On Growth and Form

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 1s ultimately due to the action of molecular forces ; but in the one case the movements of the particles of matter le 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.

The difference between the two classes gf 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*, 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.” * Cf. Aristotle, Phys. vi, 5,235 a 11, émel yap amaca ktéyynows év xpdvm, KT. Bacon emphasised, in like manner, the fact that “all motion or natural action is performed in time: some more quickly, some more slowly, but all in periods determined and fixed in the nature of things. Even those actions which seem

to be performed suddenly, and (as we say) in the twinkling of an eye, are found to admit of degree in respect of duration.” Nov. Org. XLVI. 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 httle 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,’ m which no part came into being which had not essentially existed before *. 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 altermg 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

* Of. (e.g.) Blem. Physiol. ed. 1766, vu, p. 114, “Ducimur autem ad evolutionem potissimum, quando a perfecto animale retrorsum progredimur, et incrementorum atque mutationum seriem relegimus. Ita inveniemus perfectum illud animal fuisse imperfectius, alterius figurae et fabricae, et denique rude et informe: et tamen idem semper animal sub iis diversis phasibus fuisse, quae absque ullo saltu perpetuos parvosque per gradus cohaereant.” 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 Diirer 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*, 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 outt, 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. * Beitrage zur Entwickelungsgeschichte des Hiihnchens im Hi, p. 40, 1817. Roux ascribes the same views also to Von Baer and to R. H. Lotze (Allg. Physiologie, p- 353, 1851). 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)*, and pressure, and the more subtle influences which he denominates wis derivationis et revulsionist: 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 srowth 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. 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§.”’ For Hertwig, therefore, as

* Op. cit. p. 302, **Magnum hoc naturae instrumentum, etiam in corpore animato evolvendo potenter operatur; etc.” + Ibid. p. 306. “Subtiliora ista, et aliquantum hypothesi mista, tamen magnum mihi videntur speciem veri habere.” { Cf. His, On the Principles of Animal Morphology, Proc. R. S. EH. xv, 1888, p. 294: ““My own attempts to introduce some elementary mechanical or physiological conceptions into embryology have not generally been agreed to by morphologists. -To one it seemed ridiculous to speak of the elasticity of the germinal layers; another thought that, by such considerations, we ‘put the cart before the horse’: and one more recent author states, that we have better things to do in embryology than to discuss tensions of germinal layers and similar questions, since all explanations must of necessity be of a phylogenetic nature. This opposition to the application of the fundamental principles of science to embryological questions would scarcely be intelligible had it not a dogmatic background. No other explanation of living forms is allowed than heredity, and any which is founded on another basis must be rejected....... To think that heredity will build organic beings without mechanical means is a piece of unscientific mysticism.”

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*. But in the main, Balfour, hike 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 organsy.”

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 triedt, 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§, 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

§ Ist ed. p. 444; 6th ed. p. 390. The student should not fail to consult the passage in question; for there is always a risk of misunderstanding or misinterpretation when one attempts to epitomise Darwin’s carefully condensed arguments. 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

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*. 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

* “Tn omni rerum naturalium historia utile est mensuras definiri et numeros,” Haller, Elem. Physiol. m1, p. 258, 1760. Cf. Hales, Vegetable Staticks, Introduction. 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. 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*, 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.

Fig. 3. Curve of Growth in Man, from birth to 20 yrs (3); 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 * Brussels, 1871. Cf. the same author’s Physique sociale, 1835, and Lettres sur la théorie des probabilités, 1846. See also, for the general subject, Boyd, R., Tables of weights of the Human Body, ete. Phil. Trans. vol. cit, 1861; Roberts,

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 itis, 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* 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

* Dr Johnson was not far wrong in saying that “‘life declines from thirty-five” ; though the Autocrat of the Breakfast-table, like Cicero, declares that “the furnace is in full blast for ten years longer.” of growth is in fact a diagram of activity, or “‘time-energy”’ diagram*. As the organism grows it is absorbing energy beyond its daily needs, and accumulating it at a rate depicted in our Stature, weight, and span of outstretched arms. (After Quetelet, pp. 193, 346.) Stature in metres Weight in kgm. Span of % ratio

SSS (ee SS) arms, of stature Male Female %F/M Male Female %F/M _ male to span 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, which draws us down when we would fain rise up*. 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 ft.

Side by side with the curve which repiesents growth in length, or stature, our diagram shows the curve of weightt. 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 lmear 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 lnear dimensions are varying somewhat differently from length or stature, and that consequently the growth in bulk or weight is following a more complicated Jaw.

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 hes 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

* « Tou pes, méstre de tout [Le poids, maitre de tout], méstre sénso vergougno, Que te tirasso en bas de sa brutalo pougno,” J. H. Fabre, Oubreto prouvengalo, p. 61. + The continuity of the phenomenon of growth, and the natural passage from the phase of increase to that of decrease or decay, are admirably discussed by Enriques, in “‘ La morte,” Riv. di Scienza, 1907, and in “‘Wachsthum und seine analytische Darstellung,” Biol. Centralbl. June, 1909. Haller (Hlem. vu, p. 68) recognised decrementum as a phase of growth, not less important (theoretically) than incrementum: ‘“‘tristis, sed copiosa, haec est materies.”

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 js 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 vears of life the fall is continuous and rapid,