeniana; while on the other hand, progressive dilution of the water tends to precisely opposite conditions, resulting in forms which have also been described as separate species, and even referred to a separate genus, Callaonella, closely akin to Branchipus (Fig. 33). Pari passu with these changes, there is a marked change in the relative lengths of the fore and hind portions of the body, that is to say, of the “cephalothorax” and abdomen: the latter growing relatively longer, the salter the water. In other words, not only is the rate of growth of the whole
1917 Source Illustration
Fig. 33. Brine-shrimps (Artemia), from more or less saline water. Upper figures shew tail-segment and tail-fins; lower figures, relative length of cephalothorax and abdomen. (After Abonyi.)
animal lessened by the saline concentration, but the specific rates of growth in the parts of its body are relatively changed. This latter phenomenon lends itself to numerical statement, and Abonyi has lately shewn that we may construct a very regular curve, by plotting the proportionate length of the creature’s abdomen against the salinity, or density, of the water; and the several species of Artemia, with all their other correlated specific characters, are then found to occupy successive, more or less well-defined, and more or less extended, regions of the curve (Fig. 33). In short, the density of the water is so clearly a “function” of the specific [p. 129] character, that we may briefly define the species Artemia (Callaonella) Jelskii, for instance, as the Artemia of density 1000–1010 (NaCl), or the typical A. salina, or principalis, as the Artemia of density 1018–1025, and so forth. It is a most interesting fact that these Artemiae, under the protection of their semi-permeable skins, are capable of living in waters not only of great density, but of very varied chemical composition. The natron-lakes, for instance, contain large quantities of magnesium
1917 Source Illustration
Fig. 34. Percentage ratio of length of abdomen to cephalothorax in brine-shrimps, at various salinities. (After Abonyi.)
sulphate; and the Artemiae continue to live equally well in artificial solutions where this salt, or where calcium chloride, has largely taken the place of sodium chloride in the more common habitat. Furthermore, such waters as those of the natron-lakes are subject to very great changes of chemical composition as concentration proceeds, owing to the different solubilities of the constituent salts. It appears that the forms which the Artemiae assume, and the changes which they undergo, are identical or [p. 130] indistinguishable, whichever of the above salts happen to exist, or to predominate, in their saline habitat. At the same time we still lack (so far as I know) the simple, but crucial experiments which shall tell us whether, in solutions of different chemical composition, it is at equal densities, or at “isotonic” concentrations (that is to say, under conditions where the osmotic pressure, and consequently the rate of diffusion, is identical), that the same structural changes are produced, or corresponding phases of equilibrium attained.
While Höber and others [165] have referred all these phenomena to osmosis, Abonyi is inclined to believe that the viscosity, or mechanical resistance, of the fluid also reacts upon the organism; and other possible modes of operation have been suggested. But we may take it for certain that the phenomenon as a whole is not a simple one; and that it includes besides the passive phenomena of intermolecular diffusion, some other form of activity which plays the part of a regulatory mechanism [166].
Growth and catalytic action.
In ordinary chemical reactions we have to deal (1) with a specific velocity proper to the particular reaction, (2) with variations due to temperature and other physical conditions, (3) according to van’t Hoff’s “Law of Mass,” with variations due to the actual quantities present of the reacting substances, and (4) in certain cases, with variations due to the presence of “catalysing agents.” In the simpler reactions, the law of mass involves a steady, gradual slowing-down of the process, according to a logarithmic ratio, as the reaction proceeds and as the initial amount of substance diminishes; a phenomenon, however, which need not necessarily [p. 131] occur in the organism, part of whose energies are devoted to the continual bringing-up of fresh supplies.
Catalytic action occurs when some substance, often in very minute quantity, is present, and by its presence produces or accelerates an action, by opening “a way round,” without the catalytic agent itself being diminished or used up [167]. Here the velocity curve, though quickened, is not necessarily altered in form, for gradually the law of mass exerts its effect and the rate of the reaction gradually diminishes. But in certain cases we have the very remarkable phenomenon that a body acting as a catalyser is necessarily formed as a product, or bye-product, of the main reaction, and in such a case as this the reaction-velocity will tend to be steadily accelerated. Instead of dwindling away, the reaction will continue with an ever-increasing velocity: always subject to the reservation that limiting conditions will in time make themselves felt, such as a failure of some necessary ingredient, or a development of some substance which shall antagonise or finally destroy the original reaction. Such an action as this we have learned, from Ostwald, to describe as “autocatalysis.” Now we know that certain products of protoplasmic metabolism, such as the enzymes, are very powerful catalysers, and we are entitled to speak of an autocatalytic action on the part of protoplasm itself. This catalytic activity of protoplasm is a very important phenomenon. As Blackman says, in the address already quoted, the botanists (or the zoologists) “call it growth, attribute it to a specific power of protoplasm for assimilation, and leave it alone as a fundamental phenomenon; but they are much concerned as to the distribution of new growth in innumerable specifically distinct forms.” While the chemist, on the other hand, recognises it as a familiar phenomenon, and refers it to the same category as his other known examples of autocatalysis. [p. 132]
This very important, and perhaps even fundamental phenomenon of growth would seem to have been first recognised by Professor Chodat of Geneva, as we are told by his pupil Monnier [168]. “On peut bien, ainsi que M. Chodat l’a proposé, considérer l’accroissement comme une réaction chimique complexe, dans laquelle le catalysateur est la cellule vivante, et les corps en présence sont l’eau, les sels, et l’acide carbonique.”
Very soon afterwards a similar suggestion was made by Loeb [169], in connection with the synthesis of nuclein or nuclear protoplasm; for he remarked that, as in an autocatalysed chemical reaction, the velocity of the synthesis increases during the initial stage of cell-division in proportion to the amount of nuclear matter already synthesised. In other words, one of the products of the reaction, i.e. one of the constituents of the nucleus, accelerates the production of nuclear from cytoplasmic material.
The phenomenon of autocatalysis is by no means confined to living or protoplasmic chemistry, but at the same time it is characteristically, and apparently constantly, associated therewith. And it would seem that to it we may ascribe a considerable part of the difference between the growth of the organism and the simpler growth of the crystal [170]: the fact, for instance, that the cell can grow in a very low concentration of its nutritive solution, while the crystal grows only in a supersaturated one; and the fundamental fact that the nutritive solution need only contain the more or less raw materials of the complex constituents of the cell, while the crystal grows only in a solution of its own actual substance [171].
As F. F. Blackman has pointed out, the multiplication of an organism, for instance the prodigiously rapid increase of a bacterium, [p. 133] which tends to double its numbers every few minutes, till (were it not for limiting factors) its numbers would be all but incalculable in a day [172], is a simple but most striking illustration of the potentialities of protoplasmic catalysis; and (apart from the large share taken by mere “turgescence” or imbibition of water) the same is true of the growth, or cell-multiplication, of a multicellular organism in its first stage of rapid acceleration.
It is not necessary for us to pursue this subject much further, for it is sufficiently clear that the normal “curve of growth” of an organism, in all its general features, very closely resembles the velocity-curve of chemical autocatalysis. We see in it the first and most typical phase of greater and greater acceleration; this is followed by a phase in which limiting conditions (whose details are practically unknown) lead to a falling off of the former acceleration; and in most cases we come at length to a third phase, in which retardation of growth is succeeded by actual diminution of mass. Here we may recognise the influence of processes, or of products, which have become actually deleterious; their deleterious influence is staved off for a while, as the organism draws on its accumulated reserves, but they lead ere long to the stoppage of all activity, and to the physical phenomenon of death. But when we have once admitted that the limiting conditions of growth, which cause a phase of retardation to follow a phase of acceleration, are very imperfectly known, it is plain that, ipso facto, we must admit that a resemblance rather than an identity between this phenomenon and that of chemical autocatalysis is all that we can safely assert meanwhile. Indeed, as Enriques has shewn, points of contrast between the two phenomena are not lacking; for instance, as the chemical reaction draws to a close, it is by the gradual attainment of chemical equilibrium: but when organic growth draws to a close, it is by reason of a very different kind of equilibrium, due in the main to the gradual differentiation of the organism into parts, among whose peculiar [p. 134] and specialised functions that of cell-multiplication tends to fall into abeyance [173].
It would seem to follow, as a natural consequence, from what has been said, that we could without much difficulty reduce our curves of growth to logarithmic formulae [174] akin to those which the physical chemist finds applicable to his autocatalytic reactions. This has been diligently attempted by various writers [175]; but the results, while not destructive of the hypothesis itself, are only partially successful. The difficulty arises mainly from the fact that, in the life-history of an organism, we have usually to deal (as indeed we have seen) with several recurrent periods of relative acceleration and retardation. It is easy to find a formula which shall satisfy the conditions during any one of these periodic phases, but it is very difficult to frame a comprehensive formula which shall apply to the entire period of growth, or to the whole duration of life.
But if it be meanwhile impossible to formulate or to solve in precise mathematical terms the equation to the growth of an organism, we have yet gone a very long way towards the solution of such problems when we have found a “qualitative expression,” as Poincaré puts it; that is to say, when we have gained a fair approximate knowledge of the general curve which represents the unknown function.
As soon as we have touched on such matters as the chemical phenomenon of catalysis, we are on the threshold of a subject which, if we were able to pursue it, would soon lead us far into the special domain of physiology; and there it would be necessary to follow it if we were dealing with growth as a phenomenon in itself, instead of merely as a help to our study and comprehension of form. For instance the whole question of diet, of overfeeding and underfeeding, would present itself for discussion [176]. But without attempting to open up this large subject, we may say a [p. 135] further passing word upon the essential fact that certain chemical substances have the power of accelerating or of retarding, or in some way regulating, growth, and of so influencing directly the morphological features of the organism.
Thus lecithin has been shewn by Hatai [177], Danilewsky [178] and others to have a remarkable power of stimulating growth in various animals; and the so-called “auximones,” which Professor Bottomley prepares by the action of bacteria upon peat appear to be, after a somewhat similar fashion, potent accelerators of the growth of plants. But by much the most interesting cases, from our point of view, are those where a particular substance appears to exert a differential effect, stimulating the growth of one part or organ of the body more than another.
It has been known for a number of years that a diseased condition of the pituitary body accompanies the phenomenon known as “acromegaly,” in which the bones are variously enlarged or elongated, and which is more or less exemplified in every skeleton of a “giant”; while on the other hand, disease or extirpation of the thyroid causes an arrest of skeletal development, and, if it take place early, the subject remains a dwarf. These, then, are well-known illustrations of the regulation of function by some internal glandular secretion, some enzyme or “hormone” (as Bayliss and Starling call it), or “harmozone,” as Gley calls it in the particular case where the function regulated is that of growth, with its consequent influence on form.
Among other illustrations (which are plentiful) we have, for instance the growth of the placental decidua, which Loeb has shewn to be due to a substance given off by the corpus luteum of the ovary, giving to the uterine tissues an abnormal capacity for growth, which in turn is called into action by the contact of the ovum, or even of any foreign body. And various sexual characters, such as the plumage, comb and spurs of the cock, are believed in like manner to arise in response to some particular internal secretion. When the source of such a secretion is removed by castration, well-known morphological changes take place in various animals; and when a converse change takes place, the female acquires, in greater or less degree, characters which are [p. 136] proper to the male, as in certain extreme cases, known from time immemorial, when late in life a hen assumes the plumage of the cock.
There are some very remarkable experiments by Gudernatsch, in which he has shewn that by feeding tadpoles (whether of frogs or toads) on thyroid gland substance, their legs may be made to grow out at any time, days or weeks before the normal date of their appearance [179]. No other organic food was found to produce the same effect; but since the thyroid gland is known to contain iodine [180], Morse experimented with this latter substance, and found that if the tadpoles were fed with iodised amino-acids the legs developed precociously, just as when the thyroid gland itself was used. We may take it, then, as an established fact, whose full extent and bearings are still awaiting investigation, that there exist substances both within and without the organism which have a marvellous power of accelerating growth, and of doing so in such a way as to affect not only the size but the form or proportions of the organism.
If we once admit, as we are now bound to do, the existence of such factors as these, which, by their physiological activity and apart from any direct action of the nervous system, tend towards the acceleration of growth and consequent modification of form, we are led into wide fields of speculation by an easy and a legitimate pathway. Professor Gley carries such speculations a long, long way: for he says [181] that by these chemical influences “Toute une partie de la construction des êtres parait s’expliquer d’une façon toute mécanique. La forteresse, si longtemps inaccessible, du vitalisme est entamée. Car la notion morphogénique était, suivant le mot de Dastre [182], comme ‘le dernier réduit de la force vitale.’ ”
The physiological speculations we need not discuss: but, to take a single example from morphology, we begin to understand the possibility, and to comprehend the probable meaning, of the [p. 137] all but sudden appearance on the earth of such exaggerated and almost monstrous forms as those of the great secondary reptiles and the great tertiary mammals [183]. We begin to see that it is in order to account, not for the appearance, but for the disappearance of such forms as these that natural selection must be invoked. And we then, I think, draw near to the conclusion that what is true of these is universally true, and that the great function of natural selection is not to originate, but to remove: donec ad interitum genus id natura redegit [184].
The world of things living, like the world of things inanimate, grows of itself, and pursues its ceaseless course of creative evolution. It has room, wide but not unbounded, for variety of living form and structure, as these tend towards their seemingly endless, but yet strictly limited, possibilities of permutation and degree: it has room for the great and for the small, room for the weak and for the strong. Environment and circumstance do not always make a prison, wherein perforce the organism must either live or die; for the ways of life may be changed, and many a refuge found, before the sentence of unfitness is pronounced and the penalty of extermination paid. But there comes a time when “variation,” in form, dimensions, or other qualities of the organism, goes farther than is compatible with all the means at hand of health and welfare for the individual and the stock; when, under the active and creative stimulus of forces from within and from without, the active and creative energies of growth pass the bounds of physical and physiological equilibrium: and so reach the limits which, as again Lucretius tells us, natural law has set between what may and what may not be,
Then, at last, we are entitled to use the customary metaphor, and to see in natural selection an inexorable force, whose function [p. 138] is not to create but to destroy,—to weed, to prune, to cut down and to cast into the fire [185].
Regeneration, or growth and repair.
The phenomenon of regeneration, or the restoration of lost or amputated parts, is a particular case of growth which deserves separate consideration. As we are all aware, this property is manifested in a high degree among invertebrates and many cold-blooded vertebrates, diminishing as we ascend the scale, until at length, in the warm-blooded animals, it lessens down to no more than that vis medicatrix which heals a wound. Ever since the days of Aristotle, and especially since the experiments of Trembley, Réaumur and Spallanzani in the middle of the eighteenth century, the physiologist and the psychologist have alike recognised that the phenomenon is both perplexing and important. The general phenomenon is amply discussed elsewhere, and we need only deal with it in its immediate relation to growth [186].
Regeneration, like growth in other cases, proceeds with a velocity which varies according to a definite law; the rate varies with the time, and we may study it as velocity and as acceleration.
Let us take, as an instance, Miss M. L. Durbin’s measurements of the rate of regeneration of tadpoles’ tails: the rate being here measured in terms, not of mass, but of length, or longitudinal increment [187].
From a number of tadpoles, whose average length was 34·2 mm., their tails being on an average 21·2 mm. long, about half the tail [p. 139] (11·5 mm.) was cut off, and the amounts regenerated in successive periods are shewn as follows:
Days after operation 3 7 10 14 18 24 30
(1) Amount regenerated in mm. 1·4 3·4 4·3 5·2 5·5 6·2 6·5
(2) Increment during each period 1·4 2·0 0·9 0·9 0·3 0·7 0·3
(3) (?) Rate per day during
each period 0·46 0·50 0·30 0·25 0·07 0·12 0·05
The first line of numbers in this table, if plotted as a curve against the number of days, will give us a very satisfactory view of the “curve of growth” within the period of the observations: that is to say, of the successive relations of length to time, or the velocity of the process. But the third line is not so satisfactory, and must not be plotted directly as an acceleration curve. For it is evident that the “rates” here determined do not correspond to velocities at the dates to which they are referred, but are the mean velocities over a preceding period; and moreover the periods over which these means are taken are here of very unequal length. But we may draw a good deal more information from this experiment, if we begin by drawing a smooth curve, as nearly as possible through the points corresponding to the amounts regenerated (according to the first line of the table); and if we then interpolate from this smooth curve the actual lengths attained, day by day, and derive from these, by subtraction, the successive daily increments, which are the measure of the daily mean velocities (Table, p. 141). (The more accurate and strictly correct method would be to draw successive tangents to the curve.)
In our curve of growth (Fig. 35) we cannot safely interpolate values for the first three days, that is to say for the dates between amputation and the first actual measurement of the regenerated part. What goes on in these three days is very important; but we know nothing about it, save that our curve descended to zero somewhere or other within that period. As we have already learned, we can more or less safely interpolate between known points, or actual observations; but here we have no known starting-point. In short, for all that the observations tell us, and for all that the appearance of the curve can suggest, the curve of growth may have descended evenly to the base-line, which it would then have reached about the end of the second [p. 140] day; or it may have had within the first three days a change of direction, or “point of inflection,” and may then have sprung at once from the base-line at zero. That is to say, there may
1917 Source Illustration
Fig. 35. Curve of regenerative growth in tadpoles’ tails. (From M. L. Durbin’s data.)
have been an intervening “latent period,” during which no growth
1917 Source Illustration
Fig. 36. Mean daily increments, corresponding to Fig. 35.
occurred, between the time of injury and the first measurement of regenerative growth; or, for all we yet know, regeneration may have begun at once, but with a velocity much less than that which it afterwards attained. This apparently trifling difference would correspond to a very great difference in the nature of the phenomenon, and would lead to a very striking difference in the curve which we have next to draw.
The curve already drawn (Fig. 35) illustrates, as we have seen, the relation of length to time, i.e. L/T = V. The second (Fig. 36) represents the rate of change of velocity; it sets V against T;
Total Daily
Days increment increment Logs of do.
1 —
— —
2 —
— —
3 1·40
·60 1·78
4 2·00
·52 1·72
5 2·52
·45 1·65
6 2·97
·43 1·63
7 3·40
·32 1·51
8 3·72
·30 1·48
9 4·02
·28 1·45
10 4·30
·22 1·34
11 4·52
·21 1·32
12 4·73
·19 1·28
13 4·92
·18 1·26
14 5·10
·17 1·23
15 5·27
·13 1·11
16 5·40
·14 1·15
17 5·54
·13 1·11
18 5·67
·11 1·04
19 5·78
·10 1·00
20 5·88
·10 1·00
21 5·98
·09 ·95
22 6·07
·07 ·85
23 6·14
·07 ·84
24 6·21
·08 ·90
25 6·29
·06 ·78
26 6·35
·06 ·78
27 6·41
·05 ·70
28 6·46
·04 ·60
29 6·50
·03 ·48
30 6·53
and V/T or L/T2, represents (as we have learned) the acceleration of growth, this being simply the “differential coefficient,” the first derivative of the former curve.
1917 Source Illustration
Fig. 37. Logarithms of values shewn in Fig. 36.
Now, plotting this acceleration curve from the date of the first measurement made three days after the amputation of the tail (Fig. 36), we see that it has no point of inflection, but falls steadily, only more and more slowly, till at last it comes down nearly to the base-line. The velocities of growth are continually diminishing. As regards the missing portion at the beginning of the curve, we cannot be sure whether it bent round and came down to zero, or whether, as in our ordinary acceleration curves of growth from birth onwards, it started from a maximum. The former is, in this case, obviously the more probable, but we cannot be sure.
As regards that large portion of the curve which we are acquainted with, we see that it resembles the curve known as a rectangular hyperbola, which is the form assumed when two variables (in this case V and T) vary inversely as one another. If we take the logarithms of the velocities (as given in the table) and plot them against time (Fig. 37), we see that they fall, approximately, into a straight line; and if this curve be plotted on the [p. 143] proper scale we shall find that the angle which it makes with the base is about 25°, of which the tangent is ·46, or in round numbers ½.
Had the angle been 45° (tan 45° = 1), the curve would have been actually a rectangular hyperbola, with V T = constant. As it is, we may assume, provisionally, that it belongs to the same family of curves, so that Vm Tn, or Vm/n T, or V Tn/m, are all severally constant. In other words, the velocity varies inversely as some power of the time, or vice versa. And in this particular case, the equation V T2 = constant, holds very nearly true; that is to say the velocity varies, or tends to vary, inversely as the square of the time. If some general law akin to this could be established as a general law, or even as a common rule, it would be of great importance.
1917 Source Illustration
Fig. 38. Rate of regenerative growth in larger tadpoles.
But though neither in this case nor in any other can the minute increments of growth during the first few hours, or the first couple of days, after injury, be directly measured, yet the most important point is quite capable of solution. What the foregoing curve leaves us in ignorance of, is simply whether growth starts at zero, with zero velocity, and works up quickly to a maximum velocity from which it afterwards gradually falls away; or whether after a latent period, it begins, so to speak, in full force. The answer [p. 144] to this question-depends on whether, in the days following the first actual measurement, we can or cannot detect a daily increment in velocity, before that velocity begins its normal course of diminution. Now this preliminary ascent to a maximum, or point of inflection of the curve, though not shewn in the above-quoted experiment, has been often observed: as for instance, in another similar experiment by the author of the former, the tadpoles being in this case of larger size (average 49·1 mm.) [188].
Days 3 5 7 10 12 14 17 24 28 31
Increment 0·86 2·15 3·66 5·20 5·95 6·38 7·10 7·60 8·20 8·40
Or, by graphic interpolation,
Total Daily
Days increment do.
1 ·23 ·23
2 ·53 ·30
3 ·86 ·33
4 1·30 ·44
5 2·00 ·70
6 2·78 ·78
7 3·58 ·80
8 4·30 ·72
9 4·90 ·60
10 5·29 ·39
11 5·62 ·33
12 5·90 ·28
13 6·13 ·23
14 6·38 ·25
15 6·61 ·23
16 6·81 ·20
17 7·00 ·19 etc.
The acceleration curve is drawn in Fig. 39.
Here we have just what we lacked in the former case, namely a visible point of inflection in the curve about the seventh day (Figs. 38, 39), whose existence is confirmed by successive observations on the 3rd, 5th, 7th and 10th days, and which justifies to some extent our extrapolation for the otherwise unknown period up to and ending with the third day; but even here there is a short space near the very beginning during which we are not quite sure of the precise slope of the curve.
We have now learned that, according to these experiments, with which many others are in substantial agreement, the rate of growth in the regenerative process is as follows. After a very short latent period, not yet actually proved but whose existence is highly probable, growth commences with a velocity which very [p. 145] rapidly increases to a maximum. The curve quickly,—almost suddenly,—changes its direction, as the velocity begins to fall; and the rate of fall, that is, the negative acceleration, proceeds at a slower and slower rate, which rate varies inversely as some power of the time, and is found in both of the above-quoted experiments to be very approximately as 1/T2. But it is obvious that the value which we have found for the latter portion of the curve (however closely it be conformed to) is only an empirical value; it has only a temporary usefulness, and must in time give place to a formula which shall represent the entire phenomenon, from start to finish.
1917 Source Illustration
Fig. 39. Daily increment, or amount regenerated, corresponding to Fig. 38.
While the curve of regenerative growth is apparently different from the curve of ordinary growth as usually drawn (and while this apparent difference has been commented on and treated as valid by certain writers) we are now in a position to see that it only looks different because we are able to study it, if not from the beginning, at least very nearly so: while an ordinary curve of growth, as it is usually presented to us, is one which dates, not [p. 146] from the beginning of growth, but from the comparatively late, and unimportant, and even fallacious epoch of birth. A complete curve of growth, starting from zero, has the same essential characteristics as the regeneration curve.
Indeed the more we consider the phenomenon of regeneration, the more plainly does it shew itself to us as but a particular case of the general phenomenon of growth [189], following the same lines, obeying the same laws, and merely started into activity by the special stimulus, direct or indirect, caused by the infliction of a wound. Neither more nor less than in other problems of physiology are we called upon, in the case of regeneration, to indulge in metaphysical speculation, or to dwell upon the beneficent purpose which seemingly underlies this process of healing and restoration.
It is a very general rule, though apparently not a universal one, that regeneration tends to fall somewhat short of a complete restoration of the lost part; a certain percentage only of the lost tissues is restored. This fact was well known to some of those old investigators, who, like the Abbé Trembley and like Voltaire, found a fascination in the study of artificial injury and the regeneration which followed it. Sir John Graham Dalyell, for instance, says, in the course of an admirable paragraph on regeneration [190]: “The reproductive faculty … is not confined to one portion, but may extend over many; and it may ensue even in relation to the regenerated portion more than once. Nevertheless, the faculty gradually weakens, so that in general every successive regeneration is smaller and more imperfect than the organisation preceding it; and at length it is exhausted.”
In certain minute animals, such as the Infusoria, in which the capacity for “regeneration” is so great that the entire animal may be restored from the merest fragment, it becomes of great interest to discover whether there be some definite size at which the fragment ceases to display this power. This question has [p. 147] been studied by Lillie [191], who found that in Stentor, while still smaller fragments were capable of surviving for days, the smallest portions capable of regeneration were of a size equal to a sphere of about 80 µ in diameter, that is to say of a volume equal to about one twenty-seventh of the average entire animal. He arrives at the remarkable conclusion that for this, and for all other species of animals, there is a “minimal organisation mass,” that is to say a “minimal mass of definite size consisting of nucleus and cytoplasm within which the organisation of the species can just find its latent expression.” And in like manner, Boveri [192] has shewn that the fragment of a sea-urchin’s egg capable of growing up into a new embryo, and so discharging the complete functions of an entire and uninjured ovum, reaches its limit at about one-twentieth of the original egg,—other writers having found a limit at about one-fourth. These magnitudes, small as they are, represent objects easily visible under a low power of the microscope, and so stand in a very different category to the minimal magnitudes in which life itself can be manifested, and which we have discussed in chapter II.
A number of phenomena connected with the linear rate of regeneration are illustrated and epitomised in the accompanying diagram (Fig. 40), which I have constructed from certain data given by Ellis in a paper on the relation of the amount of tail regenerated to the amount removed, in Tadpoles. These data are summarised in the next Table. The tadpoles were all very much of a size, about 40 mm.; the average length of tail was very near to 26 mm., or 65 per cent. of the whole body-length; and in four series of experiments about 10, 20, 40 and 60 per cent. of the tail were severally removed. The amount regenerated in successive intervals of three days is shewn in our table. By plotting the actual amounts regenerated against these three-day intervals of time, we may interpolate values for the time taken to regenerate definite percentage amounts, 5 per cent., 10 per cent., etc. of the [p. 148]
1917 Source Illustration
Fig. 40. Relation between the percentage amount of tail removed, the percentage restored, and the time required for its restoration. (From M. M. Ellis’s data.)
Body Tail Amount Per cent. % amount regenerated in days
length length removed of tail ────────────────────────────
Series† mm. mm. mm. removed 3 6 9 12 15 18 32
O 39·575 25·895 3·2 12·36 13 31 44 44 44 44 44
P 40·21 26·13 5·28 20·20 10 29 40 44 44 44 44
R 39·86 25·70 10·4 40·50 6 20 31 40 48 48 48
S 40·34 26·11 14·8 56·7 0 16 33 39 45 48 48
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 [p. 149] curve indicates the time taken to regenerate n per cent. of the amount removed. All the curves converge towards infinity, when the amount removed (as shewn by the ordinate) approaches 75 per cent.; and all of the curves start from zero, for nothing is regenerated where nothing had been removed. Each curve approximates in form to a cubic parabola.
The amount regenerated varies also with the age of the tadpole and with other factors, such as temperature; in other words, for any given age, or size, of tadpole and also for various specific temperatures, a similar diagram might be constructed.
The power of reproducing, or regenerating, a lost limb is particularly well developed in arthropod animals, and is sometimes accompanied by remarkable modification of the form of the regenerated limb. A case in point, which has attracted much attention, occurs in connection with the claws of certain Crustacea [193].
In many Crustacea we have an asymmetry of the great claws, one being larger than the other and also more or less different in form. For instance, in the common lobster, one claw, the larger of the two, is provided with a few great “crushing” teeth, while the smaller claw has more numerous teeth, small and serrated. Though Aristotle thought otherwise, it appears that the crushing-claw may be on the right or left side, indifferently; whether it be on one or the other is a problem of “chance.” It is otherwise in many other Crustacea, where the larger and more powerful claw is always left or right, as the case may be, according to the species: where, in other words, the “probability” of the large or the small claw being left or being right is tantamount to certainty [194].
The one claw is the larger because it has grown the faster; [p. 150] it has a higher “coefficient of growth,” and accordingly, as age advances, the disproportion between the two claws becomes more and more evident. Moreover, we must assume that the characteristic form of the claw is a “function” of its magnitude; the knobbiness is a phenomenon coincident with growth, and we never, under any circumstances, find the smaller claw with big crushing teeth and the big claw with little serrated ones. There are many other somewhat similar cases where size and form are manifestly correlated, and we have already seen, to some extent, that the phenomenon of growth is accompanied by certain ratios of velocity that lead inevitably to changes of form. Meanwhile, then, we must simply assume that the essential difference between the two claws is one of magnitude, with which a certain differentiation of form is inseparably associated.
If we amputate a claw, or if, as often happens, the crab “casts it off,” it undergoes a process of regeneration,—it grows anew, and evidently does so with an accelerated velocity, which acceleration will cease when equilibrium of the parts is once more attained: the accelerated velocity being a case in point to illustrate that vis revulsionis of Haller, to which we have already referred.
With the help of this principle, Przibram accounts for certain curious phenomena which accompany the process of regeneration. As his experiments and those of Morgan shew, if the large or knobby claw (A) be removed, there are certain cases, e.g. the common lobster, where it is directly regenerated. In other cases, e.g. Alpheus [195], the other claw (B) assumes the size and form of that which was amputated, while the latter regenerates itself in the form of the other and weaker one; A and B have apparently changed places. In a third case, as in the crabs, the A-claw regenerates itself as a small or B-claw, but the B-claw remains for a time unaltered, though slowly and in the course of repeated moults it later on assumes the large and heavily toothed A-form.
Much has been written on this phenomenon, but in essence it is very simple. It depends upon the respective rates of growth, upon a ratio between the rate of regeneration and the rate of growth of the uninjured limb: complicated a little, however, by [p. 151] the possibility of the uninjured limb growing all the faster for a time after the animal has been relieved of the other. From the time of amputation, say of A, A begins to grow from zero, with a high “regenerative” velocity; while B, starting from a definite magnitude, continues to increase, with its normal or perhaps somewhat accelerated velocity. The ratio between the two velocities of growth will determine whether, by a given time, A has equalled, outstripped, or still fallen short of the magnitude of B.
That this is the gist of the whole problem is confirmed (if confirmation be necessary) by certain experiments of Wilson’s. It is known that by section of the nerve to a crab’s claw, its growth is retarded, and as the general growth of the animal proceeds the claw comes to appear stunted or dwarfed. Now in such a case as that of Alpheus, we have seen that the rate of regenerative growth in an amputated large claw fails to let it reach or overtake the magnitude of the growing little claw: which latter, in short, now appears as the big one. But if at the same time as we amputate the big claw we also sever the nerve to the lesser one, we so far slow down the latter’s growth that the other is able to make up to it, and in this case the two claws continue to grow at approximately equal rates, or in other words continue of coequal size.
The phenomenon of regeneration goes some way towards helping us to comprehend the phenomenon of “multiplication by fission,” as it is exemplified at least in its simpler cases in many worms and worm-like animals. For physical reasons which we shall have to study in another chapter, there is a natural tendency for any tube, if it have the properties of a fluid or semi-fluid substance, to break up into segments after it comes to a certain length; and nothing can prevent its doing so, except the presence of some controlling force, such for instance as may be due to the pressure of some external support, or some superficial thickening or other intrinsic rigidity of its own substance. If we add to this natural tendency towards fission of a cylindrical or tubular worm, the ordinary phenomenon of regeneration, we have all that is essentially implied in “reproduction by fission.” And in so far [p. 152] as the process rests upon a physical principle, or natural tendency, we may account for its occurrence in a great variety of animals, zoologically dissimilar; and also for its presence here and absence there, in forms which, though materially different in a physical sense, are zoologically speaking very closely allied.
CONCLUSION AND SUMMARY.
But the phenomena of regeneration, like all the other phenomena of growth, soon carry us far afield, and we must draw this brief discussion to a close.
For the main features which appear to be common to all curves of growth we may hope to have, some day, a physical explanation. In particular we should like to know the meaning of that point of inflection, or abrupt change from an increasing to a decreasing velocity of growth which all our curves, and especially our acceleration curves, demonstrate the existence of, provided only that they include the initial stages of the whole phenomenon: just as we should also like to have a full physical or physiological explanation of the gradually diminishing velocity of growth which follows, and which (though subject to temporary interruption or abeyance) is on the whole characteristic of growth in all cases whatsoever. In short, the characteristic form of the curve of growth in length (or any other linear dimension) is a phenomenon which we are at present unable to explain, but which presents us with a definite and attractive problem for future solution. It would seem evident that the abrupt change in velocity must be due, either to a change in that pressure outwards from within, by which the “forces of growth” make themselves manifest, or to a change in the resistances against which they act, that is to say the tension of the surface; and this latter force we do not by any means limit to “surface-tension” proper, but may extend to the development of a more or less resistant membrane or “skin,” or even to the resistance of fibres or other histological elements, binding the boundary layers to the parts within. I take it that the sudden arrest of velocity is much more likely to be due to a sudden increase of resistance than to a sudden diminution of internal energies: in other words, I suspect that it is coincident with some notable event of histological differentiation, such as [p. 153] the rapid formation of a comparatively firm skin; and that the dwindling of velocities, or the negative acceleration, which follows, is the resultant or composite effect of waning forces of growth on the one hand, and increasing superficial resistance on the other. This is as much as to say that growth, while its own energy tends to increase, leads also, after a while, to the establishment of resistances which check its own further increase.
Our knowledge of the whole complex phenomenon of growth is so scanty that it may seem rash to advance even this tentative suggestion. But yet there are one or two known facts which seem to bear upon the question, and to indicate at least the manner in which a varying resistance to expansion may affect the velocity of growth. For instance, it has been shewn by Frazee [196] that electrical stimulation of tadpoles, with small current density and low voltage, increases the rate of regenerative growth. As just such an electrification would tend to lower the surface-tension, and accordingly decrease the external resistance, the experiment would seem to support, in some slight degree, the suggestion which I have made.
Delage [197] has lately made use of the principle of specific rate of growth, in considering the question of heredity itself. We know that the chromatin of the fertilised egg comes from the male and female parent alike, in equal or nearly equal shares; we know that the initial chromatin, so contributed, multiplies many thousand-fold, to supply the chromatin for every cell of the offspring’s body; and it has, therefore, a high “coefficient of growth.” If we admit, with Van Beneden and others, that the initial contributions of male and female chromatin continue to be transmitted to the succeeding generations of cells, we may then conceive these chromatins to retain each its own coefficient of growth; and if these differed ever so little, a gradual preponderance of one or other would make itself felt in time, and might conceivably explain the preponderating influence of one parent or the other upon the characters of the offspring. Indeed O. Hertwig is said (according to Delage’s interpretation) to have actually shewn that we can artificially modify the rate of growth of one or other chromatin, and so increase or diminish the influence of the maternal or paternal heredity. This theory of Delage’s has its fascination, but it calls for somewhat large assumptions; and in particular, it seems (like so many other theories relating to the chromosomes) to rest far too much upon material elements, rather than on the imponderable dynamic factors of the cell. [p. 154]
We may summarise, as follows, the main results of the foregoing discussion:
Except in certain minute organisms and minute parts of organisms, whose form is due to the direct action of molecular forces, we may look upon the form of the organism as a “function of growth,” or a direct expression of a rate of growth which varies according to its different directions.
Rate of growth is subject to definite laws, and the velocities in different directions tend to maintain a ratio which is more or less constant for each specific organism; and to this regularity is due the fact that the form of the organism is in general regular and constant.
Nevertheless, the ratio of velocities in different directions is not absolutely constant, but tends to alter or fluctuate in a regular way; and to these progressive changes are due the changes of form which accompany “development,” and the slower changes of form which continue perceptibly in after life.
The rate of growth is a function of the age of the organism, it has a maximum somewhat early in life, after which epoch of maximum it slowly declines.
The rate of growth is directly affected by temperature, and by other physical conditions.
It is markedly affected, in the way of acceleration or retardation, at certain physiological epochs of life, such as birth, puberty, or metamorphosis.
Under certain circumstances, growth may be negative, the organism growing smaller: and such negative growth is a common accompaniment of metamorphosis, and a frequent accompaniment of old age.
The phenomenon of regeneration is associated with a large temporary increase in the rate of growth (or “acceleration” of growth) of the injured surface; in other respects, regenerative growth is similar to ordinary growth in all its essential phenomena.
In this discussion of growth, we have left out of account a vast number of processes, or phenomena, by which, in the physiological mechanism of the body, growth is effected and controlled. We have dealt with growth in its relation to magnitude, and to [p. 155] that relativity of magnitudes which constitutes form; and so we have studied it as a phenomenon which stands at the beginning of a morphological, rather than at the end of a physiological enquiry. Under these restrictions, we have treated it as far as possible, or in such fashion as our present knowledge permits, on strictly physical lines.
In all its aspects, and not least in its relation to form, the growth of organisms has many analogies, some close and some perhaps more remote, among inanimate things. As the waves grow when the winds strive with the other forces which govern the movements of the surface of the sea, as the heap grows when we pour corn out of a sack, as the crystal grows when from the surrounding solution the proper molecules fall into their appropriate places: so in all these cases, very much as in the organism itself, is growth accompanied by change of form, and by a development of definite shapes and contours. And in these cases (as in all other mechanical phenomena), we are led to equate our various magnitudes with time, and so to recognise that growth is essentially a question of rate, or of velocity.
The differences of form, and changes of form, which are brought about by varying rates (or “laws”) of growth, are essentially the same phenomenon whether they be, so to speak, episodes in the life-history of the individual, or manifest themselves as the normal and distinctive characteristics of what we call separate species of the race. From one form, or ratio of magnitude, to another there is but one straight and direct road of transformation, be the journey taken fast or slow; and if the transformation take place at all, it will in all likelihood proceed in the self-same way, whether it occur within the life-time of an individual or during the long ancestral history of a race. No small part of what is known as Wolff’s or von Baer’s law, that the individual organism tends to pass through the phases characteristic of its ancestors, or that the life-history of the individual tends to recapitulate the ancestral history of its race, lies wrapped up in this simple account of the relation between rate of growth and form.
But enough of this discussion. Let us leave for a while the subject of the growth of the organism, and attempt to study the conformation, within and without, of the individual cell.
Chapter IV: On the Internal Form and Structure of the Cell¶
1917 Cambridge First Edition (pp. 156–200)
In the early days of the cell-theory, more than seventy years ago, Goodsir was wont to speak of cells as “centres of growth” or “centres of nutrition,” and to consider them as essentially “centres of force.” He looked forward to a time when the forces connected with the cell should be particularly investigated: when, that is to say, minute anatomy should be studied in its dynamical aspect. “When this branch of enquiry,” he says “shall have been opened up, we shall expect to have a science of organic forces, having direct relation to anatomy, the science of organic forms [198].” And likewise, long afterwards, Giard contemplated a science of morphodynamique,—but still looked upon it as forming so guarded and hidden a “territoire scientifique, que la plupart des naturalistes de nos jours ne le verront que comme Moïse vit la terre promise, seulement de loin et sans pouvoir y entrer [199].”
To the external forms of cells, and to the forces which produce and modify these forms, we shall pay attention in a later chapter. But there are forms and configurations of matter within the cell, which also deserve to be studied with due regard to the forces, known or unknown, of whose resultant they are the visible expression.
In the long interval since Goodsir’s day, the visible structure, the conformation and configuration, of the cell, has been studied far more abundantly than the purely dynamic problems that are associated therewith. The overwhelming progress of microscopic observation has multiplied our knowledge of cellular and intracellular structure; and to the multitude of visible structures it [p. 157] has been often easier to attribute virtues than to ascribe intelligible functions or modes of action. But here and there nevertheless, throughout the whole literature of the subject, we find recognition of the inevitable fact that dynamical problems lie behind the morphological problems of the cell.
Bütschli pointed out forty years ago, with emphatic clearness, the failure of morphological methods, and the need for physical methods, if we were to penetrate deeper into the essential nature of the cell [200]. And such men as Loeb and Whitman, Driesch and Roux, and not a few besides, have pursued the same train of thought and similar methods of enquiry.
Whitman [201], for instance, puts the case in a nutshell when, in speaking of the so-called “caryokinetic” phenomena of nuclear division, he reminds us that the leading idea in the term “caryokinesis” is motion,—“motion viewed as an exponent of forces residing in, or acting upon, the nucleus. It regards the nucleus as a seat of energy, which displays itself in phenomena of motion [202].”
In short it would seem evident that, except in relation to a dynamical investigation, the mere study of cell structure has but little value of its own. That a given cell, an ovum for instance, contains this or that visible substance or structure, germinal vesicle or germinal spot, chromatin or achromatin, chromosomes or centrosomes, obviously gives no explanation of the activities of the cell. And in all such hypotheses as that of “pangenesis,” in all the theories which attribute specific properties to micellae, [p. 158] idioplasts, ids, or other constituent particles of protoplasm or of the cell, we are apt to fall into the error of attributing to matter what is due to energy and is manifested in force: or, more strictly speaking, of attributing to material particles individually what is due to the energy of their collocation.
The tendency is a very natural one, as knowledge of structure increases, to ascribe particular virtues to the material structures themselves, and the error is one into which the disciple is likely to fall, but of which we need not suspect the master-mind. The dynamical aspect of the case was in all probability kept well in view by those who, like Goodsir himself, first attacked the problem of the cell and originated our conceptions of its nature and functions.
But if we speak, as Weismann and others speak, of an “hereditary substance,” a substance which is split off from the parent-body, and which hands on to the new generation the characteristics of the old, we can only justify our mode of speech by the assumption that that particular portion of matter is the essential vehicle of a particular charge or distribution of energy, in which is involved the capability of producing motion, or of doing “work.”
For, as Newton said, to tell us that a thing “is endowed with an occult specific quality, by which it acts and produces manifest effects, is to tell us nothing; but to derive two or three general principles of motion [203] from phenomena would be a very great step in philosophy, though the causes of these principles were not yet discovered.” The things which we see in the cell are less important than the actions which we recognise in the cell; and these latter we must especially scrutinize, in the hope of discovering how far they may be attributed to the simple and well-known physical forces, and how far they be relevant or irrelevant to the phenomena which we associate with, and deem essential to, the manifestation of life. It may be that in this way we shall in time draw nigh to the recognition of a specific and ultimate residuum. [p. 159]
And lacking, as we still do lack, direct knowledge of the actual forces inherent in the cell, we may yet learn something of their distribution, if not also of their nature, from the outward and inward configuration of the cell, and from the changes taking place in this configuration; that is to say from the movements of matter, the kinetic phenomena, which the forces in action set up.
The fact that the germ-cell develops into a very complex structure, is no absolute proof that the cell itself is structurally a very complicated mechanism: nor yet, though this is somewhat less obvious, is it sufficient to prove that the forces at work, or latent, within it are especially numerous and complex. If we blow into a bowl of soapsuds and raise a great mass of many-hued and variously shaped bubbles, if we explode a rocket and watch the regular and beautiful configuration of its falling streamers, if we consider the wonders of a limestone cavern which a filtering stream has filled with stalactites, we soon perceive that in all these cases we have begun with an initial system of very slight complexity, whose structure in no way foreshadowed the result, and whose comparatively simple intrinsic forces only play their part by complex interaction with the equally simple forces of the surrounding medium. In an earlier age, men sought for the visible embryo, even for the homunculus, within the reproductive cells; and to this day, we scrutinize these cells for visible structure, unable to free ourselves from that old doctrine of “pre-formation [204].”
Moreover, the microscope seemed to substantiate the idea (which we may trace back to Leibniz [205] and to Hobbes [206]), that there is no limit to the mechanical complexity which we may postulate in an organism, and no limit, therefore, to the hypotheses which we may rest thereon.
But no microscopical examination of a stick of sealing-wax, no study of the material of which it is composed, can enlighten [p. 160] us as to its electrical manifestations or properties. Matter of itself has no power to do, to make, or to become: it is in energy that all these potentialities reside, energy invisibly associated with the material system, and in interaction with the energies of the surrounding universe.
That “function presupposes structure” has been declared an accepted axiom of biology. Who it was that so formulated the aphorism I do not know; but as regards the structure of the cell it harks back to Brücke, with whose demand for a mechanism, or organisation, within the cell histologists have ever since been attempting to comply [207]. But unless we mean to include thereby invisible, and merely chemical or molecular, structure, we come at once on dangerous ground. For we have seen, in a former chapter, that some minute “organisms” are already known of such all but infinitesimal magnitudes that everything which the morphologist is accustomed to conceive as “structure” has become physically impossible; and moreover recent research tends generally to reduce, rather than to extend, our conceptions of the visible structure necessarily inherent in living protoplasm. The microscopic structure which, in the last resort or in the simplest cases, it seems to shew, is that of a more or less viscous colloid, or rather mixture of colloids, and nothing more. Now, as Clerk Maxwell puts it, in discussing this very problem, “one material system can differ from another only in the configuration and motion which it has at a given instant [208].” If we cannot assume differences in structure, we must assume differences in motion, that is to say, in energy. And if we cannot do this, then indeed we are thrown back upon modes of reasoning unauthorised in physical science, and shall find ourselves constrained to assume, or to “admit, that the properties of a germ are not those of a purely material system.” [p. 161]
But we are by no means necessarily in this dilemma. For though we come perilously near to it when we contemplate the lowest orders of magnitude to which life has been attributed, yet in the case of the ordinary cell, or ordinary egg or germ which is going to develop into a complex organism, if we have no reason to assume or to believe that it comprises an intricate “mechanism,” we may be quite sure, both on direct and indirect evidence, that, like the powder in our rocket, it is very heterogeneous in its structure. It is a mixture of substances of various kinds, more or less fluid, more or less mobile, influenced in various ways by chemical, electrical, osmotic, and other forces, and in their admixture separated by a multitude of surfaces, or boundaries, at which these, or certain of these forces are made manifest.
Indeed, such an arrangement as this is already enough to constitute a “mechanism”; for we must be very careful not to let our physical or physiological concept of mechanism be narrowed to an interpretation of the term derived from the delicate and complicated contrivances of human skill. From the physical point of view, we understand by a “mechanism” whatsoever checks or controls, and guides into determinate paths, the workings of energy; in other words, whatsoever leads in the degradation of energy to its manifestation in some determinate form of work, at a stage short of that ultimate degradation which lapses in uniformly diffused heat. This, as Warburg has well explained, is the general effect or function of the physiological machine, and in particular of that part of it which we call “cell-structure [209].” The normal muscle-cell is something which turns energy, derived from oxidation, into work; it is a mechanism which arrests and utilises the chemical energy of oxidation in its downward course; but the same cell when injured or disintegrated, loses its “usefulness,” and sets free a greatly increased proportion of its energy in the form of heat.
But very great and wonderful things are done after this manner by means of a mechanism (whether natural or artificial) of extreme simplicity. A pool of water, by virtue of its surface, [p. 162] is an admirable mechanism for the making of waves; with a lump of ice in it, it becomes an efficient and self-contained mechanism for the making of currents. The great cosmic mechanisms are stupendous in their simplicity; and, in point of fact, every great or little aggregate of heterogeneous matter (not identical in “phase”) involves, ipso facto, the essentials of a mechanism. Even a non-living colloid, from its intrinsic heterogeneity, is in this sense a mechanism, and one in which energy is manifested in the movement and ceaseless rearrangement of the constituent particles. For this reason Graham (if I remember rightly) speaks somewhere or other of the colloid state as “the dynamic state of matter”; or in the same philosopher’s phrase (of which Mr Hardy [210] has lately reminded us), it possesses “energia [211].”
Let us turn then to consider, briefly and diagrammatically, the structure of the cell, a fertilised germ-cell or ovum for instance, not in any vain attempt to correlate this structure with the structure or properties of the resulting and yet distant organism; but merely to see how far, by the study of its form and its changing internal configuration, we may throw light on certain forces which are for the time being at work within it.
We may say at once that we can scarcely hope to learn more of these forces, in the first instance, than a few facts regarding their direction and magnitude; the nature and specific identity of the force or forces is a very different matter. This latter problem is likely to be very difficult of elucidation, for the reason, among others, that very different forces are often very much alike in their outward and visible manifestations. So it has come to pass that we have a multitude of discordant hypotheses as to the nature of the forces acting within the cell, and producing, in cell division, the “caryokinetic” figures of which we are about to speak. One student may, like Rhumbler, choose to account for them by an hypothesis of mechanical traction, acting on a reticular web of protoplasm [212]; another, like Leduc, may shew us how in [p. 163] many of their most striking features they may be admirably simulated by the diffusion of salts in a colloid medium; others again, like Gallardo [213] and Hartog, and Rhumbler (in his earlier papers) [214], insist on their resemblance to the phenomena of electricity and magnetism [215]; while Hartog believes that the force in question is only analogous to these, and has a specific identity of its own [216]. All these conflicting views are of secondary importance, so long as we seek only to account for certain configurations which reveal the direction, rather than the nature, of a force. One and the same system of lines of force may appear in a field of magnetic or of electrical energy, of the osmotic energy of diffusion, of the gravitational energy of a flowing stream. In short, we may expect to learn something of the pure or abstract dynamics, long before we can deal with the special physics of the cell. For indeed (as Maillard has suggested), just as uniform expansion about a single centre, to whatsoever physical cause it may be due will lead to the configuration of a sphere, so will any two centres or foci of potential (of whatsoever kind) lead to the configurations with which Faraday made us familiar under the name of “lines of force [217]”; and this is as much as to say that the phenomenon, [p. 164] though physical in the concrete, is in the abstract purely mathematical, and in its very essence is neither more nor less than a property of three-dimensional space.
But as a matter of fact, in this instance, that is to say in trying to explain the leading phenomena of the caryokinetic division of the cell, we shall soon perceive that any explanation which is based, like Rhumbler’s, on mere mechanical traction, is obviously inadequate, and we shall find ourselves limited to the hypothesis of some polarised and polarising force, such as we deal with, for instance, in the phenomena of magnetism or electricity.
Let us speak first of the cell itself, as it appears in a state of rest, and let us proceed afterwards to study the more active phenomena which accompany its division.
Our typical cell is a spherical body; that is to say, the uniform surface-tension at its boundary is balanced by the outward resistance of uniform forces within. But at times the surface-tension may be a fluctuating quantity, as when it produces the rhythmical contractions or “Ransom’s waves” on the surface of a trout’s egg; or again, while the egg is in contact with other bodies, the surface-tension may be locally unequal and variable, giving rise to an amoeboid figure, as in the egg of Hydra [218].
Within the ovum is a nucleus or germinal vesicle, also spherical, and consisting as a rule of portions of “chromatin,” aggregated together within a more fluid drop. The fact has often been commented upon that, in cells generally, there is no correlation of form (though there apparently is of size) between the nucleus and the “cytoplasm,” or main body of the cell. So Whitman [219] remarks that “except during the process of division the nucleus seldom departs from its typical spherical form. It divides and sub-divides, ever returning to the same round or oval form …. How different with the cell. It preserves the spherical form as rarely as the nucleus departs from it. Variation in form marks the beginning and the end of every important chapter in its [p. 165] history.” On simple dynamical grounds, the contrast is easily explained. So long as the fluid substance of the nucleus is qualitatively different from, and incapable of mixing with, the fluid or semi-fluid protoplasm which surrounds it, we shall expect it to be, as it almost always is, of spherical form. For, on the one hand, it is bounded by a liquid film, whose surface-tension is uniform; and on the other, it is immersed in a medium which transmits on all sides a uniform fluid pressure [220]. For a similar reason the contractile vacuole of a Protozoon is spherical in form: it is just a “drop” of fluid, bounded by a uniform surface-tension and through whose boundary-film diffusion is taking place. But here, owning to the small difference between the fluid constituting, and that surrounding, the drop, the surface-tension equilibrium is unstable; it is apt to vanish, and the rounded outline of the drop, like a burst bubble, disappears in a moment [221]. The case of the spherical nucleus is closely akin to the spherical form of the yolk within the bird’s egg [222]. But if the substance of the cell acquire a greater solidity, as for instance in a muscle [p. 166] cell, or by reason of mucous accumulations in an epithelium cell, then the laws of fluid pressure no longer apply, the external pressure on the nucleus tends to become unsymmetrical, and its shape is modified accordingly. “Amoeboid” movements may be set up in the nucleus by anything which disturbs the symmetry of its own surface-tension. And the cases, as in many Rhizopods, where “nuclear material” is scattered in small portions throughout the cell instead of being aggregated in a single nucleus, are probably capable of very simple explanation by supposing that the “phase difference” (as the chemists say) between the nuclear and the protoplasmic substance is comparatively slight, and the surface-tension which tends to keep them separate is correspondingly small [223].
It has been shewn that ordinary nuclei, isolated in a living or fresh state, easily flow together; and this fact is enough to suggest that they are aggregations of a particular substance rather than bodies deserving the name of particular organs. It is by reason of the same tendency to confluence or aggregation of particles that the ordinary nucleus is itself formed, until the imposition of a new force leads to its disruption.
Apart from that invisible or ultra-microscopic heterogeneity which is inseparable from our notion of a “colloid,” there is a visible heterogeneity of structure within both the nucleus and the outer protoplasm. The former, for instance, contains a rounded nucleolus or “germinal spot,” certain conspicuous granules or strands of the peculiar substance called chromatin, and a coarse meshwork of a protoplasmic material known as “linin” or achromatin; the outer protoplasm, or cytoplasm, is generally believed to consist throughout of a sponge-work, or rather alveolar meshwork, of more and less fluid substances; and lastly, there are generally to be detected one or more very minute bodies, usually in the cytoplasm, sometimes within the nucleus, known as the centrosome or centrosomes.
The morphologist is accustomed to speak of a “polarity” of [p. 167] the cell, meaning thereby a symmetry of visible structure about a particular axis. For instance, whenever we can recognise in a cell both a nucleus and a centrosome, we may consider a line drawn through the two as the morphological axis of polarity; in an epithelium cell, it is obvious that the cell is morphologically symmetrical about a median axis passing from its free surface to its attached base. Again, by an extension of the term “polarity,” as is customary in dynamics, we may have a “radial” polarity, between centre and periphery; and lastly, we may have several apparently independent centres of polarity within the single cell. Only in cells of quite irregular, or amoeboid form, do we fail to recognise a definite and symmetrical “polarity.” The morphological “polarity” is accompanied by, and is but the outward expression (or part of it) of a true dynamical polarity, or distribution of forces; and the “lines of force” are rendered visible by concatenation of particles of matter, such as come under the influence of the forces in action.
When the lines of force stream inwards from the periphery towards a point in the interior of the cell, the particles susceptible of attraction either crowd towards the surface of the cell, or, when retarded by friction, are seen forming lines or “fibrillae” which radiate outwards from the centre and constitute a so-called “aster.” In the cells of columnar or ciliated epithelium, where the sides of the cell are symmetrically disposed to their neighbours but the free and attached surfaces are very diverse from one another in their external relations, it is these latter surfaces which constitute the opposite poles; and in accordance with the parallel lines of force so set up, we very frequently see parallel lines of granules which have ranged themselves perpendicularly to the free surface of the cell (cf. fig. 97).
A simple manifestation of “polarity” may be well illustrated by the phenomenon of diffusion, where we may conceive, and may automatically reproduce, a “field of force,” with its poles and visible lines of equipotential, very much as in Faraday’s conception of the field of force of a magnetic system. Thus, in one of Leduc’s experiments [224], if we spread a layer of salt solution over a level [p. 168] plate of glass, and let fall into the middle of it a drop of indian ink, or of blood, we shall find the coloured particles travelling outwards from the central “pole of concentration” along the lines of diffusive force, and so mapping out for us a “monopolar field” of diffusion: and if we set two such drops side by side, their lines of diffusion will oppose, and repel, one another. Or, instead of the uniform layer of salt solution, we may place at a little distance from one another a grain of salt and a drop of blood, representing two opposite poles: and so obtain a picture of a “bipolar field” of diffusion. In either case, we obtain results closely analogous to the “morphological,” but really dynamical, polarity of the organic cell. But in all probability, the dynamical polarity, or asymmetry of the cell is a very complicated phenomenon: for the obvious reason that, in any system, one asymmetry will tend to beget another. A chemical asymmetry will induce an inequality of surface-tension, which will lead directly to a modification of form; the chemical asymmetry may in turn be due to a process of electrolysis in a polarised electrical field; and again the chemical heterogeneity may be intensified into a chemical “polarity,” by the tendency of certain substances to seek a locus of greater or less surface-energy. We need not attempt to grapple with a subject so complicated, and leading to so many problems which lie beyond the sphere of interest of the morphologist. But yet the morphologist, in his study of the cell, cannot quite evade these important issues; and we shall return to them again when we have dealt somewhat with the form of the cell, and have taken account of some of the simpler phenomena of surface-tension.
We are now ready, and in some measure prepared, to study the numerous and complex phenomena which usually accompany the division of the cell, for instance of the fertilised egg.
Division of the cell is essentially accompanied, and preceded, by a change from radial or monopolar to a definitely bipolar polarity.
In the hitherto quiescent, or apparently quiescent cell, we perceive certain movements, which correspond precisely to what must accompany and result from a “polarisation” of forces within the [p. 169] cell: of forces which, whatever may be their specific nature, at least are capable of polarisation, and of producing consequent attraction or repulsion between charged particles of matter. The opposing forces which were distributed in equilibrium throughout the substance of the cell become focussed at two “centrosomes,” which may or may not be already distinguished as visible portions of matter; in the egg, one of these is always near to, and the other remote from, the “animal pole” of the egg, which pole is visibly as well as chemically different from the other, and is the region in which the more rapid and conspicuous developmental changes will presently begin. Between the two centrosomes, a spindle-shaped
1917 Source Illustration
Fig. 41. Caryokinetic figure in a dividing cell (or blastomere) of the Trout’s egg. (After Prenant, from a preparation by Prof. P. Bouin.)
figure appears, whose striking resemblance to the lines of force made visible by iron-filings between the poles of a magnet, was at once recognised by Hermann Fol, when in 1873 he witnessed for the first time the phenomenon in question. On the farther side of the centrosomes are seen star-like figures, or “asters,” in which we can without difficulty recognise the broken lines of force which run externally to those stronger lines which lie nearer to the polar axis and which constitute the “spindle.” The lines of force are rendered visible or “material,” just as in the experiment of the iron-filings, by the fact that, in the heterogeneous substance of the cell, certain portions of matter are more “permeable” to the acting force than the rest, become themselves polarised after the [p. 170] fashion of a magnetic or “paramagnetic” body, arrange themselves in an orderly way between the two poles of the field of force, cling to one another as it were in threads [225], and are only prevented by the friction of the surrounding medium from approaching and congregating around the adjacent poles.
As the field of force strengthens, the more will the lines of force be drawn in towards the interpolar axis, and the less evident will be those remoter lines which constitute the terminal, or extrapolar, asters: a clear space, free from materialised lines of force, may thus tend to be set up on either side of the spindle, the so-called “Bütschli space” of the histologists [226]. On the other hand, the lines of force constituting the spindle will be less concentrated if they find a path of less resistance at the periphery of the cell: as happens, in our experiment of the iron-filings, when we encircle the field of force with an iron ring. On this principle, the differences observed between cells in which the spindle is well developed and the asters small, and others in which the spindle is weak and the asters enormously developed, can be easily explained by variations in the potential of the field, the large, conspicuous asters being probably correlated with a marked permeability of the surface of the cell.
The visible field of force, though often called the “nuclear spindle,” is formed outside of, but usually near to, the nucleus. Let us look a little more closely into the structure of this body, and into the changes which it presently undergoes.
Within its spherical outline (Fig. 42), it contains an “alveolar” [p. 171] meshwork (often described, from its appearance in optical section, as a “reticulum”), consisting of more solid substances, with more fluid matter filling up the interalveolar meshes. This phenomenon is nothing else than what we call in ordinary language, a “froth” or a “foam.” It is a surface-tension phenomenon, due to the interacting surface-tensions of two intermixed fluids, not very different in density, as they strive to separate. Of precisely the same kind (as Bütschli was the first to shew) are the minute alveolar networks which are to be discerned in the cytoplasm of the cell [227], and which we now know to be not inherent in the nature of protoplasm, or of living matter in general, but to be due to various causes, natural as well as artificial. The microscopic honeycomb structure of cast metal under various conditions of cooling, even on a grand scale the columnar structure of basaltic rock, is an example of the same surface-tension phenomenon. [p. 172]
1917 Source Illustration
Fig. 42.
1917 Source Illustration
Fig. 43.
But here we touch the brink of a subject so important that we must not pass it by without a word, and yet so contentious that we must not enter into its details. The question involved is simply whether the great mass of recorded observations and accepted beliefs with regard to the visible structure of protoplasm and of the cell constitute a fair picture of the actual living cell, or be based on appearances which are incident to death itself and to the artificial treatment which the microscopist is accustomed to apply. The great bulk of histological work is done by methods which involve the sudden killing of the cell or organism by strong reagents, the assumption being that death is so rapid that the visible phenomena exhibited during life are retained or “fixed” in our preparations. While this assumption is reasonable and justified as regards the general outward form of small organisms or of individual cells, enough has been done of late years to shew that the case is totally different in the case of the minute internal networks, granules, etc., which represent the alleged structure of protoplasm. For, as Hardy puts it, “It is notorious that the various fixing reagents are coagulants of organic colloids, and that they produce precipitates which have a certain figure or structure, … and that the figure varies, other things being equal, according to the reagent used.” So it comes to pass that some writers [228] have altogether denied the existence in the living cell-protoplasm of a network or alveolar “foam”; others [229] have cast doubts on the main tenets of recent histology regarding nuclear structure; and Hardy, discussing the structure of certain gland-cells, declares that “there is no evidence that the structure discoverable in the cell-substance of these cells after fixation has any counterpart in the cell when living.” “A large part of it” he goes on to say “is an artefact. The profound difference in the minute structure of a secretory cell of a mucous gland according to the reagent which is used to fix it would, it seems to me, almost suffice to establish this statement in the absence of other evidence.”
Nevertheless, histological study proceeds, especially on the part of the morphologists, with but little change in theory or in method, in spite of these and many other warnings. That certain visible structures, nucleus, vacuoles, “attraction-spheres” or centrosomes, etc., are actually present in the living cell, we know for certain; and to this class belong the great majority of structures (including the nuclear “spindle” itself) with which we are at present concerned. That many other alleged structures are artificial has also been placed beyond a doubt; but where to draw the dividing line we often do not know [230]. [p. 173]
The following is a brief epitome of the visible changes undergone by a typical cell, leading up to the act of segmentation, and constituting the phenomenon of mitosis or caryokinetic division. In the egg of a sea-urchin, we see with almost diagrammatic completeness what is set forth here [231].
1917 Source Illustration
Fig. 44.
1917 Source Illustration
Fig. 45.
The chromatin, which to begin with was distributed in granules on the otherwise achromatic reticulum (Fig. 42), concentrates to form a skein or spireme, which may be a continuous thread from the first (Figs. 43, 44), or from the first segmented. In any case it divides transversely sooner or later into a number of chromosomes (Fig. 45), which as a rule have the shape of little rods, straight or curved, often bent into a V, but which may also be ovoid, or round, or even annular. Certain deeply staining masses, the nucleoli, which may be present in the resting nucleus, do not take part in the process of chromosome formation; they are either cast out of the nucleus and are dissolved in the cytoplasm, or fade away in situ.
Meanwhile, the deeply staining granule (here extra-nuclear), known as the centrosome, has divided in two. The two resulting granules travel to opposite poles of the nucleus, and [p. 174] there each becomes surrounded by a system of radiating lines, the asters; immediately around the centrosome is a clear space, the centrosphere (Figs. 43–45). Between the two centrosomes with their asters stretches a bundle of achromatic fibres, the spindle.
The surface-film bounding the nucleus has broken down, the definite nuclear boundaries are lost, and the spindle now stretches through the nuclear material, in which lie the chromosomes (Figs. 45, 46). These chromosomes now arrange themselves midway between the poles of the spindle, where they form what is called the equatorial plate (Fig. 47).
1917 Source Illustration
Fig. 46.
1917 Source Illustration
Fig. 47.
Each chromosome splits longitudinally into two: usually at this stage,—but it is to be noticed that the splitting may have taken place so early as the spireme stage (Fig. 48).
The halves of the split chromosomes now separate from one another, and travel in opposite directions towards the two poles (Fig. 49). As they move, it becomes apparent that the spindle consists of a median bundle of “fibres,” the central spindle, running from pole to pole, and a more superficial sheath of “mantle-fibres,” to which the chromosomes seem to be attached, and by which they seem to be drawn towards the asters.
The daughter chromosomes, arranged now in two groups, become closely crowded in a mass near the centre of each aster [p. 175] (Fig. 50). They fuse together and form once more an alveolar reticulum and may occasionally at this stage form another spireme.
1917 Source Illustration
Fig. 48.
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Fig. 49.
A boundary or surface wall is now developed round each reconstructed nuclear mass, and the spindle-fibres disappear (Fig. 51). The centrosome remains, as a rule, outside the nucleus.
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Fig. 50.
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Fig. 51.
On the central spindle, in the position of the equatorial plate, there has appeared during the migration of the chromosomes, a “cell-plate” of deeply staining thickenings (Figs. 50, 51). This is more conspicuous in plant-cells. [p. 176]
A constriction has meanwhile appeared in the cytoplasm, and the cell divides through the equatorial plane. In plant-cells the line of this division is foreshadowed by the “cell-plate,” which extends from the spindle across the entire cell, and splits into two layers, between which appears the membrane by which the daughter cells are cleft asunder. In animal cells the cell-plate does not attain such dimensions, and no cell-wall is formed.
The whole, or very nearly the whole of these nuclear phenomena may be brought into relation with that polarisation of forces, in the cell as a whole, whose field is made manifest by the “spindle” and “asters” of which we have already spoken: certain particular phenomena, directly attributable to surface-tension and diffusion, taking place in more or less obvious and inevitable dependence upon the polar system†.
† The reference numbers in the following account refer to the paragraphs and figures of the preceding summary of visible nuclear phenomena.
At the same time, in attempting to explain the phenomena, we cannot say too clearly, or too often, that all that we are meanwhile justified in doing is to try to shew that such and such actions lie within the range of known physical actions and phenomena, or that known physical phenomena produce effects similar to them. We want to feel sure that the whole phenomenon is not sui generis, but is somehow or other capable of being referred to dynamical laws, and to the general principles of physical science. But when we speak of some particular force or mode of action, using it as an illustrative hypothesis, we must stop far short of the implication that this or that force is necessarily the very one which is actually at work within the living cell; and certainly we need not attempt the formidable task of trying to reconcile, or to choose between, the various hypotheses which have already been enunciated, or the several assumptions on which they depend.
Any region of space within which action is manifested is a field of force; and a simple example is a bipolar field, in which the action is symmetrical with reference to the line joining two points, or poles, and also with reference to the “equatorial” plane equidistant from both. We have such a “field of force” in [p. 177] the neighbourhood of the centrosome of the ripe cell or ovum, when it is about to divide; and by the time the centrosome has divided, the field is definitely a bipolar one.
The quality of a medium filling the field of force may be uniform, or it may vary from point to point. In particular, it may depend upon the magnitude of the field; and the quality of one medium may differ from that of another. Such variation of quality, within one medium, or from one medium to another, is capable of diagrammatic representation by a variation of the direction or the strength of the field (other conditions being the same) from the state manifested in some uniform medium taken as a standard. The medium is said to be permeable to the force, in greater or less degree than the standard medium, according as the variation of the density of the lines of force from the standard case, under otherwise identical conditions, is in excess or defect. A body placed in the medium will tend to move towards regions of greater or less force according as its permeability is greater or less than that of the surrounding medium [232]. In the common experiment of placing iron-filings between the two poles of a magnetic field, the filings have a very high permeability; and not only do they themselves become polarised so as to attract one another, but they tend to be attracted from the weaker to the stronger parts of the field, and as we have seen, were it not for friction or some other resistance, they would soon gather together around the nearest pole. But if we repeat the same experiment with such a metal as bismuth, which is very little permeable to the magnetic force, then the conditions are reversed, and the particles, being repelled from the stronger to the weaker parts of the field, tend to take up their position as far from the poles as possible. The particles have become polarised, but in a sense opposite to that of the surrounding, or adjacent, field.
Now, in the field of force whose opposite poles are marked by [p. 178] the centrosomes the nucleus appears to act as a more or less permeable body, as a body more permeable than the surrounding medium, that is to say the “cytoplasm” of the cell. It is accordingly attracted by, and drawn into, the field of force, and tries, as it were, to set itself between the poles and as far as possible from both of them. In other words, the centrosome-foci will be apparently drawn over its surface, until the nucleus as a whole is involved within the field of force, which is visibly marked out by the “spindle” (par. 3, Figs. 44, 45).
If the field of force be electrical, or act in a fashion analogous to an electrical field, the charged nucleus will have its surface-tensions diminished [233]: with the double result that the inner alveolar meshwork will be broken up (par. 1), and that the spherical boundary of the whole nucleus will disappear (par. 2). The break-up of the alveoli (by thinning and rupture of their partition walls) leads to the formation of a net, and the further break-up of the net may lead to the unravelling of a thread or “spireme” (Figs. 43, 44).
Here there comes into play a fundamental principle which, in so far as we require to understand it, can be explained in simple words. The effect (and we might even say the object) of drawing the more permeable body in between the poles, is to obtain an “easier path” by which the lines of force may travel; but it is obvious that a longer route through the more permeable body may at length be found less advantageous than a shorter route through the less permeable medium. That is to say, the more permeable body will only tend to be drawn in to the field of force until a point is reached where (so to speak) the way round and the way through are equally advantageous. We should accordingly expect that (on our hypothesis) there would be found cases in which the nucleus was wholly, and others in which it was only partially, and in greater or less degree, drawn in to the field between the centrosomes. This is precisely what is found to occur in actual fact. Figs. 44 and 45 represent two so-called “types,” of a phase which follows that represented in Fig. 43. According to the usual descriptions (and in particular to Professor [p. 179] E. B. Wilson’s [234]), we are told that, in such a case as Fig. 44, the “primary spindle” disappears and the centrosomes diverge to opposite poles of the nucleus; such a condition being found in many plant-cells, and in the cleavage-stages of many eggs. In Fig. 45, on the other hand, the primary spindle persists, and subsequently comes to form the main or “central” spindle; while at the same time we see the fading away of the nuclear membrane, the breaking up of the spireme into separate chromosomes, and an ingrowth into the nuclear area of the “astral rays,”—all as in Fig. 46, which represents the next succeeding phase of Fig. 45. This condition, of Fig. 46, occurs in a variety of cases; it is well seen in the epidermal cells of the salamander, and is also on the whole characteristic of the mode of formation of the “polar bodies.” It is clear and obvious that the two “types” correspond to mere differences of degree, and are such as would naturally be brought about by differences in the relative permeabilities of the nuclear mass and of the surrounding cytoplasm, or even by differences in the magnitude of the former body.
But now an important change takes place, or rather an important difference appears; for, whereas the nucleus as a whole tended to be drawn in to the stronger parts of the field, when it comes to break up we find, on the contrary, that its contained spireme-thread or separate chromosomes tend to be repelled to the weaker parts. Whatever this difference may be due to,—whether, for instance, to actual differences of permeability, or possibly to differences in “surface-charge,”—the fact is that the chromatin substance now behaves after the fashion of a “diamagnetic” body, and is repelled from the stronger to the weaker parts of the field. In other words, its particles, lying in the inter-polar field, tend to travel towards the equatorial plane thereof (Figs. 47, 48), and further tend to move outwards towards the periphery of that plane, towards what the histologist calls the “mantle-fibres,” or outermost of the lines of force of which the spindle is made up (par. 5, Fig. 47). And if this comparatively non-permeable chromatin substance come to consist of separate portions, more or less elongated in form, these portions, or separate “chromosomes,” will adjust themselves longitudinally, [p. 180] in a peripheral equatorial circle (Figs. 48, 49). This is precisely what actually takes place. Moreover, before the breaking up of the nucleus, long before the chromatin material has broken up into separate chromosomes, and at the very time when it is being fashioned into a “spireme,” this body already lies in a polar field, and must already have a tendency to set itself in the equatorial plane thereof. But the long, continuous spireme thread is unable, so long as the nucleus retains its spherical boundary wall, to adjust itself in a simple equatorial annulus; in striving to do so, it must tend to coil and “kink” itself, and in so doing (if all this be so), it must tend to assume the characteristic convolutions of the “spireme.”
1917 Source Illustration
Fig. 52. Chromosomes, undergoing splitting and separation. (After Hatschek and Flemming, diagrammatised.)
After the spireme has broken up into separate chromosomes, these particles come into a position of temporary, and unstable, equilibrium near the periphery of the equatorial plane, and here they tend to place themselves in a symmetrical arrangement (Fig. 52). The particles are rounded, linear, sometimes annular, similar in form and size to one another; and lying as they do in a fluid, and subject to a symmetrical system of forces, it is not surprising that they arrange themselves in a symmetrical manner, the precise arrangement depending on the form of the particles themselves. This symmetry may perhaps be due, as has already been suggested, to induced electrical charges. In discussing Brauer’s observations on the splitting of the chromatic filament, and the symmetrical arrangement of the separate granules, in Ascaris megalocephala, Lillie [235] [p. 181] remarks: “This behaviour is strongly suggestive of the division of a colloidal particle under the influence of its surface electrical charge, and of the effects of mutual repulsion in keeping the products of division apart.” It is also probable that surface-tensions between the particles and the surrounding protoplasm would bring about an identical result, and would sufficiently account for the obvious, and at first sight, very curious, symmetry. We know that if we float a couple of matches in water they tend to approach one another, till they lie close together, side by side; and, if we lay upon a smooth wet plate four matches, half broken across, a precisely similar attraction brings the four matches together in the form of a symmetrical cross. Whether one of these, or some other, be the actual explanation of the phenomenon, it is at least plain that by some physical cause, some mutual and symmetrical attraction or repulsion of the particles, we must seek to account for the curious symmetry of these so-called “tetrads.” The remarkable annular chromosomes, shewn in Fig. 53, can also be easily imitated by means of loops of thread upon a soapy film when the film within the annulus is broken or its tension reduced.
1917 Source Illustration
Fig. 53. Annular chromosomes, formed in the spermatogenesis of the Mole-cricket. (From Wilson, after Vom Rath.)
So far as we have now gone, there is no great difficulty in pointing to simple and familiar phenomena of a field of force which are similar, or comparable, to the phenomena which we witness within the cell. But among these latter phenomena there are others for which it is not so easy to suggest, in accordance with known laws, a simple mode of physical causation. It is not at once obvious how, in any simple system of symmetrical forces, [p. 182] the chromosomes, which had at first been apparently repelled from the poles towards the equatorial plane, should then be split asunder, and should presently be attracted in opposite directions, some to one pole and some to the other. Remembering that it is not our purpose to assert that some one particular mode of action is at work, but merely to shew that there do exist physical forces, or distributions of force, which are capable of producing the required result, I give the following suggestive hypothesis, which I owe to my colleague Professor W. Peddie.
As we have begun by supposing that the nuclear, or chromosomal matter differs in permeability from the medium, that is to say the cytoplasm, in which it lies, let us now make the further assumption that its permeability is variable, and depends upon the strength of the field.
1917 Source Illustration
Fig. 54.
In Fig. 54, we have a field of force (representing our cell), consisting of a homogeneous medium, and including two opposite poles: lines of force are indicated by full lines, and loci of constant magnitude of force are shewn by dotted lines.
Let us now consider a body whose permeability (µ) depends on the strength of the field F. At two field-strengths, such as Fa, Fb, let the permeability of the body be equal to that of the [p. 183] medium, and let the curved line in Fig. 55 represent generally its permeability at other field-strengths; and let the outer and inner dotted curves in Fig. 54 represent respectively the loci of the field-strengths Fb 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 in the region intermediate to the two dotted curves, it will tend to move towards regions of weaker field-strength.
1917 Source Illustration
Fig. 55.
The locus Fb is therefore a locus of stable position, towards which the body tends to move; the locus Fa is a locus of unstable position, from which it tends to move. If the body were placed across Fa, it might be torn asunder into two portions, the split coinciding with the locus Fa.
Suppose a number of such bodies to be scattered throughout the medium. Let at first the regions Fa and Fb be entirely outside the space where the bodies are situated: and, in making this supposition we may, if we please, suppose that the loci which we are calling Fa and Fb are meanwhile situated somewhat farther from the axis than in our figure, that (for instance) Fa is situated where we have drawn Fb, and that Fb is still further out. The bodies then tend towards the poles; but the tendency may be very small if, in Fig. 55, the curve and its intersecting straight line do not diverge very far from one another beyond Fa; in other [p. 184] words, if, when situated in this region, the permeability of the bodies is not very much in excess of that of the medium.
Let the poles now tend to separate farther and farther from one another, the strength of each pole remaining unaltered; in other words, let the centrosome-foci recede from one another, as they actually do, drawing out the spindle-threads between them. The loci Fa, Fb, will close in to nearer relative distances from the poles. In doing so, when the locus Fa 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. 56).
1917 Source Illustration
Fig. 56.
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 Fb has crossed it, then the body again moves towards regions of stronger force, that is to say, towards the nearest pole. And, in thus moving towards the pole, it will do so, as appears actually to be the case in the dividing cell, along the course of the outer lines of force, the so-called “mantle-fibres” of the histologist [236].
Such considerations as these give general results, easily open to modification in detail by a change of any of the arbitrary postulates which have been made for the sake of simplicity. Doubtless there are many other assumptions which would more or less meet the case; for instance, that of Ida H. Hyde that, [p. 185] during the active phase of the chromatin molecule (during which it decomposes and sets free nucleic acid) it carries a charge opposite to that which it bears during its resting, or alkaline phase; and that it would accordingly move towards different poles under the influence of a current, wandering with its negative charge in an alkaline fluid during its acid phase to the anode, and to the kathode during its alkaline phase. A whole field of speculation is opened up when we begin to consider the cell not merely as a polarised electrical field, but also as an electrolytic field, full of wandering ions. Indeed it is high time we reminded ourselves that we have perhaps been dealing too much with ordinary physical analogies: and that our whole field of force within the cell is of an order of magnitude where these grosser analogies may fail to serve us, and might even play us false, or lead us astray. But our sole object meanwhile, as I have said more than once, is to demonstrate, by such illustrations as these, that, whatever be the actual and as yet unknown modus operandi, there are physical conditions and distributions of force which could produce just such phenomena of movement as we see taking place within the living cell. This, and no more, is precisely what Descartes is said to have claimed for his description of the human body as a “mechanism [237].”
The foregoing account is based on the provisional assumption that the phenomena of caryokinesis are analogous to, if not identical with those of a bipolar electrical field; and this comparison, in my opinion, offers without doubt the best available series of analogies. But we must on no account omit to mention the fact that some of Leduc’s diffusion-experiments offer very remarkable analogies to the diagrammatic phenomena of caryokinesis, as shewn in the annexed figure [238]. Here we have two identical (not opposite) poles of osmotic concentration, formed by placing a drop of indian ink in salt water, and then on either side of this central drop, a hypertonic drop of salt solution more lightly coloured. On either side the pigment of the central drop has been drawn towards the focus nearest to it; but in the middle line, the pigment [p. 186] is drawn in opposite directions by equal forces, and so tends to remain undisturbed, in the form of an “equatorial plate.”
Nor should we omit to take account (however briefly and inadequately) of a novel and elegant hypothesis put forward by A. B. Lamb. This hypothesis makes use of a theorem of Bjerknes, to the effect that synchronously vibrating or pulsating bodies in a liquid field attract or repel one another according as their oscillations are identical or opposite in phase. Under such circumstances, true currents, or hydrodynamic lines of force, are produced, identical in form with the lines of force of a magnetic field; and other particles floating, though not necessarily pulsating, in the liquid field, tend to be attracted or repelled by the pulsating bodies according as they are lighter or heavier than the surrounding fluid. Moreover (and this is the most remarkable point of all), the lines of force set up by the oppositely pulsating bodies are the same as those which are produced by opposite magnetic poles: though in the former case repulsion, and in the latter case attraction, takes place between the two poles [239].
1917 Source Illustration
Fig. 57. Artificial caryokinesis (after Leduc), for comparison with Fig. 41, p. 169.
But to return to our general discussion.
While it can scarcely be too often repeated that our enquiry is not directed towards the solution of physiological problems, save [p. 187] only in so far as they are inseparable from the problems presented by the visible configurations of form and structure, and while we try, as far as possible, to evade the difficult question of what particular forces are at work when the mere visible forms produced are such as to leave this an open question, yet in this particular case we have been drawn into the use of electrical analogies, and we are bound to justify, if possible, our resort to this particular mode of physical action. There is an important paper by R. S. Lillie, on the “Electrical Convection of certain Free Cells and Nuclei [240],” which, while I cannot quote it in direct support of the suggestions which I have made, yet gives just the evidence we need in order to shew that electrical forces act upon the constituents of the cell, and that their action discriminates between the two species of colloids represented by the cytoplasm and the nuclear chromatin. And the difference is such that, in the presence of an electrical current, the cell substance and the nuclei (including sperm-cells) tend to migrate, the former on the whole with the positive, the latter with the negative stream: a difference of electrical potential being thus indicated between the particle and the surrounding medium, just as in the case of minute suspended particles of various kinds in various feebly conducing media [241]. And the electrical difference is doubtless greatest, in the case of the cell constituents, just at the period of mitosis: when the chromatin is invariably in its most deeply staining, most strongly acid, and therefore, presumably, in its most electrically negative phase. In short, [p. 188] Lillie comes easily to the conclusion that “electrical theories of mitosis are entitled to more careful consideration than they have hitherto received.”
Among other investigations, all leading towards the same general conclusion, namely that differences of electric potential play a great part in the phenomenon of cell division, I would mention a very noteworthy paper by Ida H. Hyde [242], in which the writer shews (among other important observations) that not only is there a measurable difference of potential between the animal and vegetative poles of a fertilised egg (Fundulus, toad, turtle, etc.), but that this difference is not constant, but fluctuates, or actually reverses its direction, periodically, at epochs coinciding with successive acts of segmentation or other important phases in the development of the egg [243]; just as other physical rhythms, for instance in the production of CO2, had already been shewn to do. Hence we shall be by no means surprised to find that the “materialised” lines of force, which in the earlier stages form the convergent curves of the spindle, are replaced in the later phases of caryokinesis by divergent curves, indicating that the two foci, which are marked out within the field by the divided and reconstituted nuclei, are now alike in their polarity (Figs. 58, 59).
It is certain, to my mind, that these observations of Miss Hyde’s, and of Lillie’s, taken together with those of many writers on the behaviour of colloid particles generally in their relation to an electrical field, have a close bearing upon the physiological side of our problem, the full discussion of which lies outside our present field.
The break-up of the nucleus, already referred to and ascribed to a diminution of its surface-tension, is accompanied by certain diffusion phenomena which are sometimes visible to the eye; and we are reminded of Lord Kelvin’s view that diffusion is implicitly [p. 189] associated with surface-tension changes, of which the first step is a minute puckering of the surface-skin, a sort of interdigitation with the surrounding medium. For instance, Schewiakoff has observed in Euglypha [244] that, just before the break-up of the nucleus, a system of rays appears, concentred about it, but having nothing to do with the polar asters: and during the existence of this striation, the nucleus enlarges very considerably, evidently by imbibition of fluid from the surrounding protoplasm. In short, diffusion is at work, hand in hand with, and as it were in opposition to, the surface-tensions which define the nucleus. By diffusion, hand in hand with surface-tension, the alveoli of the nuclear meshwork are formed, enlarged, and finally ruptured: diffusion sets up the movements which give rise to the appearance of rays, or striae, around the nucleus: and through increasing diffusion, and weakening surface-tension, the rounded outline of the nucleus finally disappears. [p. 190]
1917 Source Illustration
Fig. 58. Final stage in the first segmentation of the egg of Cerebratulus. (From Prenant, after Coe.) [245]
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Fig. 59. Diagram of field of force with two similar poles.
As we study these manifold phenomena, in the individual cases of particular plants and animals, we recognise a close identity of type, coupled with almost endless variation of specific detail; and in particular, the order of succession in which certain of the phenomena occur is variable and irregular. The precise order of the phenomena, the time of longitudinal and of transverse fission of the chromatin thread, of the break-up of the nuclear wall, and so forth, will depend upon various minor contingencies and “interferences.” And it is worthy of particular note that these variations, in the order of events and in other subordinate details, while doubtless attributable to specific physical conditions, would seem to be without any obvious classificatory value or other biological significance [246].
As regards the actual mechanical division of the cell into two halves, we shall see presently that, in certain cases, such as that of a long cylindrical filament, surface-tension, and what is known as the principle of “minimal area,” go a long way to explain the mechanical process of division; and in all cells whatsoever, the process of division must somehow be explained as the result of a conflict between surface-tension and its opposing forces. But in such a case as our spherical cell, it is not very easy to see what physical cause is at work to disturb its equilibrium and its integrity.
The fact that, when actual division of the cell takes place, it does so at right angles to the polar axis and precisely in the direction of the equatorial plane, would lead us to suspect that the new surface formed in the equatorial plane sets up an annular tension, directed inwards, where it meets the outer surface layer of the cell itself. But at this point, the problem becomes more complicated. Before we could hope to comprehend it, we should have not only to enquire into the potential distribution at the surface of the cell in relation to that which we have seen to exist in its interior, but we should probably also have to take account of the differences of potential which the material arrangements along the lines of force must themselves tend to produce. Only [p. 191] thus could we approach a comprehension of the balance of forces which cohesion, friction, capillarity and electrical distribution combine to set up.
The manner in which we regard the phenomenon would seem to turn, in great measure, upon whether or no we are justified in assuming that, in the liquid surface-film of a minute spherical cell, local, and symmetrically localised, differences of surface-tension are likely to occur. If not, then changes in the conformation of the cell such as lead immediately to its division must be ascribed not to local changes in its surface-tension, but rather to direct changes in internal pressure, or to mechanical forces due to an induced surface-distribution of electrical potential.
It has seemed otherwise to many writers, and we have a number of theories of cell division which are all based directly on inequalities or asymmetry of surface-tension. For instance, Bütschli suggested, some forty years ago [247], that cell division is brought about by an increase of surface-tension in the equatorial region of the cell. This explanation, however, can scarcely hold; for it would seem that such an increase of surface-tension in the equatorial plane would lead to the cell becoming flattened out into a disc, with a sharply curved equatorial edge, and to a streaming of material towards the equator. In 1895, Loeb shewed that the streaming went on from the equator towards the divided nuclei, and he supposed that the violence of these streaming movements brought about actual division of the cell: a hypothesis which was adopted by many other physiologists [248]. This streaming movement would suggest, as Robertson has pointed out, a diminution of surface-tension in the region of the equator. Now Quincke has shewn that the formation of soaps at the surface of an oil-droplet results in a diminution of the surface-tension of the latter; and that if the saponification be local, that part of the surface tends to spread. By laying a thread moistened with a dilute solution of caustic alkali, or even merely smeared with soap, across a drop of oil, Robertson has further shewn that the drop at once divides into two: the edges of the drop, that is to say the ends of the [p. 192] diameter across which the thread lies, recede from the thread, so forming a notch at each end of the diameter, while violent streaming motions are set up at the surface, away from the thread in the direction of the two opposite poles. Robertson [249] suggests, accordingly, that the division of the cell is actually brought about by a lowering of the equatorial surface-tension, and that this in turn is due to a chemical action, such as a liberation of cholin, or of soaps of cholin, through the splitting of lecithin in nuclear synthesis.
But purely chemical changes are not of necessity the fundamental cause of alteration in the surface-tension of the egg, for the action of electrolytes on surface-tension is now well known and easily demonstrated. So, according to other views than those with which we have been dealing, electrical charges are sufficient in themselves to account for alterations of surface-tension; while these in turn account for that protoplasmic streaming which, as so many investigators agree, initiates the segmentation of the egg [250]. A great part of our difficulty arises from the fact that in such a case as this the various phenomena are so entangled and apparently concurrent that it is hard to say which initiates another, and to which this or that secondary phenomenon may be considered due. Of recent years the phenomenon of adsorption has been adduced (as we have already briefly said) in order to account for many of the events and appearances which are associated with the asymmetry, and lead towards the division, of the cell. But our short discussion of this phenomenon may be reserved for another chapter.
However, we are not directly concerned here with the phenomena of segmentation or cell division in themselves, except only in so far as visible changes of form are capable of easy and obvious correlation with the play of force. The very fact of “development” indicates that, while it lasts, the equilibrium of the egg is never complete [251]. And we may simply conclude the [p. 193] matter by saying that, if you have caryokinetic figures developing inside the cell, that of itself indicates that the dynamic system and the localised forces arising from it are in continual alteration; and, consequently, changes in the outward configuration of the system are bound to take place.
As regards the phenomena of fertilisation,—of the union of the spermatozoon with the “pronucleus” of the egg,—we might study these also in illustration, up to a certain point, of the polarised forces which are manifestly at work. But we shall merely take, as a single illustration, the paths of the male and female pronuclei, as they travel to their ultimate meeting place.
The spermatozoon, when within a very short distance of the egg-cell, is attracted by it. Of the nature of this attractive force we have no certain knowledge, though we would seem to have a pregnant hint in Loeb’s discovery that, in the neighbourhood of other substances, such even as a fragment, or bead, of glass, the spermatozoon undergoes a similar attraction. But, whatever the force may be, it is one acting normally to the surface of the ovum, and accordingly, after entry, the sperm-nucleus points straight towards the centre of the egg; from the fact that other spermatozoa, subsequent to the first, fail to effect an entry, we may safely conclude that an immediate consequence of the entry of the spermatozoon is an increase in the surface-tension of the egg [252]. Somewhere or other, near or far away, within the egg, lies its own nuclear body, the so-called female pronucleus, and we find after a while that this has fused with the head of the spermatozoon (or male pronucleus), and that the body resulting from their fusion has come to occupy the centre of the egg. This must be due (as Whitman pointed out long ago) to a force of attraction acting between the two bodies, and another force acting upon one or other or both in the direction of the centre of the cell. Did we know the magnitude of these several forces, it would be a very easy task to calculate the precise path which the two pronuclei would follow, leading to conjugation and the central [p. 194] position. As we do not know the magnitude, but only the direction, of these forces we can only make a general statement: (1) the paths of both moving bodies will lie wholly within a plane triangle drawn between the two bodies and the centre of the cell; (2) unless the two bodies happen to lie, to begin with, precisely on a diameter of the cell, their paths until they meet one another will be curved paths, the convexity of the curve being towards the straight line joining the two bodies; (3) the two bodies will meet a little before they reach the centre; and, having met and fused, will travel on to reach the centre in a straight line. The actual study and observation of the path followed is not very easy, owing to the fact that what we usually see is not the path itself, but only a projection of the path upon the plane of the microscope; but the curved path is particularly well seen in the frog’s egg, where the path of the spermatozoon is marked by a little streak of brown pigment, and the fact of the meeting of the pronuclei before reaching the centre has been repeatedly seen by many observers.
The problem is nothing else than a particular case of the famous problem of three bodies, which has so occupied the astronomers; and it is obvious that the foregoing brief description is very far from including all possible cases. Many of these are particularly described in the works of Fol, Roux, Whitman and others [253].
The intracellular phenomena of which we have now spoken have assumed immense importance in biological literature and discussion during the last forty years; but it is open to us to doubt whether they will be found in the end to possess more than a remote and secondary biological significance. Most, if not all of them, would seem to follow immediately and inevitably from very simple assumptions as to the physical constitution of the cell, and from an extremely simple distribution of polarised forces within it. We have already seen that how a thing grows, and what it grows into, is a dynamic and not a merely material problem; so far as the material substance is concerned, it is so only by reason [p. 195] of the chemical, electrical or other forces which are associated with it. But there is another consideration which would lead us to suspect that many features in the structure and configuration of the cell are of very secondary biological importance; and that is, the great variation to which these phenomena are subject in similar or closely related organisms, and the apparent impossibility of correlating them with the peculiarities of the organism as a whole. “Comparative study has shewn that almost every detail of the processes (of mitosis) described above is subject to variation in different forms of cells [254].” A multitude of cells divide to the accompaniment of caryokinetic phenomena; but others do so without any visible caryokinesis at all. Sometimes the polarised field of force is within, sometimes it is adjacent to, and at other times it lies remote from the nucleus. The distribution of potential is very often symmetrical and bipolar, as in the case described; but a less symmetrical distribution often occurs, with the result that we have, for a time at least, numerous centres of force, instead of the two main correlated poles: this is the simple explanation of the numerous stellate figures, or “Strahlungen,” which have been described in certain eggs, such as those of Chaetopterus. In one and the same species of worm (Ascaris megalocephala), one group or two groups of chromosomes may be present. And remarkably constant, in general, as the number of chromosomes in any one species undoubtedly is, yet we must not forget that, in plants and animals alike, the whole range of observed numbers is but a small one; for (as regards the germ-nuclei) few organisms have less than six chromosomes, and fewer still have more than sixteen [255]. In closely related animals, such as various species of Copepods, and even in the same species of worm or insect, the form of the chromosomes, and their arrangement in relation to the nuclear spindle, have been found to differ in the various ways alluded to above. In short, there seem to be strong grounds for believing that these and many similar phenomena are in no way specifically related to the particular organism in which they have [p. 196] been observed, and are not even specially and indisputably connected with the organism as such. They include such manifestations of the physical forces, in their various permutations and combinations, as may also be witnessed, under appropriate conditions, in non-living things.
When we attempt to separate our purely morphological or “purely embryological” studies from physiological and physical investigations, we tend ipso facto to regard each particular structure and configuration as an attribute, or a particular “character,” of this or that particular organism. From this assumption we are apt to go on to the drawing of new conclusions or the framing of new theories as to the ancestral history, the classificatory position, the natural affinities of the several organisms: in fact, to apply our embryological knowledge mainly, and at times exclusively, to the study of phylogeny. When we find, as we are not long of finding, that our phylogenetic hypotheses, as drawn from embryology, become complex and unwieldy, we are nevertheless reluctant to admit that the whole method, with its fundamental postulates, is at fault. And yet nothing short of this would seem to be the case, in regard to the earlier phases at least of embryonic development. All the evidence at hand goes, as it seems to me, to shew that embryological data, prior to and even long after the epoch of segmentation, are essentially a subject for physiological and physical investigation and have but the very slightest link with the problems of systematic or zoological classification. Comparative embryology has its own facts to classify, and its own methods and principles of classification. Thus we may classify eggs according to the presence or absence, the paucity or abundance, of their associated food-yolk, the chromosomes according to their form and their number, the segmentation according to its various “types,” radial, bilateral, spiral, and so forth. But we have little right to expect, and in point of fact we shall very seldom and (as it were) only accidentally find, that these embryological categories coincide with the lines of “natural” or “phylogenetic” classification which have been arrived at by the systematic zoologist.
The cell, which Goodsir spoke of as a “centre of force,” is in [p. 197] reality a “sphere of action” of certain more or less localised forces; and of these, surface-tension is the particular force which is especially responsible for giving to the cell its outline and its morphological individuality. The partially segmented differs from the totally segmented egg, the unicellular Infusorian from the minute multicellular Turbellarian, in the intensity and the range of those surface-tensions which in the one case succeed and in the other fail to form a visible separation between the “cells.” Adam Sedgwick used to call attention to the fact that very often, even in eggs that appear to be totally segmented, it is yet impossible to discover an actual separation or cleavage, through and through between the cells which on the surface of the egg are so clearly delimited; so far and no farther have the physical forces effectuated a visible “cleavage.” The vacuolation of the protoplasm in Actinophrys or Actinosphaerium is due to localised surface-tensions, quite irrespective of the multinuclear nature of the latter organism. In short, the boundary walls due to surface-tension may be present or may be absent with or without the delimination of the other specific fields of force which are usually correlated with these boundaries and with the independent individuality of the cells. What we may safely admit, however, is that one effect of these circumscribed fields of force is usually such a separation or segregation of the protoplasmic constituents, the more fluid from the less fluid and so forth, as to give a field where surface-tension may do its work and bring a visible boundary into being. When the formation of a “surface” is once effected, its physical condition, or phase, will be bound to differ notably from that of the interior of the cell, and under appropriate chemical conditions the formation of an actual cell-wall, cellulose or other, is easily intelligible. To this subject we shall return again, in another chapter.
From the moment that we enter on a dynamical conception of the cell, we perceive that the old debates were in vain as to what visible portions of the cell were active or passive, living or non-living. For the manifestations of force can only be due to the interaction of the various parts, to the transference of energy from one to another. Certain properties may be manifested, certain functions may be carried on, by the protoplasm apart [p. 198] from the nucleus; but the interaction of the two is necessary, that other and more important properties or functions may be manifested. We know, for instance, that portions of an Infusorian are incapable of regenerating lost parts in the absence of a nucleus, while nucleated pieces soon regain the specific form of the organism: and we are told that reproduction by fission cannot be initiated, though apparently all its later steps can be carried on, independently of nuclear action. Nor, as Verworn pointed out, can the nucleus possibly be regarded as the “sole vehicle of inheritance,” since only in the conjunction of cell and nucleus do we find the essentials of cell-life. “Kern und Protoplasma sind nur vereint lebensfähig,” as Nussbaum said. Indeed we may, with E. B. Wilson, go further, and say that “the terms ‘nucleus’ and ‘cell-body’ should probably be regarded as only topographical expressions denoting two differentiated areas in a common structural basis.”
Endless discussion has taken place regarding the centrosome, some holding that it is a specific and essential structure, a permanent corpuscle derived from a similar pre-existing corpuscle, a “fertilising element” in the spermatozoon, a special “organ of cell-division,” a material “dynamic centre” of the cell (as Van Beneden and Boveri call it); while on the other hand, it is pointed out that many cells live and multiply without any visible centrosomes, that a centrosome may disappear and be created anew, and even that under artificial conditions abnormal chemical stimuli may lead to the formation of new centrosomes. We may safely take it that the centrosome, or the “attraction sphere,” is essentially a “centre of force,” and that this dynamic centre may or may not be constituted by (but will be very apt to produce) a concrete and visible concentration of matter.
It is far from correct to say, as is often done, that the cell-wall, or cell-membrane, belongs “to the passive products of protoplasm rather than to the living cell itself”; or to say that in the animal cell, the cell-wall, because it is “slightly developed,” is relatively unimportant compared with the important role which it assumes in plants. On the contrary, it is quite certain that, whether visibly differentiated into a semi-permeable membrane, or merely constituted by a liquid film, the surface of the cell is the seat of [p. 199] important forces, capillary and electrical, which play an essential part in the dynamics of the cell. Even in the thickened, largely solidified cellulose wall of the plant-cell, apart from the mechanical resistances which it affords, the osmotic forces developed in connection with it are of essential importance.
But if the cell acts, after this fashion, as a whole, each part interacting of necessity with the rest, the same is certainly true of the entire multicellular organism: as Schwann said of old, in very precise and adequate words, “the whole organism subsists only by means of the reciprocal action of the single elementary parts [256].”
As Wilson says again, “the physiological autonomy of the individual cell falls into the background … and the apparently composite character which the multicellular organism may exhibit is owing to a secondary distribution of its energies among local centres of action [257].”
It is here that the homology breaks down which is so often drawn, and overdrawn, between the unicellular organism and the individual cell of the metazoon [258].
Whitman, Adam Sedgwick [259], and others have lost no opportunity of warning us against a too literal acceptation of the cell-theory, against the view that the multicellular organism is a colony (or, as Haeckel called it (in the case of the plant), a “republic”) of independent units of life [260]. As Goethe said long ago, “Das lebendige ist zwar in Elemente [p. 200] zerlegt, aber man kann es aus diesen nicht wieder zusammenstellen und beleben;” the dictum of the Cellularpathologie being just the opposite, “Jedes Thier erscheint als eine Summe vitaler Einheiten, von denen jede den vollen Charakter des Lebens an sich trägt.”
Hofmeister and Sachs have taught us that in the plant the growth of the mass, the growth of the organ, is the primary fact, that “cell formation is a phenomenon very general in organic life, but still only of secondary significance.” “Comparative embryology” says Whitman, “reminds us at every turn that the organism dominates cell-formation, using for the same purpose one, several, or many cells, massing its material and directing its movements and shaping its organs, as if cells did not exist [261].” So Rauber declared that, in the whole world of organisms, “das Ganze liefert die Theile, nicht die Theile das Ganze: letzteres setzt die Theile zusammen, nicht diese jenes [262].” And on the botanical side De Bary has summed up the matter in an aphorism, “Die Pflanze bildet Zellen, nicht die Zelle bildet Pflanzen.”
Discussed almost wholly from the concrete, or morphological point of view, the question has for the most part been made to turn on whether actual protoplasmic continuity can be demonstrated between one cell and another, whether the organism be an actual reticulum, or syncytium. But from the dynamical point of view the question is much simpler. We then deal not with material continuity, not with little bridges of connecting protoplasm, but with a continuity of forces, a comprehensive field of force, which runs through and through the entire organism and is by no means restricted in its passage to a protoplasmic continuum. And such a continuous field of force, somehow shaping the whole organism, independently of the number, magnitude and form of the individual cells, which enter, like a froth, into its fabric, seems to me certainly and obviously to exist. As Whitman says, “the fact that physiological unity is not broken by cell-boundaries is confirmed in so many ways that it must be accepted as one of the fundamental truths of biology [263].”