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. [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:
| 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 ⁄ L3 = 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.)
| 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 | — |