The synthesis which is motion is obviously not a new deploying in another homogeneous medium, of the same positions that have been perceived in space; for if it were such an act, the necessity for resynthesis would be indefinitely repeated. The synthesis which is motion is a qualitative synthesis, a gradual organization of our successive sensations with each other, a unity analogous to that of a melodic phrase. The space traversed is a quantity, indefinitely divisible; the act by which space is traversed is a quality, and indivisible. Again that endosmotic exchange takes place, as between the melodically organized perception of the series of the pendulum’s motions and its distinct objective presence at each instant. That is, we attribute to the motion the divisibility of the space traversed; and we project the act upon space, implying that outside as well as inside of consciousness the past coexists with the present. In space are only parts of space. In any point of space where the movable may be considered, there is only a position. You would search space in vain for motion.
From the fact that motion cannot be in space, Zeno concluded wrongly that motion is impossible. But those who try to answer his arguments by seeking it also in space, find it no more than he. Achilles overtakes the tortoise because each Achilles step and each tortoise step is not a space but a duration, whose nature is not addible nor divisible, and whose production therefore does not presuppose productions of parts of themselves, _ad infinitum_. Their development is not construction. They are entire while they are at all, and since the intersections of their terminal moments with space are not at equal distances, these intersections will coincide, or their spatial relations will be inverted, after a certain number of these simultaneities--whether of Achilles’ steps or of the tortoise’s--with points of the road have been counted; in other words, Achilles will have overtaken or outrun the tortoise after a certain number of steps.
To measure the velocity of a motion is simply to find a simultaneity; to introduce this simultaneity into calculation is to use a convenient means of foreseeing a simultaneity. Just as in duration there is nothing homogeneous except what does not lapse, to wit space in which simultaneities are aligned, so the homogeneous element of motion is that which least pertains to it, to wit the space traversed, which is immobility.