5 0 2 D O C . 5 0 6 A N N I V E R S A R Y O F N E W T O N ’ S D E A T H the path covered during time IJ. If one imagines the distance P–G as rectilinearly extended beyond G, and imagines the length plotted as longer than P–G by as much as the unit of time is greater than IJ, then one obtains the velocity of the pointlike body at point P in the form of an arrow of a specific length, a so-called vector. But this is not entirely correct. The arbitrary choice of the length of the short span of time will have an influence on the result, small though it may be. One will have to define it more precisely: The smaller the difference in time IJ is chosen to be, the more precisely the arrow constructed in this way will represent the velocity. This is an exact mathematical definition of the arrow or vector of the velocity by means of a limiting process. The acceleration is then similarly defined by the velocity as the latter had been by the given motion. The velocity is defined by an arrow for each time. Imagine the velocity existing at one time as represented by an arrow L of a particular size and direction. The velocity has changed after a short time IJ has elapsed this means that the new arrow M has a somewhat different direction and length. Now imagine the arrows L and M plotted from one point in space. The tips S and T of the arrows then do not coincide exactly. The directed distance S–T, the connection between the arrow tips, then represents the change in velocity during time IJ. If one extends S–T beyond T inasmuch as the unit of time is larger than IJ, one obtains the change in velocity with reference to the unit of time, that is, the ac- celeration as an arrow or vector. This, too, is a limiting process. For, the smaller the time span T chosen, the more accurate the representation will be. According to Newton, the acceleration thus defined is directly determined by the force acting on the point mass. However, it is not the case that the arrow represent- ing the force is the same as the one representing the acceleration. Because, in order to move a point mass weighing 2 kg in a particular way, forces are obviously re- quired that are double the strength necessary for a point mass of 1 kg. This is how Newton came to introduce the mass of a body and lay out his famous law of motion. Mass times vector of acceleration = vector of force. This is the foundation of all mechanics, indeed of theoretical physics as a whole. Assuming that the force acting on a point mass were given for all times, then its acceleration is also given for all times. Then finding out its velocities and its loca- tions for all times is a purely mathematical problem, not a scientific one anymore. But how would Newton be able to discover the forces acting on the celestial bod- ies? He surely could not simply invent their correct formulation. He could only have proceeded the other way, by obtaining these forces from known motions of the planets and the moon, by deducing them from the accelerations. He accom- plished all of this essentially as a 23-year-old youth, with concentrated effort in rural seclusion. [p. 39]