On Growth and Form

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 yrs. Fig. 4. Mean annual increments of stature (3), Belgian and American. ,and very marked acceleration of growth beginning at about twelve years of age, and lasting for three or four years; when this period of acceleration is over, the rate begins to fall again, and does so with great rapidity. We do not know how far the absence of this striking feature in the Belgian curve is due to the imperfections of Quetelet’s data, or whether it is a real and significant feature in the small-statured race which he investigated.

Even apart from these data of Quetelet’s (which seem to ~ constitute an extreme case), it is evident that there are very — ‘L-9 ‘9-G fo sade oy} usaMyoq :1vak 07 rvok WoIZ oMyRYS JO OSaIOUT ‘past[eNPLATpUr JO ‘[enzov UO suOeAIasqo yoo +f sim = shog sa shog quoUa1ouy JUSTO FT ss Vs (gg d ‘QOUN*BY) 40 JOMRA,) y.SIIeq ‘SOUSYDIY UDriIaMp pun unibjag mors (“md Ur) ainjory fo JuamaiouT jonuup 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*.) 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 +

* 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. 7+ As Haller observed it to do in the chick (Hlem. vu, p. 294): ‘‘Hoe iterum incrementum miro ordine ita distribuitur, ut in principio incubationis maximum est: inde perpetuo minuatur.” during the first two or three years of hfe. 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

O}- foe aie Sees 1 | n 1 n nl | StS CS Se ie Oe ee Le ee ee eee 0 5 10 is 20 years 25 Fig. 5. Mean annual increments of weight, in man and woman; from Quetelet’s data. 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 accelerationcurves 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,

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 stmilar 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 (/.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 bane and girls respectively, and the annual increments being as follows: The result of these differences (which are essentially phasedifferences) 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. Tull 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

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

Fig. 6. Percentage ratio, throughout life, of female weight to male; from Quetelet’s data. same phases of growth*. 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 coldblooded 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 * There is a famous passage in Lucretius (Vv, 883) where he compares the course of life, or rate of growth, in the horse and his boyish master: Principio circum tribus actis impiger annis Floret equus, puer hautquaquam, etc. 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* introduced a curious, and (as it seems to me) an unhappy method of representing erowth, 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:

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/7’, we have dL/dT’, which is (of course) velocity; and from this, by a second differentiation, we obtain d?L/dT?, that is to say, the acceleration. But when you take percentages of y, you are determining dy/y, and when you plot this against dx, you have

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.

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 Riissow’s*, who gives the stature of the infant month by month during the first year of its life, as follows:

If we multiply these monthly differences, or mean monthly velocities, by 12, to bring them into a form comparable with the * Quoted in Vierordt’s Anatomische...Daten und Tabellen, 1906. p. Ss 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 accelerationcurve 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 Fig. 7. Curve of growth (in length or stature) of child, before and after birth. (From His and Riissow’s data.) 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.

According to His*, the following are the mean lengths of the unborn human embryo, from month to month. These data link on very well to those of Riissow, which we have just considered, and (though His’s measurements for the Fig. 8. Mean monthly increments of length or stature of child (in cms.). pre-natal months are more detailed than are those of Riissow 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: up to that date growth proceeds with a continually increasing

+ No such point of inflection appears in the curve of weight according to C. M. Jackson’s data (On the Prenatal Growth of the Human Body, etc., Amer. Journ. of Anat. 1x, 1909, pp. 126 156), nor in those quoted by him from Ahlfeld, 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

Fig. 9. Curve of pre-natal growth (length or stature) of child; and corresponding curve of mean monthly increments (mm.). Fehling and others. But it is plain that the very rapid increase of the monthly weights, approximately in the ratio of the cubes of the corresponding lengths, would tend to conceal any such breach of continuity, unless it happened to be very marked indeed. Moreover in the case of Jackson’s data (and probably also in the others) the actual age of the embryos was not determined, but was estimated from their lengths. The following is Jackson’s estimate of average weights at intervals of a lunar month: Months 0 ya t24 53) rt: 5 6 7 8 9 10 Wtin gms. 0 -04 3 36 120 330 600 1000 1500 2200 3200

method of nutrition has its inevitable effect; but this slight temporary set-back is ew 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

days Fig. 10. Curve of growth of bamboo (from Ostwald, after Kraus), 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*, we see (so far as it goes) the same essential features,

the slow beginning, the rapid increase of velocity, the point of inflection, and the subsequent slow negative acceleration *. , 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 le far beyond the scope of this elementary book; but we:may very briefly illustrate the nature of the process, in connection with the phenomena of growth which we have just been studying.

It was in connection with these phenomena, in the case of man, that Quetelet first conceived the statistical study of variation, on lines which were afterwards expounded and developed by Galton, and which have grown, in the hands of Karl Pearson and others, into the modern science of Biometrics. When Quetelet tells us, for instance, that the mean stature of the ten-year old boy is 1-273 metres, this implies, according to the law of error, or law of probabilities, that all the individual measurements of ten-year-old boys group themselves in an orderly way, that is to say according to a certain definite law, about this mean value of 1-273. When these individual measurements are grouped and plotted as a curve, so as to show the number of individual cases at each individual length, we obtain a characteristic curve of error or curve of frequency; and the “spread” of this curve is a measure of the amount of variability in this particular case. A certain mathematical measure of this “spread,” as described in works upon statistics, is called the Index of Variability, or Standard Deviation, and is usually denominated by the letter o. It is practically equivalent to a determination of the point upon the frequency curve where it changes its curvature on either side of the mean, and where, from being concave towards the middle line, it spreads out to be convex thereto. When we divide this

value by the mean, we get a figure which is independent of any particular units, and which is called the Coefficient of Variability. (It is usually multiplied by 100, to make it of a more convenient amount; and we may then define this coefficient, CU, as = o/M x 100.) In regard to the growth of man, Pearson has determined this coefliicient of variability as follows: in male new-born infants, the coefficient in regard to weight is 15-66, and in regard to stature, 6:50; in male adults, for weight 10-83, and for stature, 3°66. The amount of variability tends, therefore, to decrease with growth or age.

Similar determinations have been elaborated by Bowditch, by Boas and Wissler, and by other writers for intermediate ages, especially from about five years old to eighteen, so covering a great part of the whole period of growth in man*. Coefficient of Variability (o/M x 100) in Man, at various ages. Age 5 6 sy 8 9 The result is very curious indeed. We see, from Fig. 11, that the curve of variability is very similar to what we have called the acceleration-curve (Fig. 4): that is to say, it descends when the rate of growth diminishes, and rises very markedly again when, in late boyhood, the rate of growth is temporarily accelerated. We

see, in short, that the amount of variability in stature or in weight is a function of the rate of growth in these magnitudes, though we are not yet in a position to equate the terms precisely, one with another. If we take not merely the variability of stature or weight at a given age, but the variability of the actual successive increments in each yearly period, we see that this latter coefficient of variability tends to increase steadily, and more and more rapidly, within

yrs. Fig. 11. Coefficients of variability of stature in Man (3). from Boas and Wissler’s data. the limits of age for which we have information; and this phenomenon is, in the main, easy of explanation. For a great part of the difference, in regard to rate of growth, between one individual and another is a difference of phase,—a difference in the epochs of acceleration and retardation, and finally in the epoch when growth comes to an end. And it follows that the variability of rate will be more and more marked, as we approach and reach the period when some individuals still continue, and others have aiready ceased, to grow. In the following epitomised table,

I have taken Boas’s determinations of variability (co) (op. cit. . 1548), converted them into the corresponding coefficients of variability (o/M x 100), and then smoothed the resulting numbers. Coefficients of Variability in Annual. Increment of Stature. The greater variability of annual increment in the girls, as conipared with the boys, is very marked, and is easily explained by the more rapid rate at which the girls run through the several phases of the phenomenon.

Just as there is a marked difference in “phase” between the growthcurves of the tworsexes, that is to say a difference in the periods when growth is rapid or the reverse, so also, within each sex, will there be room for similar, but individual phase-differences. Thus we may have children of accelerated development, who at a given epoch after birth are both rapidly growing and already “‘big for their age”; and others of retarded development who are comparatively small and have not reached the period of acceleration which, in greater or less degree, will come to them in turn. In other words, there must under such circumstances be a strong positive “coefficient of correlation” between stature and rate of growth, and also between the rate of growth in one year and the next. But it does not by any means follow that a child who is precociously big will continue to grow rapidly, and become a man or woman of exceptional stature. On the contrary, when in the case of the precocious or ‘‘accelerated”’ children growth has begun to slow down, the backward ones may still be growing rapidly, and so making up (more or less completely) to the others. In other words, the period of high positive correlation between stature and increment will tend to be followed by one of negative correlation. This interesting and important point, due to Boas and Wissler*, is confirmed by the following table :—