But on the other hand, the question of strength of materials comes in once more, and the factors of stress and strain and bending moment make it, so to speak, more and more difficult for nature to endow the larger animal with the length of lever with which she has provided the flea or the grasshopper.
To Kirby and Spence it seemed that ãThis wonderful strength of insects is doubtless the result of something peculiar in the structure and arrangement of their muscles, and principally their extraordinary power of contraction.ã This hypothesis, which is so easily seen, on physical grounds, to be unnecessary, has been amply disproved in a series of excellent papers by F. Plateau27.
A somewhat simple problem is presented to us by the act of walking. It is obvious that there will be a great economy of work, if the leg swing at its normal pendulum-rate; and, though this rate is hard to calculate, owing to the shape and the jointing of the limb, we may easily convince ourselves, by counting our steps, that the leg does actually swing, or tend to swing, just as a pendulum does, at a certain definite rate28. When we walk quicker, we cause the leg-pendulum to describe a greater arc, but we do not appreciably cause it to swing, or vibrate, quicker, until we shorten the pendulum and begin to run. Now let two individuals, A and B, walk in a similar fashion, that is to say, with a similar angle of swing. The arc through which the leg swings, or the amplitude of each step, will therefore vary as the length of leg, or say as aã₤ãã₤b; but the time of swing will vary as the square {31} root of the pendulum-length, or ãÿ£¢aã₤ãã₤ãÿ£¢b. Therefore the velocity, which is measured by amplitudeã₤ãã₤time, will also vary as the square-roots of the length of leg: that is to say, the average velocities of A and B are in the ratio of ãÿ£¢aã₤:ã₤ãÿ£¢b.
The smaller man, or smaller animal, is so far at a disadvantage compared with the larger in speed, but only to the extent of the ratio between the square roots of their linear dimensions: whereas, if the rate of movement of the limb were identical, irrespective of the size of the animal,ãif the limbs of the mouse for instance swung at the same rate as those of the horse,ãthen, as F. Plateau said, the mouse would be as slow or slower in its gait than the tortoise. M. Delisle29 observed a ãminute flyã walk three inches in half-a-second. This was good steady walking. When we walk five miles an hour we go about 88 inches in a second, or 88ã₤ãã₤6 =ã₤14ôñ7 times the pace of M. Delisleãs fly. We should walk at just about the flyãs pace if our stature were 1ã₤ãã₤(14ôñ7)ÿ£¢2ã₤, or 1ã₤ãã₤216 of our present height,ãsay 72ã₤ãã₤216 inches, or one-third of an inch high.
But the leg comprises a complicated system of levers, by whose various exercise we shall obtain very different results. For instance, by being careful to rise upon our instep, we considerably increase the length or amplitude of our stride, and very considerably increase our speed accordingly. On the other hand, in running, we bend and so shorten the leg, in order to accommodate it to a quicker rate of pendulum-swing30. In short, the jointed structure of the leg permits us to use it as the shortest possible pendulum when it is swinging, and as the longest possible lever when it is exerting its propulsive force.
Apart from such modifications as that described in the last paragraph,ãapart, that is to say, from differences in mechanical construction or in the manner in which the mechanism is used,ãwe have now arrived at a curiously simple and uniform result. For in all the three forms of locomotion which we have attempted {32} to study, alike in swimming, in flight and in walking, the general result, attained under very different conditions and arrived at by very different modes of reasoning, is in every case that the velocity tends to vary as the square root of the linear dimensions of the organism.