What is a velocity? It is the space traversed in the course of a second. But the space thus measured by our moving observer, which he believes to be of a certain length, is in reality, for us who are stationary, smaller than he thinks, because the rules he uses are, as Einstein has shown, shortened by velocity without his perceiving it. Therefore the velocities are not added together in equal proportions and indefinitely for a given observer, as classical mechanics maintained. Under the action of the same force, the old mechanics said, a body will always experience the same acceleration, whatever be the velocity already acquired. Under the action of the same force, the new mechanics says, the motion of the body will be accelerated less and less in proportion to its velocity. Take, for instance, some movable object having, relatively to me, a velocity of 200,000 kilometres a second. Let us place an observer on this object. The observer will then start, in the same direction and under the same conditions as we have done, a second movable object, which will thus have, _relatively to him_, a speed of 200,000 kilometres. The Relativist says that the resultant velocity of the second object relatively to us will not be, as the classical addition of velocities would make it, 200,000 + 200,000 = 400,000 kilometres a second. It will be only 277,000 kilometres a second. What the second moving observer took to be 200,000 kilometres (because his measuring rod was shortened owing to velocity) was really only 77,000 of our kilometres. How is it possible to calculate that? Simply by using the formula of Lorentz which I gave in Chapter II, which gives us the value of the contraction due to velocity. We then easily find that, if we have two velocities, _v_ and _v_₂, and if we call the resultant _w_, classical mechanics stated that _w_ = _v_₁ + _v_₂ The Einstein mechanics says that this is not correct, and that what we really have (C being the velocity of light) is _v_₁ + _v_₂ _w_ = ————————————— ( _v_₁_v_₂ ) (1 + —————————) ( C² ) I apologise for again introducing—it shall be the last time—an algebraical formula into my work. But it spares me a large number of words, and it is so simple that every reader who has even a tincture of elementary mathematics will at once see its great significance and the consequences of it. The formula expresses in the first place the fact that the resultant of the velocities, however great it may be, cannot be greater than the speed of light. It conveys also that, if one of the component velocities is that of light, the resultant velocity must have the same value. It means, in fine, that in the case of the slight velocities we have to do with in actual life (that is to say, when the component velocities are much smaller than that of light) the resultant is very nearly equal to the sum of the two components, as the classical mechanics says.