reconstructs the movement of Achilles according to the same law as the movement of the tortoise, forgetting that space alone can be divided and put together again in any way we like, and thus confusing space with motion. Hence we do not think it necessary to admit, even after the acute and profound analysis of a contemporary thinker,[2] that the meeting of the two moving bodies implies a discrepancy between real and imaginary motion, between _space in itself_ and indefinitely divisible space, between concrete time and abstract time. Why resort to a metaphysical hypothesis, however ingenious, about the nature of space, time, and motion, when immediate intuition shows us motion within duration, and duration outside space? There is no need to assume a limit to the divisibility of concrete space; we can admit that it is infinitely divisible, provided that we make a distinction between the simultaneous positions of the two moving bodies, which are in fact in space, and their movements, which cannot occupy space, being duration rather than extent, quality and not quantity. To measure the velocity of a movement, as we shall see, is simply to ascertain a simultaneity; to introduce this velocity into calculations is simply to use a convenient means of anticipating a simultaneity. Thus mathematics confines itself to its own province as long as it is occupied with determining the simultaneous positions of Achilles and the tortoise at a given moment, or when it admits _à priori_ that the two moving bodies meet at a point _X_--a meeting which is itself a simultaneity. But it goes beyond its province when it claims to reconstruct what takes place in the interval between two simultaneities; or rather it is inevitably led, even then, to consider simultaneities once more, fresh simultaneities, the indefinitely increasing number of which ought to be a warning that we cannot make movement out of immobilities, nor time out of space. In short, just as nothing will be found homogeneous in duration except a symbolical medium with no duration at all, namely space, in which simultaneities are set out in line, in the same way no homogeneous element will be found in motion except that which least belongs to it, the traversed space, which is motionless. [Sidenote: Science has to eliminate duration from time and mobility from motion before it can deal with them.] Now, just for this reason, science cannot deal with time and motion except on condition of first eliminating the essential and qualitative element--of time, duration, and of motion, mobility. We may easily convince ourselves of this by examining the part played in astronomy and mechanics by considerations of time, motion, and velocity.