Читать книгу On Growth and Form - D'Arcy Wentworth Thompson - Страница 13

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The differences in regard to rate of growth between various parts or organs of the body, internal and external, can be amply illustrated in the case of man, and also, but chiefly in regard to external form, in some few other creatures118. It is obvious that there lies herein an endless field for the math­e­mat­i­cal study of correlation and of variability, but with this aspect of the case we cannot deal.

In the accompanying table, I shew, from some of Vierordt’s data, the relative weights, at various ages, compared with the weight at birth, of the entire body, of the brain, heart and liver; {89} and also the percentage relation which each of these organs bears, at the several ages, to the weight of the whole body.

Weight of Various Organs, compared with the Total Weight of the Human Body (male). (After Vierordt, Anatom. Tabellen, pp. 38, 39.)
	Weight of body†	Relative weights of	Percentage weights compared with total body-weights
Age	in kg.	Body	Brain	Heart	Liver	Body	Brain	Heart	Liver
0	3·1	1	1	1	1	100	12·29	0·76	4·57
1	9·0	2·90	2·48	1·75	2·35	100	10·50	0·46	3·70
2	11·0	3·55	2·69	2·20	3·02	100	9·32	0·47	3·89
3	12·5	4·03	2·91	2·75	3·42	100	8·86	0·52	3·88
4	14·0	4·52	3·49	3·14	4·15	100	9·50	0·53	4·20
5	15·9	5·13	3·32	3·43	3·80	100	7·94	0·51	3·39
6	17·8	5·74	3·57	3·60	4·34	100	7·63	0·48	3·45
7	19·7	6·35	3·54	3·95	4·86	100	6·84	0·47	3·49
8	21·6	6·97	3·62	4·02	4·59	100	6·38	0·44	3·01
9	23·5	7·58	3·74	4·59	4·95	100	6·06	0·46	2·99
10	25·2	8·13	3·70	5·41	5·90	100	5·59	0·51	3·32
11	27·0	8·71	3·57	5·97	6·14	100	5·04	0·52	3·22
12	29·0	9·35	3·78	(4·13)	6·21	100	4·88	(0·34)	3·03
13	33·1	10·68	3·90	6·95	7·31	100	4·49	0·50	3·13
14	37·1	11·97	3·38	9·16	8·39	100	3·47	0·58	3·20
15	41·2	13·29	3·91	8·45	9·22	100	3·62	0·48	3·17
16	45·9	14·81	3·77	9·76	9·45	100	3·16	0·51	2·95
17	49·7	16·03	3·70	10·63	10·46	100	2·84	0·51	2·98
18	53·9	17·39	3·73	10·33	10·65	100	2·64	0·46	2·80
19	57·6	18·58	3·67	11·42	11·61	100	2·43	0·51	2·86
20	59·5	19·19	3·79	12·94	11·01	100	2·43	0·51	2·62
21	61·2	19·74	3·71	12·59	11·48	100	2·31	0·49	2·66
22	62·9	20·29	3·54	13·24	11·82	100	2·14	0·50	2·66
23	64·5	20·81	3·66	12·42	10·79	100	2·16	0·46	2·37
24	—	—	3·74	13·09	13·04	100	—	—	—
25	66·2	21·36	3·76	12·74	12·84	100	2·16	0·46	2·75

† From Quetelet.

From the first portion of the table, it will be seen that none of these organs by any means keep pace with the body as a whole in regard to growth in weight; in other words, there must be some other part of the fabric, doubtless the muscles and the bones, which increase more rapidly than the average increase of the body. Heart and liver both grow nearly at the same rate, and by the {90} age of twenty-five they have multiplied their weight at birth by about thirteen times, while the weight of the entire body has been multiplied by about twenty-one; but the weight of the brain has meanwhile been multiplied only about three and a quarter times. In the next place, we see the very remarkable phenomenon that the brain, growing rapidly till the child is about four years old, then grows more much slowly till about eight or nine years old, and after that time there is scarcely any further perceptible increase. These phenomena are dia­gram­ma­ti­cally illustrated in Fig. 18.

Fig. 18. Relative growth in weight (in Man) of Brain, Heart, and whole Body.

Many statistics indicate a decrease of brain-weight during adult life. Boas119 was inclined to attribute this apparent phenomenon to our statistical methods, and to hold that it could “hardly be explained in any other way than by assuming an increased death-rate among men with very large brains, at an age of about twenty years.” But Raymond Pearl has shewn that there is evidence of a steady and very gradual decline in the weight of the brain with advancing age, beginning at or before the twentieth year, and continuing throughout adult life120. {91}

The second part of the table shews the steadily decreasing weights of the organs in question as compared with the body; the brain falling from over 12 per cent. at birth to little over 2 per cent. at five and twenty; the heart from ·75 to ·46 per cent.; and the liver from 4·57 to 2·75 per cent. of the whole bodily weight.

It is plain, then, that there is no simple and direct relation, holding good throughout life, between the size of the body as a whole and that of the organs we have just discussed; and the changing ratio of magnitude is especially marked in the case of the brain, which, as we have just seen, constitutes about one-eighth of the whole bodily weight at birth, and but one-fiftieth at five and twenty. The same change of ratio is observed in other animals, in equal or even greater degree. For instance, Max Weber121 tells us that in the lion, at five weeks, four months, eleven months, and lastly when full-grown, the brain-weight represents the following fractions of the weight of the whole body, viz. 1 ⁄ 18, 1 ⁄ 80, 1 ⁄ 184, and 1 ⁄ 546. And Kellicott has, in like manner, shewn that in the dogfish, while some organs (e.g. rectal gland, pancreas, etc.) increase steadily and very nearly proportionately to the body as a whole, the brain, and some other organs also, grow in a diminishing ratio, which is capable of representation, ap­prox­i­mate­ly, by a logarithmic curve122.

But if we confine ourselves to the adult, then, as Raymond Pearl has shewn in the case of man, the relation of brain-weight to age, to stature, or to weight, becomes a comparatively simple one, and may be sensibly expressed by a straight line, or simple equation.

Thus, if W be the brain-weight (in grammes), and A be the age, or S the stature, of the individual, then (in the case of Swedish males) the following simple equations suffice to give the required ratios:

W = 1487·8 − 1·94 A = 915·06 + 2·86 S.

These equations are applicable to ages between fifteen and eighty; if we take narrower limits, say between fifteen and fifty, we can get a closer agreement by using somewhat altered constants. In the two sexes, and in different races, these empirical constants will be greatly changed123. Donaldson has further shewn that the correlation between brain-weight and body-weight is very much closer in the rat than in man124.

The falling ratio of weight of brain to body with increase of size or age finds its parallel in comparative anatomy, in the general law that the larger the animal the less is the relative weight of the brain.

	Weight of entire animal gms.	Weight of brain gms.	Ratio
Marmoset	335	12·5	1 : 26
Spider monkey	1845	126	1 : 15
Felis minuta	1234	23·6	1 : 56
F. domestica	3300	31	1 : 107
Leopard	27,700	164	1 : 168
Lion	119,500	219	1 : 546
Elephant	3,048,000	5430	1 : 560
Whale (Globiocephalus)	1,000,000	2511	1 : 400

For much information on this subject, see Dubois, “Abhängigkeit des Hirngewichtes von der Körpergrösse bei den Säugethieren,” Arch. f. Anthropol. XXV, 1897. Dubois has attempted, but I think with very doubtful success, to equate the weight of the brain with that of the animal. We may do this, in a very simple way, by representing the weight of the body as a power of that of the brain; thus, in the above table of the weights of brain and body in four species of cat, if we call W the weight of the body (in grammes), and w the weight of the brain, then if in all four cases we express the ratio by W = wⁿ , we find that n is almost constant, and differs little from 2·24 in all four species: the values being respectively, in the order of the table 2·36, 2·24, 2·18, and 2·17. But this evidently amounts to no more than an empirical rule; for we can easily see that it depends on the particular scale which we have used, and that if the weights had been taken, for instance, in kilogrammes or in milligrammes, the agreement or coincidence would not have occurred125. {93}

The Length of the Head in Man at various Ages. (After Quetelet, p. 207.)
Age	Men	Women
Total height m.	Head m.	Ratio	Height m.	Head† m.	Ratio
Birth	0·500	0·111	4·50	0·494	0·111	4·45
1 year	0·698	0·154	4·53	0·690	0·154	4·48
2 years	0·791	0·173	4·57	0·781	0·172	4·54
3 years	0·864	0·182	4·74	0·854	0·180	4·74
5 years	0·987	0·192	5·14	0·974	0·188	5·18
10 years	1·273	0·205	6·21	1·249	0·201	6·21
15 years	1·513	0·215	7·04	1·488	0·213	6·99
20 years	1·669	0·227	7·35	1·574	0·220	7·15
30 years	1·686	0·228	7·39	1·580	0·221	7·15
40 years	1·686	0·228	7·39	1·580	0·221	7·15

† A smooth curve, very similar to this, for the growth in “auricular height” of the girl’s head, is given by Pearson, in Biometrika, III, p. 141. 1904.

As regards external form, very similar differences exist, which however we must express in terms not of weight but of length. Thus the annexed table shews the changing ratios of the vertical length of the head to the entire stature; and while this ratio constantly diminishes, it will be seen that the rate of change is greatest (or the coefficient of acceleration highest) between the ages of about two and five years.

In one of Quetelet’s tables (supra, p. 63), he gives measurements of the total span of the outstretched arms in man, from year to year, compared with the vertical stature. The two measurements are so nearly identical in actual magnitude that a direct comparison by means of curves becomes unsatisfactory; but I have reduced Quetelet’s data to percentages, and it will be seen from Fig. 19 that the percentage proportion of span to height undergoes a remarkable and steady change from birth to the age of twenty years; the man grows more rapidly in stretch of arms than he does in height, and the span which was less than {94} the stature at birth by about 1 per cent. exceeds it at the age of twenty by about 4 per cent. After the age of twenty, Quetelet’s data are few and irregular, but it is clear that the span goes on for a long while increasing in proportion to the stature. How far the phenomenon is due to actual growth of the arms and how far to the increasing breadth of the chest is not yet ascertained.

Fig. 19. Ratio of stature in Man, to span of outstretched arms.

(From Quetelet’s data.)

The differences of rate of growth in different parts of the body are very simply brought out by the following table, which shews the relative growth of certain parts and organs of a young trout, at intervals of a few days during the period of most rapid development. It would not be difficult, from a picture of the little trout at any one of these stages, to draw its ap­prox­i­mate form at any other, by the help of the numerical data here set forth126. {95}

Trout (Salmo fario): proportionate growth of various organs. (From Jenkinson’s data.)
Days old	Total length	Eye	Head	1st dorsal	Ventral fin	2nd dorsal	Tail-fin	Breadth of tail
49	100	100	100	100	100	100	100	100
63	129·9	129·4	148·3	148·6	148·5	108·4	173·8	155·9
77	154·9	147·3	189·2	(203·6)	(193·6)	139·2	257·9	220·4
92	173·4	179·4	220·0	(193·2)	(182·1)	154·5	307·6	272·2
106	194·6	192·5	242·5	173·2	165·3	173·4	337·3	287·7

While it is inequality of growth in different directions that we can most easily comprehend as a phenomenon leading to gradual change of outward form, we shall see in another chapter127 that differences of rate at different parts of a longitudinal system, though always in the same direction, also lead to very notable and regular trans­for­ma­tions. Of this phenomenon, the difference in rate of longitudinal growth between head and body is a simple case, and the difference which accompanies and results from it in the bodily form of the child and the man is easy to see. A like phenomenon has been studied in much greater detail in the case of plants, by Sachs and certain other botanists, after a method in use by Stephen Hales a hundred and fifty years before128.

On the growing root of a bean, ten narrow zones were marked off, starting from the apex, each zone a millimetre in breadth. After twenty-four hours’ growth, at a certain constant temperature, the whole marked portion had grown from 10 mm. to 33 mm. in length; but the individual zones had grown at very unequal rates, as shewn in the annexed table129.

Zone	Increment mm.		Zone	Increment mm.
Apex	1·5		6th	1·3
2nd	5·8		7th	0·5
3rd	8·2		8th	0·3
4th	3·5		9th	0·2
5th	1·6		10th	0·1

{96}

Fig. 20. Rate of growth in successive zones near the tip of the bean-root.

The several values in this table lie very nearly (as we see by Fig. 20) in a smooth curve; in other words a definite law, or principle of continuity, connects the rates of growth at successive points along the growing axis of the root. Moreover this curve, in its general features, is singularly like those acceleration-curves which we have already studied, in which we plotted the rate of growth against successive intervals of time, as here we have plotted it against successive spatial intervals of an actual growing structure. If we suppose for a moment that the velocities of growth had been transverse to the axis, instead of, as in this case, longitudinal and parallel with it, it is obvious that these same velocities would have given us a leaf-shaped structure, of which our curve in Fig. 20 (if drawn to a suitable scale) would represent the actual outline on either side of the median axis; or, again, if growth had been not confined to one plane but symmetrical about the axis, we should have had a sort of turnip-shaped root, {97} having the form of a surface of revolution generated by the same curve. This then is a simple and not unimportant illustration of the direct and easy passage from velocity to form.

A kindred problem occurs when, instead of “zones” artificially marked out in a stem, we deal with the rates of growth in successive actual “internodes”; and an interesting variation of this problem occurs when we consider, not the actual growth of the internodes, but the varying number of leaves which they successively produce. Where we have whorls of leaves at each node, as in Equisetum and in many water-weeds, then the problem presents itself in a simple form, and in one such case, namely in Ceratophyllum, it has been carefully investigated by Mr. Raymond Pearl130.

It is found that the mean number of leaves per whorl increases with each successive whorl; but that the rate of increment diminishes from whorl to whorl, as we ascend the axis. In other words, the increase in the number of leaves per whorl follows a logarithmic ratio; and if y be the mean number of leaves per whorl, and x the successional number of the whorl from the root or main stem upwards, then

y = A + C log(x − a),

where A, C, and a are certain specific constants, varying with the part of the plant which we happen to be considering. On the main stem, the rate of change in the number of leaves per whorl is very slow; when we come to the small twigs, or “tertiary branches,” it has become rapid, as we see from the following abbreviated table:

Number of leaves per whorl on the tertiary branches of Ceratophyllum.
Position of whorl	1	2	3	4	5	6
Mean number of leaves	6·55	8·07	9·00	9·20	9·75	10·00
Increment	—	1·52	·93	·20	(·55)	(·25)

We have seen that a slow but definite change of form is a common accompaniment of increasing age, and is brought about as the simple and natural result of an altered ratio between the rates of growth in different dimensions: or rather by the progressive change necessarily brought about by the difference in their accelerations. There are many cases however in which the change is all but imperceptible to ordinary measurement, and many others in which some one dimension is easily measured, but others are hard to measure with cor­re­spon­ding accuracy. {98} For instance, in any ordinary fish, such as a plaice or a haddock, the length is not difficult to measure, but measurements of breadth or depth are very much more uncertain. In cases such as these, while it remains difficult to define the precise nature of the change of form, it is easy to shew that such a change is taking place if we make use of that ratio of length to weight which we have spoken of in the preceding chapter. Assuming, as we may fairly do, that weight is directly proportional to bulk or volume, we may express this relation in the form W ⁄ L³ = k, where k is a constant, to be determined for each particular case. (W and L are expressed in grammes and centimetres, and it is usual to multiply the result by some figure, such as 1000, so as to give the constant k a value near to unity.)