D'Arcy Wentworth Thompson

“ In the case of the growing embryo we know from the beginning that surface tension is only one of the physical forces at work; and that other forces, including those displayed within the interior of each living cell, play their part in the determination of the system. But we have no evidence whatsoever that at this point, or that point, or at any, the dominion of the physical forces over the material system gives place to a new condition where agencies at present unknown to the physicist impose themselves on the living matter, and become responsible for the conformation of its material fabric. ”

“ 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. ”

“ 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 ”

“ In it is reflected a field of force: and keeping pace, as it were, in action and interaction with this field of force, the whole skeleton and every part thereof, down to the minute intrinsic structure of the bones themselves, is related in form and in position to the lines of force, to the resistances it has to encounter; for by one of the mysteries of biology, resistance begets resistance, and where pressure falls there growth springs up in strength to meet it. And, pursuing the same train of thought, we see that all this is true not of the skeleton alone but of the whole fabric of the body. ”

“ 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 ”

“ 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 exist261.” ”

“ 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. ”

“ The living organism represents, or occupies, a field of force which is never simple, and which as a rule is of immense complexity. And just as in the very simplest of actual cases we meet with a departure from such symmetry as could only exist under conditions of ideal simplicity, so do we pass quickly to cases where the interference of numerous, though still perhaps very simple, causes leads to a resultant which lies far beyond our powers of analysis. Nor must we forget that the biologist is much more exacting in his requirements, as regards form, than the physicist ”

“ Partly associated with the same phenomenon, and partly to be looked upon (meanwhile at least) as a fact apart, is the very important physiological truth that a condition of strain, the result of a stress, is a direct stimulus to growth itself. This indeed is no less than one of the cardinal facts of theoretical biology. The soles of our boots wear thin, but the soles of our feet grow thick, the more we walk upon them: for it would seem that the living cells are “stimulated” by pressure, or by what we call “exercise,” to increase and multiply. ”

“ 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. ”

“ In the great majority of cases, when we consider an organism in part or whole, when we look (for instance) at our own hand or foot, or contemplate an insect or a worm, we have no reason (or very little) to consider one part of the existing structure as older than another; through and through, the newer particles have been merged and commingled, by intussusception, among the old; the whole outline, such as it is, is due to forces which for the most part are still at work to shape it, and which in shaping it have shaped it as a whole. But the horn, or the snail-shell, is curiously different ”

“ 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 God hath bestowed upon this humble architect the practical skill of a learned geometrician, and he makes this provision with admirable precision in that curvature of the logarithmic spiral which he gives to the section of the shell. This curvature obtaining, he has only to turn his operculum slightly round in its own plane as he advances it into each newly formed portion of his chamber, to adapt one margin of it to a new and larger surface and a different curvature, leaving the space to be filled up by increasing the operculum wholly on the other margin. ”

“ Let us now imagine a spicule, whose natural tendency is to grow into a straight linear element, either by reason of its own molecular anisotropy, or because it is deposited about a thread-like axis; and let us suppose that it is confined either within a cell-wall or in adhesion thereto; it at once follows that its line of growth will be simply a geodetic to the surface of the cell. And if the cell be an imperfect sphere, or a more or less regular ellipsoid, the spicule will tend to grow into one or other of three forms: either a plane curve of circular arc ”

“ 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. ”

“ 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. ”

“ 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. ”

“ It is evident that by the combined action of appropriate forces any material form can be transformed into any other: just as out of a “shapeless” mass of clay the potter or the sculptor models his artistic product; or just as we attribute to Nature herself the power to effect the gradual and successive transformation of the simplest into the most complex organism. In like manner it is possible, at least theoretically, to cause the outline of any closed curve to appear as a projection of any other whatsoever. ”

“ Wherever the tooth is very long indeed, as in the mammoth or the beaver, the effect of some slight and all but inevitable lateral asymmetry in the rate of growth begins to shew itself: in other words, the spiral is seen to lie not absolutely in a plane, but to be a curve of double curvature, like a twisted horn. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 we also saw, according to Gauss’s equation, that the potential energy of the system will be diminished (and its diminution will accordingly be manifested in work) if from any cause the specific surface energy be diminished, that is to say if it be brought more nearly to an equality with the specific energy of the molecules in the interior of the liquid mass. This latter is a phenomenon of great moment in modern physiology, and, while we need not attempt to deal with it in detail, it has a bearing on cell-form and cell-structure which we cannot afford to overlook. ”