628 VIII. ON THE STRUCTURE OF MATHEMATICS 

represented by the second derivative of the distance with respect to the 'time'; symbolically,
It is important to notice that in spacetime an accelerated motion is represented by a curved line. In uniform (constant velocity) motion the distances are proportional to the 'times', and the line is straight and its equation is of the first degree. In accelerated motion the distances are not proportional to the 'times', the lines are curved and the 'time' element dt enters in the second degree at least; namely, as dt*.
For example, let us study the graph of the motion represented by the equation which means that the distance x is proportional to the
square of the 'time'. 







Similarly, in threedimensional spacetime, a point moving uniformly in the plane XY would be represented in the plane XY by the line AB, and in threedimensional spacetime by the static line AB', where the 'times' are proportional to the distance.
As we have already seen, nonrectilinear motion may be considered as accelerated motion. We will generalize the above to the case where any curved path is traversed with constant velocity. In this case the direction of the velocity is changed. If we take the motion of a point which describes a circular orbit with constant velocity, it is easy to find its acceleration, which is called in this case centripetal acceleration.
Let us consider a point P, moving in a circular orbit with a constant velocity v, as given in Fig. 16. If at a certain 'time' it is at A, after a short interval /, it will be at B. The direction of the velocity will be changed from AA' to BB'.

