Now suppose that a ball is rolling along the moving train with a speed [Pg 119] , and suppose that owing to the train’s motion the train observer is carried past the embankment observer at the instant of time when the clocks mark noon. If the ball leaves the train observer at this precise instant of time, its distance from the train observer at a time (i.e. seconds after the noon instant) will be (since and are identical). But then its distance from the embankment observer will be the sum of the distance between the two observers, and of this distance we have just expressed (distances being absolute). It will be, therefore, (where is the velocity of the train along the embankment).[38] This last deduction is obvious provided we assume that a distance is an absolute concept which would turn out the same whether measured by one observer or another, and such was precisely the assumption that classical science always made. In short, knowing the point where an event takes place at a given instant as referred to the train we can deduce immediately the point where the same event takes place when referred to the embankment. All we have to do is to replace by , and the trick is done. These space and time transformations permitting us to pass from one Galilean frame to another were the celebrated Galilean transformations of classical science, and they were, as we have seen, the necessary consequences of our belief in absolute duration and distance. They were given by: or again by Applying them to particular problems, as, e.g., to the change in the shape of the trajectory of a falling body when viewed successively from a moving train and from the embankment, or to the change in the colour of a monochromatic source of light, or in the pitch of a musical note, the anticipations of the transformations were always found to be verified, to the order of precision of our experiments. Corroborations of this sort, among others, were considered to prove the correctness of the transformations, hence also of our traditional belief in the absoluteness of duration and distance.