A direct analysis of the notion of velocity will bring us to the same conclusion. Mechanics gets this notion through a series of ideas, the connexion of which it is easy enough to trace. It first builds up the idea of uniform motion by picturing, on the one hand, the path AB of a certain moving body, and, on the other, a physical phenomenon which is repeated indefinitely under the same conditions, e.g., a stone always falling from the same height on to the same spot. If we mark on the path AB the points M, Ν, P ... reached by the moving body at each of the moments when the stone touches the ground, and if the intervals AM, MN and NP are found to be equal to one another, the motion will be said to be uniform: and any one of these intervals will be called the velocity of the moving body, provided that it is agreed to adopt as unit of duration the physical phenomenon which has been chosen as the term of comparison. Thus, the velocity of a uniform motion is defined by mechanics without appealing to any other notions than those of space and simultaneity. Now let us turn to the case of a variable motion, that is, to the case when the elements AM, MN, NP ... are found to be unequal. In order to define the velocity of the moving body A at the point M, we shall only have to imagine an unlimited number of moving bodies A*1, A*2, A*3 ... all moving uniformly with velocities _v_*1, _v_*2, _v_*3 ... which are arranged, e.g., in an ascending scale and which correspond to all possible magnitudes. Let us then consider on the path of the moving body _A_ two points M' and M", situated on either side of the point M but very near it. At the same time as this moving body reaches the points M', M, M", the other moving bodies reach points M'*1 M*1 M"*1, M'*2 M*2 M"*2 ... on their respective paths; and there must be two moving bodies Ah and Ap such that we have on the one hand M' M= M'*h M*h and on the other hand M M"= M*p M"*p. We shall then agree to say that the velocity of the moving body A at the point M lies between _v_*h and _v_*p. But nothing prevents our assuming that the points M' and M" are still nearer the point M, and it will then be necessary to replace _v_*h and _v_*p by two fresh velocities _v_*i and _v_*n, the one greater than _v_*h and the other less than _v_*p. And in proportion as we reduce the two intervals M'M and MM", we shall lessen the difference between the velocities of the uniform corresponding movements. Now, the two intervals being capable of decreasing right down to zero, there evidently exists between _v_*i and _v_*n a certain velocity _v_*m, such that the difference between this velocity and _v_*h, _v_*i ... on the one hand, and _v_*p, _v_*n ... on the other, can become smaller than any given quantity. It is this common limit _v_*m which we shall call the velocity of the moving body A at the point M.--Now, in this analysis of variable motion, as in that of uniform motion, it is a question only of spaces once traversed