Chapter III: The Rate of Growth (1)

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” {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]. {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. {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.” {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 {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. {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 {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.

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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 {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 {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.

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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 {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. {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.

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 {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 {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, weight, and span of outstretched arms._

(_After Quetelet_, _pp._ 193, 346.)

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, {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 {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, {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

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.

_Annual Increment of Stature (in cm.) from Belgian and American Statistics._

D: Belgian (Quetelet, p. 344) E: Paris* (Variot et Chaumet, p. 55) F: Toronto† (Boas, p. 1547) G: Worcester‡, Mass. (Boas, p. 1548) H: Ann. increment I: Increment J: Boys K: Girls V: Variability of do.

────[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 — — — — — — — — — — —

* Ages from 1–2, 2–3, etc.

† The epochs are, in this table, 5·5, 6·5, years, etc.

‡ 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 {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.

_Annual Increment of Weight in Man_ (_kgm._).

(After Quetelet, _Anthropométrie_, p. 346*.)

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] {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.

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, {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 {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).

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 {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

Minot would state the percentage growth in each of the four annual periods at 39·6, 13·3, 9·6 and 7·3 per cent. respectively.

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 _d_^2 _L_/_dT_^2, that is to say, the acceleration. {73}

But when you take percentages of _y_, you are determining _dy_/_y_, and when you plot this against _dx_, you have

(_dy_/_y_)/_dx_, or _dy_/(_y_ ⋅ _dx_), or (1/_y_) ⋅ (_dy_/_dx_),

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 {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.

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. {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

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 {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 {77} method of nutrition has its inevitable effect; but this slight temporary set-back is immediately followed by a secondary, and temporary, acceleration.

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, {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.