The rate of growth of various parts or organs

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. 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. .

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. Boas 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. {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 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.

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

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 = 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. {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. 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 forth. {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 chapter 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.

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.

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. ) 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. (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.

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 = 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
× 10,000 W ⁄ L
(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.

Fig. 21. Changes in the weight-length ratio of Plaice, with increasing size.

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. ). 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.

Fig. 22. Periodic annual change in the weight-length ratio of Plaice.

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. When for simplicity’s sake (as in the accompanying table and Fig. ) 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
× 100 W ⁄ L
(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.

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†.)

Age in
months W, wt.
in kg. L,
length
of back H,
height L ⁄ H k
= W ⁄ L
× 10 k ′
= W ⁄ H
× 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 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.

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. 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}

Fig. 23. Variability of length of tail-forceps in a sample of Earwigs. (After Bateson, P. Z. S. 1892, p. 588.)

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. ); 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. , 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.

Fig. 24. Variability of length of body in a sample of Plaice.

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.

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. {107}

Full-fed Rats.

Age in
weeks Length
of body
(mm.) Length
of tail
(mm.) Total
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.