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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 Anthropométrie94, 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 ();) 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 {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 fifty95 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” diagram96. 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 arms, male	% ratio of stature to span
Age	Male	Female	% F ⁄ M	Male	Female	% F ⁄ M
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 up97. 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 grave98.

Side by side with the curve which represents growth in length, or stature, our diagram shows the curve of weight99. 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

Fig. 4. Mean annual increments of stature (), 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.

Annual Increment of Stature (in cm.) from Belgian and American Statistics.
	Belgian (Quetelet, p. 344)	Paris* (Variot et Chau­met, p. 55)	Toronto† (Boas, p. 1547)	Worcester‡, Mass. (Boas, p. 1548)
Age	Height (Boys)	Ann. in­cre­ment	Height	In­cre­ment	Height (Boys)	Var­i­a­bil­i­ty of do. (6)	Ann. in­cre­ment	Ann. in­cre­ment (Boys)	Var­i­a­bil­i­ty of do.	Ann. in­cre­ment (Girls)	Var­i­a­bil­i­ty of do.
Boys	Girls	Boys	Girls
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			Increment
Age	Male	Female		Age	Male	Female
0–1	5·9	5·6		12–13	4·1	3·5
1–2	2·0	2·4		13–14	4·0	3·8
2–3	1·5	1·4		14–15	4·1	3·7
3–4	1·5	1·5		15–16	4·2	3·5
4–5	1·9	1·4		16–17	4·3	3·3
5–6	1·9	1·4		17–18	4·2	3·0
6–7	1·9	1·1		18–19	3·7	2·3
7–8	1·9	1·2		19–20	1·9	1·1
8–9	1·9	2·0		20–21	1·7	1·1
9–10	1·7	2·1		21–22	1·7	0·5
10–11	1·8	2·4		22–23	1·6	0·4
11–12	2·0	3·5		23–24	0·9	−0·2
12–13	4·1	3·5		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 rapidly100 {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.

Fig. 5. Mean annual increments of weight, in man and woman; from Quetelet’s data.

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).

Fig. 6. Percentage ratio, throughout life, of female weight to male; from Quetelet’s data.

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 growth101. 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 Minot102 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 d² L ⁄ dT² , 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.