 Figure 2: Neumann’s Time-Scale By Newton’s first law, a particle not subject to forces travels equal distances in equal times. But which particles are free of forces? This might appear to be a matter of convention.  (a) Either \(P_1\) or \(P_2\) can be arbitrarily stipulated to be at the origin of a system of coordinates, and to serve as the measure of equal times  (b) But one can say of two particles with different velocities: in intervals of time in which one moves a given distance \(d_1\), the other moves a proportional distance \(d_2 = kd_1\) (where k is a constant; i.e., \(d_1/d_2 = k\)). Or one can compare a particle to a freely rotating planet: in intervals of time through which the planet rotates through equal angles, the particle moves equal distances. 