But when you take percentages of y, you are determining dy ⁄ y, and when you plot this against dx, you have
(dy ⁄ y) ⁄ dx, or dy ⁄ (y · dx), or (1 ⁄ y) · (dy ⁄ dx),
that is to say, you are multiplying the thing you wish to represent by another quantity which is itself continually varying; and the result is that you are dealing with something very much less easily grasped by the mind than the original factors. Professor Minot is, of course, dealing with a perfectly legitimate function of x and y; and his method is practically tantamount to plotting log y against x, that is to say, the logarithm of the increment against the time. This could only be defended and justified if it led to some simple result, for instance if it gave us a straight line, or some other simpler curve than our usual curves of growth. As a matter of fact, it is manifest that it does nothing of the kind.
Pre-natal and post-natal growth.
In the acceleration-curves which we have shown above (Figs. [2], 3), it will be seen that the curve starts at a considerable interval from the actual date of birth; for the first two increments which we can as yet compare with one another are those attained during the first and second complete years of life. Now we can in many cases “interpolate” with safety between known points upon a curve, but it is very much less safe, and is not very often justifiable (at least until we understand the physical principle involved, and its mathematical expression), to “extrapolate” beyond the limits of our observations. In short, we do not yet know whether our curve continued to ascend as we go backwards to the date of birth, or whether it may not have changed its direction, and descended, perhaps, to zero-value. In regard to length, or stature, however, we can obtain the requisite information from certain tables of Rüssow’s[103], who gives the stature of the infant month by month during the first year of its life, as follows:
| Age in months | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Length in cm. | (50) | 54 | 58 | 60 | 62 | 64 | 65 | 66 | 67·5 | 68 | 69 | 70·5 | 72 |
| [Differences (in cm.) | 4 | 4 | 2 | 2 | 2 | 1 | 1 | 1·5 | ·5 | 1 | 1·5 | 1·5] |
If we multiply these monthly differences, or mean monthly velocities, by 12, to bring them into a form comparable with the {74} annual velocities already represented on our acceleration-curves, we shall see that the one series of observations joins on very well with the other; and in short we see at once that our acceleration-curve rises steadily and rapidly as we pass back towards the date of birth.
Fig. 7. Curve of growth (in length or stature) of child, before and after birth. (From His and Rüssow’s data.)
But birth itself, in the case of a viviparous animal, is but an unimportant epoch in the history of growth. It is an epoch whose relative date varies according to the particular animal: the foal and the lamb are born relatively later, that is to say when development has advanced much farther, than in the case of man; the kitten and the puppy are born earlier and therefore more helpless than we are; and the mouse comes into the world still earlier and more inchoate, so much so that even the little marsupial is scarcely more unformed and embryonic. In all these cases alike, we must, in order to study the curve of growth in its entirety, take full account of prenatal or intra-uterine growth. {75}