Читать книгу On Growth and Form - D'Arcy Wentworth Thompson - Страница 14
Оглавление| 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 ⁄ L3 × 10,000 | W ⁄ L3 (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 | — |
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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. 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.
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 year131. 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 ⁄ L3 × 100 | W ⁄ L3 (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 ⁄ L3 × 10 | k′ = W ⁄ H3 × 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 ⁄ H3 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 form132. 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. 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.
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 it133.
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 manifest134. {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.
