dragon-flies expel from their intestines a very strong jet of liquid, and acquire, by this means, a rapid and forcible impulse.

The oar is found in many insects which move on the surface of the water. A contrivance is employed by other animals, which resembles the action of an oar used at the stern of a boat in the process called sculling. To this latter motive power may be referred all those movements in which an inclined plane is displaced in the liquid, and finds in the resistance of the water, which it presses obliquely, two component forces, of which one furnishes a movement of propulsion. This mechanism will require some explanation; it will be found in its proper place, with all the developments which it affords.

Aerial locomotion. This mechanism is still the same; the motion of an inclined plane, which causes motion through the air. The wing, in fact, in the insect as well as in the bird, strikes the air in an oblique manner, repels it in a certain direction, and gives the body a motion directly opposite. With the exception of certain birds which spread their wings to the wind, and which, hovering thus without any other effort than simply steering, have received the picturesque name of hovering or sailing birds (oiseaux voiliers), all animals move forward only by an effort exerted between two masses unequally movable. It can be easily understood that if one of these points where the force is applied is absolutely fixed, the other alone will receive without diminution the motive work developed; such is the condition of terrestrial locomotion on soil perfectly solid. But we can understand also that the softness of the ground constitutes a condition unfavourable to the utilization of the force employed, and that the extreme mobility both of the air and the water offer still less favourable conditions for swimming or flight.

But this mobility of the point of resistance varies with the rapidity of the movement; so that a certain stroke of the wing or the oar, which would be without effect if produced slowly, would become efficacious by its very rapidity.

In different kinds of locomotion, the resistance which it is necessary to overcome in order to displace the body, does not vary less than that which serves as an external point of

resistance. This variability depends on many causes. Thus, different kinds of animals, when they move, have not to struggle with the same effort against their weight. The fish, which is of nearly the same specific gravity as water, finds itself suspended in it without having to exert any force; and if it wishes to move in any direction, it has only to overcome the resistance of the fluid which it is necessary to displace. The bird, on the contrary, if it desires to sustain itself in the air, must make an effort capable of neutralizing the action of its weight. If it moves forward at the same time, it must perform, in addition, the work which is consumed in overcoming the resistance of the air.

Partition of muscular force between the points of resistance and the mass of the body. When, in physiology, we seek to estimate the work of a muscle, we fix it firmly by one of its attachments, and we ascertain the extent passed through by its movable extremity. If we know the weight which this muscle can raise as it contracts, and the extent through which that weight is raised, we have elements by which we can estimate the work effected. But these are almost ideal conditions, which are scarcely ever found in terrestrial locomotion; nor can we observe them in animals which move in the water, and more especially in those which fly through the air. Let us only compare the effort necessary to walk on a movable soil, on sandy dunes, for instance, with that required in walking on firm soil. We shall see that the mobility of the resisting surface presented by the sand destroys a part of the effort necessary for the contraction of our muscles; in other words, that a greater effort is necessary to produce the same useful work, when the point of resistance is not stable. This amount of work is easy to be understood, and even to be measured.

When a man, while walking, places one of his feet on the ground, the corresponding leg, slightly bent, draws itself up, and pressing on the ground below, gives at the same time an upward impulse to the body. If the ground entirely resist this pressure, all the movement produced will be in the direction of the trunk of the body, which will be raised to a certain height, three centimetres for example. But if

the ground sink two centimetres under the pressure of the foot, it is evident that the body will only be raised one centimetre, and the useful work will be diminished by twothirds.

ance.

The compression of the soil under the foot certainly constitutes work, according to the mechanical definition of this word. In fact, the soil, as it yields, offers a certain resist This resistance must be multiplied by the extent to which the soil is indented, in order to ascertain the value of the work accomplished in this direction. But this work is absolutely useless with respect to locomotion: it is an entire loss of the motive force expended.

When a fish strikes the water with his tail, in order to drive himself forward, he executes a double work; a part tends to drive behind him a certain mass of fluid with a certain velocity, and the other to drive the animal forward in spite of the resistance of the surrounding water. This last work alone is utilized; it would be much more considerable if the tail of the animal met with a solid point of resistance instead of the water which flies from before it.

Is it possible to measure the diminution of useful work in locomotion, according to the greater or less mobility of the point of resistance?

If the ground on which we walk resist perfectly, it must be admitted that no part of the muscular work is lost; but in every case in which a displacement of the resisting surface exists at the same time as that of the body, it is necessary to determine the law according to which this partition is made. A principle established by Newton regulates the science of mechanics; this is that "action and re-action are equal." Does this mean, in the case before us, that half of the work is expended on the resisting surface, and the other half on the displacement of the body of the animal? This cannot be true, if we may judge by the many cases in which a force acts on two bodies at the same time.

Thus, in the science of projectiles, the motive force of the powder-that is to say, the pressure of the gases which are disengaged in the cannon, acts at the same time on the projectile and on the piece, giving these masses a velocity in

opposite directions. Thus, the momentum (M.V.) is equally divided between the two projectiles, so that the mass of the cannon and of its carriage, multiplied by the velocity of the recoil which is communicated to it, is equal to the mass of the projectile multiplied by the velocity of propulsion which it receives. As the cannon weighs much more than the ball, the velocity of its recoil is much less than that communicated to the projectile.

As to the work developed by the powder against the cannon and against the ball, it is divided very unequally between these two masses.

In fact, the work produced by an active force being proportional to the square of the velocity of the mass in motion (its formula is my2), calculation shows that this work, when the piece weighs 300 times more than the ball, would be 300 times greater for the ball than for the cannon.

We shall return to these questions, when in considering the particular kinds of animal motion, we enter on the investigation of human locomotion.

CHAPTER II.

TERRESTRIAL LOCOMOTION (BIPEDS).

Choice of certain types in order to study terrestrial locomotion-Human locomotion-Walking-Pressure exerted on the ground, its duration and intensity-Re-actions on the body during walking-Graphic method of studying them-Vertical oscillations of the bodyHorizontal oscillations-Attempt to represent the trajectory of the pubis-Forward movement of the body-Inequalities of its velocity during the time occupied by a pace.

ACT OF WALKING IN MAN.

THE types of terrestrial locomotion are so various that we must, for a time at least, confine ourselves to the study of the most important among them. For locomotion among bipeds we will take as a type that of man. The horse will be chosen

as the most important representative of the method of walking adopted by quadrupeds. As to other animals, they will be studied in an accessory manner, and especially with reference to the resemblances and differences which the modes of their locomotion present when compared with the types which we have chosen.

Many authors have already treated on this subject; from the time of Borelli to that of modern physiologists, science has slowly advanced: it seems to us that it can now resolve all obscure questions, and determine them definitely, by the employment of the graphic method.

While observation employed alone furnishes only incomplete and sometimes false data, the graphic method carries its precision into the analysis of the very complex movements concerned in locomotion. We shall see, when we treat of the paces of the horse, that the disagreement we find among writers on this subject shows clearly the insufficiency of the methods hitherto employed.

Human locomotion, though much more simple in its mechan-. ism, is still very difficult to analyse; the works of the two Webers, though considered as the deepest investigation of human locomotion that have yet been made, show many omissions and some errors.

The most simple and usual pace is walking, which, according to the received definition, consists in that mode of locomotion in which the body never quits the ground. In running and leaping, on the contrary, we shall see that the body is entirely raised above the ground, and remains suspended during a certain time.

In walking, the weight of the body passes alternately from one leg to the other, and as each of these limbs places itself in turn before the other, the body is thus continually carried forward. This action appears very simple at first sight, but its complexity is soon observed when we seek to ascertain what are the movements which concur in producing this motion.

We see, in fact, that each movement of the limbs brings under consideration a phase of impact and one of support in each of these; the different articulations bend and extend