the fact that all animals have an even number of feet, and the reasons for the fact have been set forth. what follows will explain that if there were no point at rest flexion and straightening would be impossible. flexion is a change from a right line to an arc or an angle, straightening a change from either of these to a right line. now in all such changes the flexion or the straightening must be relative to one point. moreover, without flexion there could not be walking or swimming or flying. for since limbed creatures stand and take their weight alternately on one or other of the opposite legs, if one be thrust forward the other of necessity must be bent. for the opposite limbs are naturally of equal length, and the one which is under the weight must be a kind of perpendicular at right angles to the ground.

when then one leg is advanced it becomes the hypotenuse of a right-angled triangle. its square then is equal to the square on the other side together with the square on the base. as the legs then are equal, the one at rest must bend either at the knee or, if there were any kneeless animal which walked, at some other articulation. the following experiment exhibits the fact. if a man were to walk parallel to a wall in sunshine, the line described (by the shadow of his head> would be not straight but zigzag, becoming lower as he bends, and higher when he stands and lifts himself up.

it is, indeed, possible to move oneself even if the leg be not bent, in the way in which children crawl. this was the old though erroneous account of the movement of elephants. but these kinds of movements involve a flexion in the shoulders or in the hips. nothing at any rate could walk upright continuously and securely without flexions at the knee, but would have to move like men in the wrestling schools who crawl forward through the sand on their knees. for the upper part of the upright creature is long so that its leg has to be correspondingly long; in consequence there must be flexion. for since a stationary position is perpendicular, if that which moves cannot bend it will either fall forward as the right angle becomes acute or will not be able to progress. for if one leg is at right angles to the ground and the other is advanced, the latter will be at once equal and greater. for it will be equal to the stationary leg and also equivalent to the hypotenuse of a right-angled triangle. that which goes forward therefore must bend, and while bending one, extend the other leg simultaneously, so as to incline forward and make a stride and still remain above the perpendicular; for the legs form an isosceles triangle, and the head sinks lower when it is perpendicularly above the base on which it stands.

of limbless animals, some progress by undulations (and this happens in two ways, either they undulate on the ground, like snakes, or up and down, like caterpillars), and undulation is a flexion; others by a telescopic action, like what are called earthworms and leeches. these go forward, first one part leading and then drawing the whole of the rest of the body up to this, and so they change from place to place. it is plain too that if the two curves were not greater than the one line which subtends them undulating animals could not move themselves; when the flexure is extended they would not have moved forward at all if the flexure or arc were equal to the chord subtended; as it is, it reaches further when it is straightened out, and then this part stays still and it draws up what is left behind.

in all the changes described that which moves now extends itself in a straight line to progress, and now is hooped; it straightens itself in its leading part, and is hooped in what follows behind. even jumping animals all make a flexion in the part of the body which is underneath, and after this fashion make their leaps. so too flying and swimming things progress, the one straightening and bending their wings to fly, the other their fins to swim. of the latter some have four fins, others which are rather long, for example eels, have only two. these swim by substituting a flexion of the rest of their body for the (missing) pair of fins to complete the movement, as we have said before. flat fish use two fins, and the flat of their body as a substitute for the absent pair of fins. quite flat fish, like the ray, produce their swimming movement with the actual fins and with the two extremes or semicircles of their body, bending and straightening themselves alternately.