CLASSICAL MECHANICS AND THE NEWTONIAN PRINCIPLE OF RELATIVITY WE mentioned, when discussing amorphous mathematical space, that position and hence motion could have no meaning other than that of expressing conditions with respect to some particular frame of reference. In short, the visual principle of relativity would have to be adhered to. We will accordingly select some arbitrary system of reference, which we will assume to be defined by three mutually perpendicular planes meeting at a common point . The instantaneous position of a body will henceforth be defined when we have measured with rigid rods its distances from the three planes of our frame; and the only type of position and of motion of the body which we can define in an unambiguous way will be its position and motion as referred to our frame. The measurement of the velocity of the body with respect to our frame necessitates the addition of a clock. We will assume that this clock beats out congruent intervals of time, where congruence is, of course, defined by the requirements of practical congruence for time, that is, by the beatings of some isolated periodic mechanism. Now, it is perfectly apparent that unless referred to some particular frame, the path followed by a body and its motion along this path can have no determinate meaning. Thus, if in a uniformly moving train we throw a ball into the air and catch it again in our hands, then, as referred to the train, the ball will have followed an up-and-down motion along the same vertical; on the other hand, as referred to the embankment, the ball will have described a parabola. Again, if a stone is allowed to fall from a great height, its motion with respect to the earth will be accelerated, whereas, with respect to a frame of reference falling together with the stone, it will have remained at rest. It would thus appear that we might credit to the path followed by a body any arbitrary shape we wished; all we should have to do would be to select some appropriate frame of reference. The same indeterminateness would, of course, apply to the motion of the body along its path. Thus the shape of a path in space (rectilinear or curved) or the species of a motion (uniform or accelerated) could have no absolute significance. From the standpoint of mathematical space these would all be essentially relative to the reference frame we had selected.