Had Leonardo da Vinci known what Galileo knew, he would not have spent a great part of his life on vain efforts to make to himself wings. Borelli had learned the lesson thoroughly, and {28} in one of his chapters he deals with the proposition, ãEst impossible, ut homines propriis viribus artificiose volare possint21.ã
But just as it is easier to swim than to fly, so is it obvious that, in a denser atmosphere, the conditions of flight would be altered, and flight facilitated. We know that in the carboniferous epoch there lived giant dragon-flies, with wings of a span far greater than nowadays they ever attain; and the small bodies and huge extended wings of the fossil pterodactyles would seem in like manner to be quite abnormal according to our present standards, and to be beyond the limits of mechanical efficiency under present conditions. But as Harlûˋ suggests22, following upon a suggestion of Arrhenius, we have only to suppose that in carboniferous and jurassic days the terrestrial atmosphere was notably denser than it is at present, by reason, for instance, of its containing a much larger proportion of carbonic acid, and we have at once a means of reconciling the apparent mechanical discrepancy.
Very similar problems, involving in various ways the principle of dynamical similitude, occur all through the physiology of locomotion: as, for instance, when we see that a cockchafer can carry a plate, many times his own weight, upon his back, or that a flea can jump many inches high.
Problems of this latter class have been admirably treated both by Galileo and by Borelli, but many later writers have remained ignorant of their work. Linnaeus, for instance, remarked that, if an elephant were as strong in proportion as a stag-beetle, it would be able to pull up rocks by the root, and to level mountains. And Kirby and Spence have a well-known passage directed to shew that such powers as have been conferred upon the insect have been withheld from the higher animals, for the reason that had these latter been endued therewith they would have ãcaused the early desolation of the world23.ã
Such problems as that which is presented by the fleaãs jumping powers, though essentially physiological in their nature, have their interest for us here: because a steady, progressive diminution of activity with increasing size would tend to set limits to the possible growth in magnitude of an animal just as surely as those factors which tend to break and crush the living fabric under its own weight. In the case of a leap, we have to do rather with a sudden impulse than with a continued strain, and this impulse should be measured in terms of the velocity imparted. The velocity is proportional to the impulse (x), and inversely proportional to the mass (M) moved: V =ã₤xã₤ãã₤M. But, according to what we still speak of as ãBorelliãs law,ã the impulse (i.e. the work of the impulse) is proportional to the volume of the muscle by which it is produced24, that is to say (in similarly constructed animals) to the mass of the whole body; for the impulse is proportional on the one hand to the cross-section of the muscle, and on the other to the distance through which it contracts. It follows at once from this that the velocity is constant, whatever be the size of the animals: in other words, that all animals, provided always that they are similarly fashioned, with their various levers etc., in like proportion, ought to jump, not to the same relative, but to the same actual height25. According to this, then, the flea is not a better, but rather a worse jumper than a horse or a man. As a matter of fact, Borelli is careful to point out that in the act of leaping the impulse is not actually instantaneous, as in the blow of a hammer, but takes some little time, during which the levers are being extended by which the centre of gravity of the animal is being propelled forwards; and this interval of time will be longer in the case of the longer levers of the larger animal. To some extent, then, this principle acts as a corrective to the more general one, {30} and tends to leave a certain balance of advantage, in regard to leaping power, on the side of the larger animal26.