The Law of Motion.—50. As we have already gone thus far, we will venture upon a further digression which will enable us to formulate the most important law of physics, the law of motion of a free body. By the attribute free we denote a body or material particle which does not (or in so far as it does not) meet any other material particle in its course. Given any two non-contemporaneous elements of experience forming this particle, the law of motion has to determine the remaining elements of Experience forming it. In current terminology, given any two points in the course of a material point, the law of motion has to determine the remaining points of this course. What will this course be? Can it be determined a priori? I believe it can, if we exclude the possible intervention of some unknown power (e.g., so-called force). It is found by the following process of reasoning: There is an infinite number of possible paths by which two elements of Experience may be joined, and the course which a particle of matter chooses must itself afford the ground for its choice. There must be something in this course which distinguishes it from all other possible courses, otherwise there would be no a priori reason why it did not choose one of the remaining courses. What property can we find in this course to distinguish it from all other possible ones? A clue is afforded us in this direction by the fact of the multiplicity of observers; a course is a course, i.e., a something, some condition of the “external world,” which has no reference to who observes it, and must be the same for all. Its affinity with the interval between two elements of experience is evident; and we shall find that what distinguishes one material particle from others (one path connecting two points from other possible paths) is the interval between its various elements, and that the path which a free material point will take, is determined by the interval between any two of its elements.
If we reverse the sequence of our reasoning and postulate the length of the course of a free body as the measure of the distance between two elements of Experience, we have introduced into the physical continuum a geometry, in which a straight line (a geodetic line) is defined by the course of a free material particle, that is, where the simplest law of motion to which we can attain (“a free material particle moves in a geodetic line”) is at the same time a determinant of the metrical properties of Experience, which will make the geometry of our continuum the simplest.
Physics and Geometry.—51. At this point it will not be inappropriate to touch upon the relation of physics to geometry—a question which has given rise to so many violent controversies in the past forty years. In the first place, it will be
