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ther secured by a pendulum movement in the arms as well as in the legs, the right arm swinging before the body when the right leg swings behind it, and the converse. The right leg and left arm advance simultaneously, and alternate with the left leg and right arm, which likewise advance together. This gives rise to a double twisting of the body at the shoulders and loins. The legs and arms when advancing move in curves, the convexities of the curves made by the right leg and left arm, which advance together when a step is being made, being directed outwards, and forming, when placed together, a more or less symmetrical ellipse. If the curves formed by the legs and arms respectively be united, they form waved lines which intersect at every step. This arises from the fact that the curves formed by the right and left legs are found alternately on either side of a given line, the same holding true of the right and left arms. Walking is consequently to be regarded as the result of a twisting diagonal movement in the trunk and in the extremities. Without this movement, the momentum acquired by the different portions of the moving mass could not be utilized. As the momentum acquired by animals in walking, swimming, and flying forms an important factor in those movements, it is necessary that we should have a just conception of the value to be attached to weight when in motion. In the horse when walking, the stride is something like five feet, in trotting ten feet, but in galloping eighteen or more feet. The stride is in fact determined by the speed acquired by the mass of the body of the horse; the momentum at which the mass is moving carrying the limbs forward.1

In the swimming of the fish, the body is thrown into double or figure-of-8 curves, as in the walking of the biped. The twisting of the body, and the continuity of movement which that twisting begets, reappear. The curves formed in the swimming

1 "According to Sainbell, the celebrated horse Eclipse, when galloping at liberty, and with its greatest speed, passed over the space of twenty-five feet at each stride, which he repeated 24 times in a second, being nearly four miles in six minutes and two seconds. The race-horse Flying Childers was computed to have passed over eighty-two feet and a half in a second, or nearly a mile in a minute."

of the fish are never less than two, a caudal and a cephalic one. They may and do exceed this number in the long-bodied fishes. The tail of the fish is made to vibrate pendulum fashion on either side of the spine, when it is lashed to and fro in the act of swimming. It is made to rotate upon one or more of the vertebræ of the spine, the vertebra or vertebræ forming the centre of a lemniscate, which is described by the caudal fin. There is, therefore, an obvious analogy between the tail of the fish and the extremity of the biped. This is proved by the conformation and swimming of the seal,-an animal in which the posterior extremities are modified to resemble the tail of the fish. In the swimming of the seal the hind legs are applied to the water by a sculling figure-of-8 motion, in the same manner as the tail of the fish. Similar remarks might be made with regard to the swimming of the whale, dugong, manatee, and porpoise, sea mammals, which still more closely resemble the fish in shape. The double curve into which the fish throws its body in swimming, and which gives continuity of motion, also supplies the requisite degree of steadiness. When the tail is lashed from side to side there is a tendency to produce a corresponding movement in the head, which is at once corrected by the complementary curve. Nor is this all; the cephalic curve, in conjunction with the water contained within it, forms the point d'appui for the caudal curve, and vice versa. When a fish swims, the anterior and posterior portions of its body (supposing it to be a shortbodied fish) form curves, the convexities of which are directed on opposite sides of a given line, as is the case in the extremities of the biped when walking. The mass of the fish, like the mass of the biped, when once set in motion, contributes to progression by augmenting the rate of speed. The velocity acquired by certain fishes is very great. A shark can gambol around the bows of a ship in full sail; and a sword-fish (such is the momentum acquired by it) has been known to thrust its tusk through the copper sheathing of a vessel, a layer of felt, four inches of deal, and fourteen inches of oaken plank.1

The wing of the bird does not materially differ from the

1 A portion of the timbers, etc., of one of Her Majesty's ships, having the

By comparing the flipper of the seal, sea-bear, and walrus with the fin and tail of the fish, whale, porpoise, etc.; and the wing of the penguin (a bird which is incapable of flight, and can only swim and dive) with the wing of the insect, bat, and bird, I have been able to show that a close analogy exists between the flippers, fins, and tails of sea mammals and fishes on the one hand, and the wings of insects, bats, and birds on the other; in fact, that theoretically and practically these organs, one and all, form flexible helices or screws, which, in virtue of their rapid reciprocating movements, operate upon the water and air by a wedge-action after the manner of twisted or double inclined planes. The twisted inclined planes act upon the air and water by means of curved surfaces, the curved surfaces reversing, reciprocating, and engendering a wave pressure, which can be continued indefinitely at the will of the animal. The wave pressure emanates in the one instance mainly from the tail of the fish, whale, porpoise, etc., and in the other from the wing of the insect, bat, or bird-the reciprocating and opposite curves into which the tail and wing are thrown in swimming and flying constituting the mobile helices, or screws, which, during their action, produce the precise kind and degree of pressure adapted to fluid media, and to which they respond with the greatest readiness.

In order to prove that sea mammals and fishes swim, and insects, bats, and birds fly, by the aid of curved figure of-8 surfaces, which exert an intermittent wave pressure, I constructed artificial fish-tails, fins, flippers, and wings, which curve and taper in every direction, and which are flexible and elastic, particularly towards the tips and posterior margins. These artificial fish-tails, fins, flippers, and wings are slightly twisted upon themselves, and when applied to the water and air by a sculling or figure-of-8 motion, curiously enough reproduce the curved surfaces and movements peculiar to real fish-tails, fins, flippers, and wings, in swimming, and flying.

Propellers formed on the fish-tail and wing model are, I find, the most effective that can be devised, whether for navigating the water or the air. To operate efficiently on

fluid, i.e. yielding media, the propeller itself must yield. Of this I am fully satisfied from observation and experiment. The propellers at present employed in navigation are, in my opinion, faulty both in principle and application.

The observations and experiments recorded in the present volume date from 1864. In 1867 I lectured on the subject of animal mechanics at the Royal Institution of Great Britain :1 in June of the same year (1867) I read a memoir "On the Mechanism of Flight" to the Linnean Society of London ;2 and in August of 1870 I communicated a memoir "On the Physiology of Wings" to the Royal Society of Edinburgh.3 These memoirs extend to 200 pages quarto, and are illustrated by 190 original drawings. The conclusions at which I arrived, after a careful study of the movements of walking, swimming, and flying, are briefly set forth in a letter addressed to the French Academy of Sciences in March 1870. This the Academy did me the honour of publishing in April of that year (1870) in the Comptes Rendus, p. 875. In it I claim to have been the first to describe and illustrate the following points, viz. :

That quadrupeds walk, and fishes swim, and insects, bats, and birds fly by figure-of-8 movements.

That the flipper of the sea bear, the swimming wing of the penguin, and the wing of the insect, bat, and bird, are screws structurally, and resemble the blade of an ordinary screwpropeller.

That those organs are screws functionally, from their twisting and untwisting, and from their rotating in the direction of their length, when they are made to oscillate.

That they have a reciprocating action, and reverse their planes more or less completely at every stroke.

That the wing describes a figure-of-8 track in space when the flying animal is artificially fixed.

That the wing, when the flying animal is progressing at 1 "On the various modes of Flight in relation to Aeronautics."-Proceedings of the Royal Institution of Great Britain, March 22, 1867.

"On the Mechanical Appliances by which Flight is attained in the Animal Kingdom."-Transactions of the Linnean Society, vol. xxvi.

3" On the Physiology of Wings."-Transactions of the Royal Society of Edinburgh, vol. xxvi.

a high speed in a horizontal direction, describes a looped and then a waved track, from the fact that the figure of 8 is gradually opened out or unravelled as the animal advances.

That the wing acts after the manner of a kite, both during the down and up strokes.

I was induced to address the above to the French Academy from finding that, nearly two years after I had published my views on the figure of 8, looped and wave movements made by the wing, etc., Professor E. J. Marey (College of France, Paris) published a course of lectures, in which the peculiar figure-of-8 movements, first described and figured by me, were put forth as a new discovery. The accuracy of this statement will be abundantly evident when I mention that my first lecture, "On the various modes of Flight in relation to Aëronautics," was published in the Proceedings of the Royal Institution of Great Britain on the 22d of March 1867, and translated into French (Revue des cours scientifiques de la France et de l'Étranger) on the 21st of September 1867; whereas Professor Marey's first lecture, "On the Movements of the Wing in the Insect" (Revue des cours scientifiques de la France et de l'Étranger), did not appear until the 13th of February 1869.

Professor Marey, in a letter addressed to the French Academy in reply to mine, admits my claim to priority in the following terms :—

"J'ai constaté qu'effectivement M. Pettigrew a vu avant moi, et représenté dans son Mémoire, la forme en 8 du parcours de l'aile de l'insecte : que la méthode optique à laquelle j'avais recours est à peu près identique à la sienne.

Je

m'empresse de satisfaire à cette demande légitime, et de laisser entièrement la priorité sur moi à M. Pettigrew relativement à la question ainsi restreinte."-(Comptes Rendus, May 16, 1870, p. 1093).

The figure-of-8 theory of walking, swimming, and flying, as originally propounded in the lectures, papers, and memoirs referred to, has been confirmed not only by the researches and experiments of Professor Marey, but also by those of M. Senecal, M. de Fastes, M. Ciotti, and others. Its accuracy is

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