Fig. 6. Percentage ratio, throughout life, of female weight to male; from Quetelet’s data.
While careful observations on the rate of growth in other animals are somewhat scanty, they tend to show so far as they go that the general features of the phenomenon are always much the same. Whether the animal be long-lived, as man or the elephant, or short-lived, like horse or dog, it passes through the same phases of growth[101]. In all cases growth begins slowly; it attains a maximum velocity early in its course, and afterwards slows down (subject to temporary accelerations) towards a point where growth ceases altogether. But especially in the cold-blooded animals, such as fishes, the slowing-down period is very greatly protracted, and the size of the creature would seem never actually to reach, but only to approach asymptotically, to a maximal limit.
The size ultimately attained is a resultant of the rate, and of {72} the duration, of growth. It is in the main true, as Minot has said, that the rabbit is bigger than the guinea-pig because he grows the faster; but that man is bigger than the rabbit because he goes on growing for a longer time.
In ordinary physical investigations dealing with velocities, as for instance with the course of a projectile, we pass at once from the study of acceleration to that of momentum and so to that of force; for change of momentum, which is proportional to force, is the product of the mass of a body into its acceleration or change of velocity. But we can take no such easy road of kinematical investigation in this case. The “velocity” of growth is a very different thing from the “velocity” of the projectile. The forces at work in our case are not susceptible of direct and easy treatment; they are too varied in their nature and too indirect in their action for us to be justified in equating them directly with the mass of the growing structure.
It was apparently from a feeling that the velocity of growth ought in some way to be equated with the mass of the growing structure that Minot[102] introduced a curious, and (as it seems to me) an unhappy method of representing growth, in the form of what he called “percentage-curves”; a method which has been followed by a number of other writers and experimenters. Minot’s method was to deal, not with the actual increments added in successive periods, such as years or days, but with these increments represented as percentages of the amount which had been reached at the end of the former period. For instance, taking Quetelet’s values for the height in centimetres of a male infant from birth to four years old, as follows:
| Years | 0 | 1 | 2 | 3 | 4 |
| cm. | 50·0 | 69·8 | 79·1 | 86·4 | 92·7 |
Minot would state the percentage growth in each of the four annual periods at 39·6, 13·3, 9·6 and 7·3 per cent. respectively.
Now when we plot actual length against time, we have a perfectly definite thing. When we differentiate this L ⁄ T, we have dL ⁄ dT, which is (of course) velocity; and from this, by a second differentiation, we obtain d2 L ⁄ dT2 , that is to say, the acceleration. {73}