 Neumann’s definition of a time-scale directly inspired Ludwig Lange’s conception of “inertial system” (Lange 1885). An inertial coordinate system ought to be one in which free particles move in straight lines. But any trajectory may be stipulated to be rectilinear, and a coordinate system can always be constructed in which it is rectilinear. And so, as in the case of the time-scale, we cannot adequately define an inertial system by the motion of one particle. Indeed, for any two particles moving anyhow, a coordinate system may be found in which both their trajectories are rectilinear. So far the claim that either particle, or some third particle, is moving in a straight line may be said to be a matter of convention. We must define an inertial system as one in which at least three free particles move in straight lines. Then we can state the law of inertia as the claim that, relative to an inertial system so defined, the motion of any fourth particle, or arbitrarily many particles, will be rectilinear. The notions of inertial system and Neumann’s time-scale, which Lange called an “inertial time-scale,” may be combined as follows: relative to a coordinate system in which three free particles move in straight lines and travel mutually-proportional distances, the motion of any fourth free particle will be rectilinear and uniform. The questionable Newtonian concepts of absolute rotation and acceleration, Lange proposed, could now be replaced by the concepts of “inertial rotation” and “inertial acceleration,” i.e. rotation and acceleration relative to an inertial system and inertial time-scale. See Figures 2 and 3.