On Growth And Form - free ebook by D'Arcy Wentworth Thompson

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On Growth And Form

D'Arcy Wentworth Thompson

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GROWTH AND FORM

GROWTH AND FORM

“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....

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Prefatory Note.

Prefatory Note.

This book of mine has little need of preface, for indeed it is “all preface” from beginning to end. I have written it as an easy introduction to the study of organic Form, by methods which are the common-places of physical science, which are by no means novel in their application to natural history, but which nevertheless naturalists are little accustomed to employ. It is not the biologist with an inkling of mathematics, but the skilled and learned mathematician who must ultimately deal with suc

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I. Introductory.

I. Introductory.

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 possibl

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II. On Magnitude.

II. On Magnitude.

To terms of magnitude, and of direction, must we refer all our conceptions of form. For the form of an object is defined when we know its magnitude, actual or relative, in various directions; and growth involves the same 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 o

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The rate of growth in Man.

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

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Pre-natal and post-natal growth.

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 physic

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Variability and Correlation of Growth.

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

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Rate of growth in other organisms.

Rate of growth in other organisms.

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 i

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The rate of growth of various parts or organs.

The rate of growth of various parts or organs.

The differences in regard to rate of growth between various parts or organs of the body, internal and external, can be amply illustrated in the case of man, and also, but chiefly in regard to external form, in some few other 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

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The effect of temperature*.

The effect of temperature*.

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 stat

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Osmotic factors in growth.

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 wit

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Growth and catalytic action.

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

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Regeneration, or growth and repair.

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 T

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CONCLUSION AND SUMMARY.

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, demonstrat

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IV. On the Internal Form and Structure of the Cell.

IV. On the Internal Form and Structure of the Cell.

In the early days of the cell-theory, more than seventy years ago, Goodsir was wont to speak of cells as “centres of growth” or “centres of nutrition,” and to consider them as essentially “centres of force.” He looked forward to a time when the forces connected with the cell should be particularly investigated: when, that is to say, minute anatomy should be studied in its dynamical aspect. “When this branch of enquiry,” he says “shall have been opened up, we shall expect to have a science of org

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V. The Forms of Cells

V. The Forms of Cells

Protoplasm, as we have already said, is a fluid or rather a semifluid substance, and we need not pause here to attempt to describe the particular properties of the semifluid, colloid, or jelly-like substances to which it is allied; we should find it no easy matter. Nor need we appeal to precise theoretical definitions of fluidity, lest we come into a debateable land. It is in the most general sense that protoplasm is “fluid.” As Graham said (of colloid matter in general), “its softness partakes

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VI. A Note on Adsorption.

VI. A Note on Adsorption.

A very important corollary to, or amplification of the theory of surface tension is to be found in the modern chemico-physical doctrine of Adsorption 324 . In its full statement this subject soon becomes complicated, and involves physical conceptions and mathematical treatment which go beyond our range. But it is necessary for us to take account of the phenomenon, though it be in the most elementary way. In the brief account of the theory of surface tension with which our last chapter began,

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VII. The Forms of Tissues Or Cell-aggregates.

VII. The Forms of Tissues Or Cell-aggregates.

We now pass from the consideration of the solitary cell to that of cells in contact with one another,—to what we may call in the first instance “cell-aggregates,”—through which we shall be led ultimately to the study of complex tissues. In this part of our subject, as in the preceding chapters, we shall have to give some consideration to the effects of various forces; but, as in the case of the conformation of the solitary cell, we shall probably find, and we may at least begin by assuming, that

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VIII. The Forms of Tissues or Cell-aggregates (continued)

VIII. The Forms of Tissues or Cell-aggregates (continued)

The problems which we have been considering, and especially that of the bee’s cell, belong to a class of “isoperimetrical” problems, which deal with figures whose surface is a minimum for a definite content or volume. Such problems soon become difficult, but we may find many easy examples which lead us towards the explanation of biological phenomena; and the particular subject which we shall find most easy of approach is that of the division, in definite proportions, of some definite portion of

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IX. On Concretions, Spicules, and Spicular Skeletons.

IX. On Concretions, Spicules, and Spicular Skeletons.

The deposition of inorganic material in the living body, usually in the form of calcium salts or of silica, is a very common and wide-spread phenomenon. It begins in simple ways, by the appearance of small isolated particles, crystalline or non-crystalline, whose form has little relation or sometimes none to the structure of the organism; it culminates in the complex skeletons of the vertebrate animals, in the massive skeletons of the corals, or in the polished, sculptured and mathematically

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X. A Parenthetic Note on Geodetics.

X. A Parenthetic Note on Geodetics.

We have made use in the last chapter of the mathematical principle of Geodetics (or Geodesics) in order to explain the conformation of a certain class of sponge-spicules; but the principle is of much wider application in morphology, and would seem to deserve attention which it has not yet received. Defining, meanwhile, our geodetic line (as we have already done) as the shortest distance between two points on the surface of a solid of revolution, we find that the geodetics of the cylinder giv

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The Univalve Shell: a summary.

The Univalve Shell: a summary.

The surface of any shell, whether discoid or turbinate, may be imagined to be generated by the revolution about a fixed axis of a closed curve, which, remaining always geometrically similar to itself, increases continually its dimensions: and, since the rate of growth of the generating curve and its velocity of rotation follow the same law, the curve traced in space by corresponding points {554} in the generating curve is, in all cases, a logarithmic spiral. In discoid shells, the generating

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Of Bivalve Shells.

Of Bivalve Shells.

Hitherto we have dealt only with univalve shells, and it is in these that all the mathematical problems connected with the spiral, or helico-spiral, are best illustrated. But the case of the bivalve shell, of Lamellibranchs or of Brachiopods, presents no essential difference, save only that we have here to do with two conjugate spirals, whose two axes have a definite relation to one another, and some freedom of rotatory movement relatively to one another. The generating curve is particularly

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The Shells of Pteropods.

The Shells of Pteropods.

While mathematically speaking we are entitled to look upon the bivalve shell of the Lamellibranch as consisting of two distinct elements, each comparable to the entire shell of the univalve, we have no biological grounds for such a statement; for the shell arises from a single embryonic origin, and afterwards becomes split into portions which constitute the two separate valves. We can perhaps throw some indirect light upon this phenomenon, and upon several other phenomena connected with shel

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Of Septa.

Of Septa.

Before we leave the subject of the molluscan shell, we have still another problem to deal with, in regard to the form and arrangement of the septa which divide up the tubular shell into chambers, in the Nautilus, the Ammonite and their allies (Fig. 304 , etc.). The existence of septa in a Nautiloid shell may probably be accounted for as follows. We have seen that it is a property of a cone that, while growing by increments at one end only, it conserves its original shape: therefore the animal wi

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Conclusion.

Conclusion.

If we can comprehend and interpret on some such lines as these the form and mode of growth of the foraminiferal shell, we may also begin to understand two striking features of the group, namely, on the one hand the large number of diverse types or families which exist and the large number of species and varieties within each, and on the other the persistence of forms which in many cases seem to have undergone little change or none at all from the Cretaceous or even from earlier periods to the pr

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A further Note upon Torsion.

A further Note upon Torsion.

The phenomenon of torsion, to which we have been thus introduced, opens up many wide questions in connection with form. Some of the associated phenomena are admirably illustrated in the case of climbing plants; but we can only deal with these still more briefly and parenthetically. The subject of climbing plants has been elaborately dealt with not only in Darwin’s books 566 , but also by a very large number of earlier and later writers. In “twining” plants, which constitute the greater number of

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Of Deer’s Antlers.

Of Deer’s Antlers.

But let us return to our subject of the shapes of horns, and consider briefly our last class of these structures, namely the bony antlers of the various species of elk and deer 569 . The problems which these present to us are very different from those which we have had to do with in the antelope or the sheep. With regard to its structure, it is plain that the bony antler corresponds, upon the whole, to the bony core of the antelope’s horn; while in place of the hard horny sheath of the latter, w

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Of Teeth, and of Beak and Claw.

Of Teeth, and of Beak and Claw.

In a fashion similar to that manifested in the shell or the horn, we find the logarithmic spiral to be implicit in a great many other organic structures where the phenomena of growth proceed in a similar way: that is to say, where about an axis there is some asymmetry leading to unequal rates of longitudinal growth, and where the structure is of such a kind that each new increment is added on as a permanent and unchanging part of the entire conformation. Nail and claw, beak and tooth, all come u

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XIV. On Leaf-arrangement, Or Phyllotaxis.

XIV. On Leaf-arrangement, Or Phyllotaxis.

The beautiful configurations produced by the orderly arrangement of leaves or florets on a stem have long been an object of admiration and curiosity. Leonardo da Vinci would seem, as Sir Theodore Cook tells us, to have been the first to record his thoughts upon this subject; but the old Greek and Egyptian geometers are not likely to have left unstudied or unobserved the spiral traces of the leaves upon a palm-stem, or the spiral curves of the petals of a lotus or the florets in a sunfl

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On the Form of Sea-urchins

On the Form of Sea-urchins

As a corollary to the problem of the bird’s egg, we may consider for a moment the forms assumed by the shells of the sea-urchins. These latter are commonly divided into two classes, the Regular and the Irregular Echinids. The regular sea-urchins, save in {662} slight details which do not affect our problem, have a complete radial symmetry. The axis of the animal’s body is vertical, with mouth below and the intestinal outlet above; and around this axis the shell is built as a symmetrical system.

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On the Form and Branching of Blood-vessels

On the Form and Branching of Blood-vessels

Passing to what may seem a very different subject, we may investigate a number of interesting points in connection with the form and structure of the blood-vessels, on the same principle and by help of the same equations as those we have used, for instance, in studying the egg-shell. We know that the fluid pressure ( P ) within the vessel is balanced by (1) the tension ( T ) of the wall, divided by the radius of curvature, and (2) the external pressure ( p n ), normal to the wall: according

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XVI. On Form and Mechanical Efficiency.

XVI. On Form and Mechanical Efficiency.

There is a certain large class of morphological problems of which we have not yet spoken, and of which we shall be able to say but little. Nevertheless they are so important, so full of deep theoretical significance, and are so bound up with the general question of form and of its determination as a result of growth, that an essay on growth and form is bound to take account of them, however imperfectly and briefly. The phenomena which I have in mind are just those many cases where adaptation , i

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XVII. On the Theory of Transformations, Or the Comparison of Related Forms.

XVII. On the Theory of Transformations, Or the Comparison of Related Forms.

In the foregoing chapters of this book we have attempted to study the inter-relations of growth and form, and the part which certain of the physical forces play in this complex interaction; and, as part of the same enquiry, we have tried in comparatively simple cases to use mathematical methods and mathematical terminology in order to describe and define the forms of organisms. We have learned in so doing that our own study of organic form, which we call by Goethe’s name of Morphology, i

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Epilogue.

Epilogue.

In the beginning of this book I said that its scope and treatment were of so prefatory a kind that of other preface it had no need; and now, for the same reason, with no formal and elaborate conclusion do I bring it to a close. The fact that I set little store by certain postulates (often deemed to be fundamental) of our present-day biology the reader will have discovered and I have not endeavoured to conceal. But it is not for the sake of polemical argument that I have written, and the doctrine

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NOTES:

NOTES:

1 These sayings of Kant and of Du Bois, and others like to them, have been the text of many discourses: see, for instance, Stallo’s Concepts , p. 21, 1882; Höber, Biol. Centralbl. XIX, p. 284, 1890, etc. Cf. also Jellett, Rep. Brit. Ass. 1874, p. 1. 2 “Quum enim mundi universi fabrica sit perfectissima, atque a Creatore sapientissimo absoluta, nihil omnino in mundo contingit in quo non maximi minimive ratio quaepiam eluceat; quamobrem dubium prorsus est nullum quin omnes mundi effectus ex causis

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