The inverse square law of gravitation

1—In a previous paper the "cosmical" acceleration of a free particle in the presence of the substratum or smoothed-out universe was obtained, and shown to be of the nature of gravitation. In the present paper, the abstract problem of "local" gravitation is considered. The simplest problem of "local" gravitation is the Kepler-Newton problem., which in classical mechanics is the problem of ascertaining the acceleration undergone by a free test-particle in the presence of an isolated point-mass in an otherwise empty universe. Our object is to derive the value of this acceleration by purely kinematic arguments, that is to say, arguments which rely for their empirical premises only on the existence of a temporal experience for each individual observer; as was implicit in the thinking of Zeno, such a temporal experience has to be taken as given before motion can be described at all. However, the concept of the isolated gravitating, mass-particle in an otherwise empty universe is essentially an illegitimate one. In the first place it ignores Mach's principle. We must introduce an array of observers before a relativistic account of gravitation can have a meaning, and these observers must have positions and velocities in order that they may describe the position and velocity of the isolated mass-particle. They must therefore be associated with the presence of particle. They must therefore be associated with the presence of particles other than the massive particle under consideration, and these will play their part in determining the acceleration of a free test particle. In the second place, it has been shown in the previous paper, in general accordance with many modern views, that the phenomenon of gravitation, as summed up in the existence of a "constant" of gravitation, depends itself on the mean matter and motion in the substratum. If we abolish the substratum, as in the classical formulation of the Kepler-Newton problem, we abolish the elements of existence which lead to the isolation of a constant of gravitation. We must therefore retain the substratum.