Chapter III: The Rate of Growth (2)

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 σ. 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 {79} 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, _C_, as = (σ/_M_) × 100.)

In regard to the growth of man, Pearson has determined this coefficient 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[108].

_Coefficient of Variability (σ/_M_ × 100) in Man, at various ages._

Age 5 6 7 8 9 Stature (Bowditch) 4·76 4·60 4·42 4·49 4·40 Stature (Boas and Wissler) 4·15 4·14 4·22 4·37 4·33 Weight (Bowditch) 11·56 10·28 11·08 9·92 11·04

Age 10 11 12 13 14 Stature (Bowditch) 4·55 4·70 4·90 5·47 5·79 Stature (Boas and Wissler) 4·36 4·54 4·73 5·16 5·57 Weight (Bowditch) 11·60 11·76 13·72 13·60 16·80

Age 15 16 17 18 Stature (Bowditch) 5·57 4·50 4·55 3·69 Stature (Boas and Wissler) 5·50 4·69 4·27 3·94 Weight (Bowditch) 15·32 13·28 12·96 10·40

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 {80} 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 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 already ceased, to grow. In the following epitomised table, {81} I have taken Boas’s determinations of variability (σ) (_op. cit._ p. 1548), converted them into the corresponding coefficients of variability ((σ/_M_) × 100), and then smoothed the resulting numbers.

_Coefficients of Variability in Annual Increment of Stature._

Age 7 8 9 10 11 12 13 14 15 Boys 17·3 15·8 18·6 19·1 21·0 24·7 29·0 36·2 46·1 Girls 17·1 17·8 19·2 22·7 25·9 29·3 37·0 44·8 —

The greater variability of annual increment in the girls, as compared 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 growth-curves of the two sexes, 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[109], is confirmed by the following table:―

_Correlation of Stature and Increment in Boys and Girls._

(_From Boas and Wissler._)

Age 6 7 8 9 10 11 12 13 14 15 Stature (B) 112·7 115·5 123·2 127·4 133·2 136·8 142·7 147·3 155·9 162·2 (G) 111·4 117·7 121·4 127·9 131·8 136·7 144·6 149·7 153·8 157·2 Increment (B) 5·7 5·3 4·9 5·1 5·0 4·7 5·9 7·5 6·2 5·2 (G) 5·9 5·5 5·5 5·9 6·2 7·2 6·5 5·4 3·3 1·7 Correlation (B) ·25 ·11 ·08 ·25 ·18 ·18 ·48 ·29 −·42 −·44 (G) ·44 ·14 ·24 ·47 ·18 −·18 −·42 −·39 −·63 ·11

{82}

A minor, but very curious point brought out by the same investigators is that, if instead of stature we deal with height in the sitting posture (or, practically speaking, with length of trunk or back), then the correlations between this height and its annual increment are throughout negative. In other words, there would seem to be a general tendency for the long trunks to grow slowly throughout the whole period under investigation. It is a well-known anatomical fact that tallness is in the main due not to length of body but to length of limb.

The whole phenomenon of variability in regard to magnitude and to rate of increment is in the highest degree suggestive: inasmuch as it helps further to remind and to impress upon us that specific rate of growth is the real physiological factor which we want to get at, of which specific magnitude, dimensions and form, and all the variations of these, are merely the concrete and visible resultant. But the problems of variability, though they are intimately related to the general problem of growth, carry us very soon beyond our present limitations.

_Rate of growth in other organisms[110]._

Just as the human curve of growth has its slight but well-marked interruptions, or variations in rate, coinciding with such epochs as birth and puberty, so is it with other animals, and this phenomenon is particularly striking in the case of animals which undergo a regular metamorphosis.

In the accompanying curve of growth in weight of the mouse (Fig. 12), based on W. Ostwald’s observations[111], we see a distinct slackening of the rate when the mouse is about a fortnight old, at which period it opens its eyes and very soon afterwards is weaned. At about six weeks old there is another well-marked retardation of growth, following on a very rapid period, and coinciding with the epoch of puberty. {83}

Fig. 13 shews the curve of growth of the silkworm[112], during its whole larval life, up to the time of its entering the chrysalis stage.

The silkworm moults four times, at intervals of about a week, the first moult being on the sixth or seventh day after hatching. A distinct retardation of growth is exhibited on our curve in the case of the third and fourth moults; while a similar retardation accompanies the first and second moults also, but the scale of our diagram does not render it visible. When the worm is about seven weeks old, a remarkable process of “purgation” takes place, as a preliminary to entering on the pupal, or chrysalis, stage; and the great and sudden loss of weight which accompanies this process is the most marked feature of our curve.

The rate of growth in the tadpole[113] (Fig. 14) is likewise marked by epochs of retardation, and finally by a sudden and drastic change. There is a slight diminution in weight immediately after {84} the little larva frees itself from the egg; there is a retardation of growth about ten days later, when the external gills disappear; and finally, the complete metamorphosis, with the loss of the tail, the growth of the legs and the cessation of branchial respiration, is accompanied by a loss of weight amounting to wellnigh half the weight of the full-grown larva. {85}

While as a general rule, the better the animals be fed the quicker they grow and the sooner they metamorphose, Barfürth has pointed out the curious fact that a short spell of starvation, just before metamorphosis is due, appears to hasten the change.

The negative growth, or actual loss of bulk and weight which often, and perhaps always, accompanies metamorphosis, is well shewn in the case of the eel[114]. The contrast of size is great between {87} the flattened, lancet-shaped Leptocephalus larva and the little black cylindrical, almost thread-like elver, whose magnitude is less than that of the Leptocephalus in every dimension, even, at first, in length (Fig. 15).

From the higher study of the physiology of growth we learn that such fluctuations as we have described are but special interruptions in a process which is never actually continuous, but is perpetually interrupted in a rhythmic manner[115]. Hofmeister shewed, for instance, that the growth of Spirogyra proceeds by fits and starts, by periods of activity and rest, which alternate with one another at intervals of so many minutes (Fig. 16). And Bose, by very refined methods of experiment, has shewn that plant-growth really proceeds by tiny and perfectly rhythmical pulsations recurring at regular intervals of a few seconds of time. Fig. 17 shews, according to Bose’s observations[116], the growth of a crocus, under a very high magnification. The stalk grows by little jerks, each with an amplitude of about ·002 mm., every {88} twenty seconds or so, and after each little increment there is a partial recoil.

_The rate of growth of various parts or organs[117]._

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 creatures[118]. 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._)

Percentage weights compared Weight Relative weights of with total body-weights of body† ───────────────────────── ─────────────────────────── 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.

Many statistics indicate a decrease of brain-weight during adult life. Boas[119] 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 life[120]. {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 Weber[121] 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 curve[122].

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_.

{ 92}

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 changed[123]. Donaldson has further shewn that the correlation between brain-weight and body-weight is very much closer in the rat than in man[124].

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 Weight of entire animal brain gms. 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_ = _w_^{_n_}, 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 occurred[125]. {93}

_The Length of the Head in Man at various Ages._

(_After Quetelet, p. 207._)

Men Women
────────────────────────── ──────────────────────
Age Total height Head Ratio Height Head† Ratio
m. m. m. m.
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.

(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 forth[126]. {95}

_Trout (Salmo fario): proportionate growth of various organs._

(_From Jenkinson’s data._)

Days Total 1st Ventral 2nd Breadth
old length Eye Head dorsal fin dorsal Tail-fin 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 chapter[127] 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 before[128].

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 table[129].

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

{96}

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 Pearl[130].

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_/_L_^3 = _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.)

_Plaice caught in a certain area, March, 1907. Variation of k (the weight-length coefficient) with size. (Data taken from the Department of Agriculture and Fisheries’ Plaice-Report, vol._ I, _p._ 107, 1908.)

Size in cm. Weight in gm. _W_/_L_^3 × 10,000 _W_/_L_^3 (smoothed)
23 113 92·8 —
24 128 92·6 94·3
25 152 97·3 96·1
26 173 98·4 97·9
27 193 98·1 99·0
28 221 100·6 100·4
29 250 102·5 101·2
30 271 100·4 101·2
31 300 100·7 100·4
32 328 100·1 99·8
33 354 98·5 98·8
34 384 97·7 98·0
35 419 97·7 97·6
36 454 97·3 96·7
37 492 95·2 96·3
38 529 96·4 95·6
39 564 95·1 95·0
40 614 95·9 95·0
41 647 93·9 93·8
42 679 91·6 92·5
43 732 92·1 92·5
44 800 93·9 94·0
45 875 96·0 —

{99}

Now while this _k_ may be spoken of as a “constant,” having a certain mean value specific to each species of organism, and depending on the form of the organism, any change to which it may be subject will be a very delicate index of progressive changes of form; for we know that our measurements of length are, on the average, very accurate, and weighing is a still more delicate method of comparison than any linear measurement.

Thus, in the case of plaice, when we deal with the mean values for a large number of specimens, and when we are careful to deal only with such as are caught in a particular locality and at a particular time, we see that _k_ is by no means constant, but steadily increases to a maximum, and afterwards slowly declines with the increasing size of the fish (Fig. 21). To begin with, therefore, the weight is increasing more rapidly than the cube of the length, and it follows that the length itself is increasing less rapidly than some other linear dimension; while in later life this condition is reversed. The maximum is reached when the length of the fish is somewhere near to 30 cm., and it is tempting to suppose that with this “point of inflection” there is associated some well-marked epoch in the fish’s life. As a matter of fact, the size of 30 cm. is approximately that at which sexual maturity may be said to begin, or is at least near enough to suggest a close connection between the two phenomena. The first step towards further investigation of the {100} apparent coincidence would be to determine the coefficient _k_ of the two sexes separately, and to discover whether or not the point of inflection is reached (or sexual maturity is reached) at a smaller size in the male than in the female plaice; but the material for this investigation is at present scanty.

A still more curious and more unexpected result appears when we compare the values of _k_ for the same fish at different seasons of the year[131]. When for simplicity’s sake (as in the accompanying table and Fig. 22) we restrict ourselves to fish of one particular size, it is not necessary to determine the value of _k_, because a change in the ratio of length to weight is obvious enough; but when we have small numbers, and various sizes, to deal with, the determination of _k_ may help us very much. It will be seen, then, that in the case of plaice the ratio of weight to length exhibits a regular periodic variation with the course of the seasons. {101}

_Relation of Weight to Length in Plaice of 55 cm. long, from Month to Month. (Data taken from the Department of Agriculture and Fisheries Plaice-Report, vol._ II, _p._ 92, 1909.)

Average weight in grammes _W_/_L_^3 × 100 _W_/_L_^3 (smoothed) Jan. 2039 1·226 1·157 Feb. 1735 1·043 1·080 March 1616 0·971 0·989 April 1585 0·953 0·967 May 1624 0·976 0·985 June 1707 1·026 1·005 July 1686 1·013 1·037 August 1783 1·072 1·042 Sept. 1733 1·042 1·111 Oct. 2029 1·220 1·160 Nov. 2026 1·218 1·213 Dec. 1998 1·201 1·215

With unchanging length, the weight and therefore the bulk of the fish falls off from about November to March or April, and again between May or June and November the bulk and weight are gradually restored. The explanation is simple, and depends wholly on the process of spawning, and on the subsequent building up again of the tissues and the reproductive organs. It follows that, by this method, without ever seeing a fish spawn, and without ever dissecting one to see the state of its reproductive system, we can ascertain its spawning season, and determine the beginning and end thereof, with great accuracy.

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As a final illustration of the rate of growth, and of unequal growth in various directions, I give the following table of data regarding the ox, extending over the first three years, or nearly so, of the animal’s life. The observed data are (1) the weight of the animal, month by month, (2) the length of the back, from the occiput to the root of the tail, and (3) the height to the withers. To these data I have added (1) the ratio of length to height, (2) the coefficient (_k_) expressing the ratio of weight to the cube of the length, and (3) a similar coefficient (_k′_) for the height of the animal. It will be seen that, while all these ratios tend to alter continuously, shewing that the animal’s form is steadily altering as it approaches maturity, the ratio between length and weight {102} changes comparatively little. The simple ratio between length and height increases considerably, as indeed we should expect; for we know that in all Ungulate animals the legs are remarkably

_Relations between the Weight and certain Linear Dimensions of the Ox. (Data from Przibram, after Cornevin†.)_

_k_ = _k′_ =
Age in _W_, wt. _L_, length _H_, _W_/_L_^3 _W_/_H_^3
months in kg. of back height _L_/_H_  × 10  × 10
0 37 ·78 ·70 1·114 ·779 1·079
1 55·3 ·94 ·77 1·221 ·665 1·210
2 86·3 1·09 ·85 1·282 ·666 1·406
3 121·3 1·207 ·94 1·284 ·690 1·460
4 150·3 1·314 ·95 1·383 ·662 1·754
5 179·3 1·404 1·040 1·350 ·649 1·600
6 210·3 1·484 1·087 1·365 ·644 1·638
7 247·3 1·524 1·122 1·358 ·699 1·751
8 267·3 1·581 1·147 1·378 ·677 1·791
9 282·8 1·621 1·162 1·395 ·664 1·802
10 303·7 1·651 1·192 1·385 ·675 1·793
11 327·7 1·694 1·215 1·394 ·674 1·794
12 350·7 1·740 1·238 1·405 ·666 1·849
13 374·7 1·765 1·254 1·407 ·682 1·900
14 391·3 1·785 1·264 1·412 ·688 1·938
15 405·9 1·804 1·270 1·420 ·692 1·982
16 417·9 1·814 1·280 1·417 ·700 2·092
17 423·9 1·832 1·290 1·420 ·689 1·974
18 423·9 1·859 1·297 1·433 ·660 1·943
19 427·9 1·875 1·307 1·435 ·649 1·916
20 437·9 1·884 1·311 1·437 ·655 1·944
21 447·9 1·893 1·321 1·433 ·661 1·943
22 464·4 1·901 1·333 1·426 ·676 1·960
23 480·9 1·909 1·345 1·419 ·691 1·977
24 500·9 1·914 1·352 1·416 ·714 2·027
25 520·9 1·919 1·359 1·412 ·737 2·075
26 534·1 1·924 1·361 1·414 ·750 2·119
27 547·3 1·929 1·363 1·415 ·762 2·162
28 554·5 1·929 1·363 1·415 ·772 2·190
29 561·7 1·929 1·363 1·415 ·782 2·218
30 586·2 1·949 1·383 1·409 ·792 2·216
31 610·7 1·969 1·403 1·403 ·800 2·211
32 625·7 1·983 1·420 1·396 ·803 2·186
33 640·7 1·997 1·437 1·390 ·805 2·159
34 655·7 2·011 1·454 1·383 ·806 2·133

† Cornevin, Ch., Études sur la croissance, _Arch. de
Physiol. norm. et pathol._ (5), IV, p. 477, 1892.

{103}

long at birth in comparison with other dimensions of the body. It is somewhat curious, however, that this ratio seems to fall off a little in the third year of growth, the animal continuing to grow in height to a marked degree after growth in length has become very slow. The ratio between height and weight is by much the most variable of our three ratios; the coefficient _W_/_H_^3 steadily increases, and is more than twice as great at three years old as it was at birth. This illustrates the important, but obvious fact, that the coefficient _k_ is most variable in the case of that dimension which grows most uniformly, that is to say most nearly in proportion to the general bulk of the animal. In short, the successive values of _k_, as determined (at successive epochs) for one dimension, are a measure of the _variability_ of the others.

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From the whole of the foregoing discussion we see that a certain definite rate of growth is a characteristic or specific phenomenon, deep-seated in the physiology of the organism; and that a very large part of the specific morphology of the organism depends upon the fact that there is not only an average, or aggregate, rate of growth common to the whole, but also a variation of rate in different parts of the organism, tending towards a specific rate characteristic of each different part or organ. The smallest change in the relative magnitudes of these partial or localised velocities of growth will be soon manifested in more and more striking differences of form. This is as much as to say that the time-element, which is implicit in the idea of growth, can never (or very seldom) be wholly neglected in our consideration of form[132]. It is scarcely necessary to enlarge here upon our statement, for not only is the truth of it self-evident, but it will find illustration again and again throughout this book. Nevertheless, let us go out of our way for a moment to consider it in reference to a particular case, and to enquire whether it helps to remove any of the difficulties which that case appears to present. {104}

In a very well-known paper, Bateson shewed that, among a large number of earwigs, collected in a particular locality, the males fell into two groups, characterised by large or by small tail-forceps, with very few instances of intermediate magnitude. This distribution into two groups, according to magnitude, is illustrated in the accompanying diagram (Fig. 23); and the phenomenon was described, and has been often quoted, as one of dimorphism, or discontinuous variation. In this diagram the time-element does not appear; but it is certain, and evident, that it lies close behind. Suppose we take some organism which is born not at all times of the year (as man is) but at some one particular season (for instance a fish), then any random sample will consist of individuals whose _ages_, and therefore whose _magnitudes_, will form a discontinuous series; and by plotting these magnitudes on a curve in relation to the number of individuals of each particular magnitude, we obtain a curve such as that shewn in Fig. 24, the first practical use of which is to enable us to analyse our sample into its constituent “age-groups,” or in other words to determine approximately the age, or ages of the fish. And if, instead of measuring the whole length of our fish, we had confined ourselves to particular parts, such as head, or {105} tail or fin, we should have obtained discontinuous curves of distribution, precisely analogous to those for the entire animal. Now we know that the differences with which Bateson was dealing were entirely a question of magnitude, and we cannot help seeing that the discontinuous distributions of magnitude represented by his earwigs’ tails are just such as are illustrated by the magnitudes of the older and younger fish; we may indeed go so far as to say that the curves are precisely comparable, for in both cases we see a characteristic feature of detail, namely that the “spread” of the curve is greater in the second wave than in the first, that is to say (in the case of the fish) in the older as well as larger series. Over the reason for this phenomenon, which is simple and all but obvious, we need not pause.

It is evident, then, that in this case of “dimorphism,” the tails of the one group of earwigs (which Bateson calls the “high males”) have either grown _faster_, or have been growing for a longer period of time, than those of the “low males.” If we could be certain that the whole random sample of earwigs were of one and the same age, then we should have to refer the phenomenon of dimorphism to a physiological phenomenon, simple in kind (however remarkable and unexpected); viz. that there were two alternative {106} values, very different from one another, for the mean velocity of growth, and that the individual earwigs varied around one or other of these mean values, in each case according to the law of probabilities. But on the other hand, if we could believe that the two groups of earwigs were _of different ages_, then the phenomenon would be simplicity itself, and there would be no more to be said about it[133].

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Before we pass from the subject of the relative rate of growth of different parts or organs, we may take brief note of the fact that various experiments have been made to determine whether the normal ratios are maintained under altered circumstances of nutrition, and especially in the case of partial starvation. For instance, it has been found possible to keep young rats alive for many weeks on a diet such as is just sufficient to maintain life without permitting any increase of weight. The rat of three weeks old weighs about 25 gms., and under a normal diet should weigh at ten weeks old about 150 gms., in the male, or 115 gms. in the female; but the underfed rat is still kept at ten weeks old to the weight of 25 gms. Under normal diet the proportions of the body change very considerably between the ages of three and ten weeks. For instance the tail gets relatively longer; and even when the _total_ growth of the rat is prevented by underfeeding, the _form_ continues to alter so that this increasing length of the tail is still manifest[134]. {107}

_Full-fed Rats._

Age in Length of Length of Total
weeks body (mm.) tail (mm.) length % of tail
0 48·7 16·9 65·6 25·8
1 64·5 29·4 93·9 31·3
3 90·4 59·1 149·5 39·5
6 128·0 110·0 238·0 46·2
10 173·0 150·0 323·0 46·4

_Underfed Rats._

6 98·0 72·3 170·3 42·5
10 99·6 83·9 183·5 45·7

Again as physiologists have long been aware, there is a marked difference in the variation of weight of the different organs, according to whether the animal’s total weight remain constant, or be caused to diminish by actual starvation; and further striking differences appear when the diet is not only scanty, but ill-balanced. But these phenomena of abnormal growth, however interesting from the physiological view, are of little practical importance to the morphologist.

_The effect of temperature[135]._

The rates of growth which we have hitherto dealt with are based on special investigations, conducted under particular local conditions. For instance, Quetelet’s data, so far as we have used them to illustrate the rate of growth in man, are drawn from his study of the population of Belgium. But apart from that “fortuitous” individual variation which we have already considered, it is obvious that the normal rate of growth will be found to vary, in man and in other animals, just as the average stature varies, in different localities, and in different “races.” This phenomenon is a very complex one, and is doubtless a resultant of many undefined contributory causes; but we at least gain something in regard to it, when we discover that the rate of growth is directly affected by temperature, and probably by other physical {108} conditions. Réaumur was the first to shew, and the observation was repeated by Bonnet[136], that the rate of growth or development of the chick was dependent on temperature, being retarded at temperatures below and somewhat accelerated at temperatures above the normal temperature of incubation, that is to say the temperature of the sitting hen. In the case of plants the fact that growth is greatly affected by temperature is a matter of familiar knowledge; the subject was first carefully studied by Alphonse De Candolle, and his results and those of his followers are discussed in the textbooks of Botany[137].

That variation of temperature constitutes only one factor in determining the rate of growth is admirably illustrated in the case of the Bamboo. It has been stated (by Lock) that in Ceylon the rate of growth of the Bamboo is directly proportional to the humidity of the atmosphere: and again (by Shibata) that in Japan it is directly proportional to the temperature. The two statements have been ingeniously and satisfactorily reconciled by Blackman[138], who suggests that in Ceylon the temperature-conditions are all that can be desired, but moisture is apt to be deficient: while in Japan there is rain in abundance but the average temperature is somewhat too low. So that in the one country it is the one factor, and in the other country it is the other, which is _essentially_ variable.

The annexed diagram (Fig. 25), shewing the growth in length of the roots of some common plants during an identical period of forty-eight hours, at temperatures varying from about 14° to 37° C., is a sufficient illustration of the phenomenon. We see that in all cases there is a certain optimum temperature at which the rate of growth is a maximum, and we can also see that on either side of this optimum temperature the acceleration of growth, positive or negative, with increase of temperature is rapid, while at a distance from the optimum it is very slow. From the data given by Sachs and others, we see further that this optimum temperature is very much the same for all the common plants of our own climate which have as yet been studied; in them it is {109} somewhere about 26° C. (or say 77° F.), or about the temperature of a warm summer’s day; while it is found, very naturally, to be considerably higher in the case of plants such as the melon or the maize, which are at home in warmer regions that our own.

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In a large number of physical phenomena, and in a very marked degree in all chemical reactions, it is found that rate of action is affected, and for the most part accelerated, by rise of temperature; and this effect of temperature tends to follow a definite “exponential” law, which holds good within a considerable range of temperature, but is altered or departed from when we pass beyond certain normal limits. The law, as laid down by van’t Hoff for chemical reactions, is, that for an interval of _n_ degrees the velocity varies as _x_^{_n_}, _x_ being called the “temperature coefficient”[139] for the reaction in question. {110}

Van’t Hoff’s law, which has become a fundamental principle of chemical mechanics, is likewise applicable (with certain qualifications) to the phenomena of vital chemistry; and it follows that, on very much the same lines, we may speak of the “temperature coefficient” of growth. At the same time we must remember that there is a very important difference (though we can scarcely call it a _fundamental_ one) between the purely physical and the physiological phenomenon, in that in the former we study (or seek and profess to study) one thing at a time, while in the latter we have always to do with various factors which intersect and interfere; increase in the one case (or change of any kind) tends to be continuous, in the other case it tends to be brought to arrest. This is the simple meaning of that _Law of Optimum_, laid down by Errera and by Sachs as a general principle of physiology: namely that _every_ physiological process which varies (like growth itself) with the amount or intensity of some external influence, does so according to a law in which progressive increase is followed by progressive decrease; in other words the function has its _optimum_ condition, and its curve shews a definite _maximum_. In the case of temperature, as Jost puts it, it has on the one hand its accelerating effect which tends to follow van’t Hoff’s law. But it has also another and a cumulative effect upon the organism: “Sie schädigt oder sie ermüdet ihn, und je höher sie steigt, desto rascher macht sie die Schädigung geltend und desto schneller schreitet sie voran.” It would seem to be this double effect of temperature in the case of the organism which gives us our “optimum” curves, which are the expression, accordingly, not of a primary phenomenon, but of a more or less complex resultant. Moreover, as Blackman and others have pointed out, our “optimum” temperature is very ill-defined until we take account also of the _duration_ of our experiment; for obviously, a high temperature may lead to a short, but exhausting, spell of rapid growth, while the slower rate manifested at a lower temperature may be the best in the end. {111} The mile and the hundred yards are won by different runners; and maximum rate of working, and maximum amount of work done, are two very different things[140].

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In the case of maize, a certain series of experiments shewed that the growth in length of the roots varied with the temperature as follows[141]:

Temperature Growth in 48 hours
°C. mm.
18·0 1·1
23·5 10·8
26·6 29·6
28·5 26·5
30·2 64·6
33·5 69·5
36·5 20·7

Let us write our formula in the form

_V__{(_t_+_n_)}/_V__{_t_} = _x_^{_n_}.

Then choosing two values out of the above experimental series (say the second and the second-last), we have _t_ = 23·5, _n_ = 10, and _V_, _V′_ = 10·8 and 69·5 respectively.

Accordingly 69·5/10·8 = 6·4 = _x_^{10}.

Therefore (log 6·4)/10, or ·0806 = log _x_.

And, _x_ = 1·204 (for an interval of 1° C.).

This first approximation might be considerably improved by taking account of all the experimental values, two only of which we have as yet made use of; but even as it is, we see by Fig. 26 that it is in very fair accordance with the actual results of observation, _within those particular limits_ of temperature to which the experiment is confined. {112}

For an experiment on _Lupinus albus_, quoted by Asa Gray[142], I have worked out the corresponding coefficient, but a little more carefully. Its value I find to be 1·16, or very nearly identical with that we have just found for the maize; and the correspondence between the calculated curve and the actual observations is now a close one.

Since the above paragraphs were written, new data have come to hand. Miss I. Leitch has made careful observations of the rate of growth of rootlets of the Pea; and I have attempted a further analysis of her principal results[143]. In Fig. 27 are shewn the mean rates of growth (based on about a hundred experiments) at some thirty-four different temperatures between 0·8° and 29·3°, each experiment lasting rather less than twenty-four hours. Working out the mean temperature coefficient for a great many combinations of these values, I obtain a value of 1·092 per C.°, or 2·41 for an interval of 10°, and a mean value for the whole series showing a rate of growth of just about 1 mm. per hour at a temperature of 20°. My curve in Fig. 27 is drawn from these determinations; and it will be seen that, while it is by no means exact at the lower temperatures, and will of course fail us altogether at very high {113} temperatures, yet it serves as a very satisfactory guide to the relations between rate and temperature within the ordinary limits of healthy growth. Miss Leitch holds that the curve is _not_ a van’t Hoff curve; and this, in strict accuracy, we need not dispute. But the phenomenon seems to me to be one into which the van’t Hoff ratio enters largely, though doubtless combined with other factors which we cannot at present determine or eliminate.