We are now able to state formally our proposed law of motion—Every particle moves so as to take the track of greatest interval-length between two events, except in so far as it is disturbed by impacts of other particles or electrical forces.
This cannot be construed into a truism like Newton's first law of motion. The reservation is not an undefined agency like force, whose meaning can be extended to cover any breakdown of the law. We reserve only direct material impacts and electromagnetic causes, the latter being outside our present field of discussion.
Consider, for example, two events in space-time, viz. the position of the earth at the present moment, and its position a hundred years ago. Call these events and . In the interim the earth (being undisturbed by impacts) has moved so as to take the longest track from to —or, if we prefer, so as to take the longest possible proper-time over the journey. In the weird geometry of the part of space-time through which it passes (a geometry which is no doubt associated in some way with our perception of the existence of a massive body, the sun) this longest track is a spiral—a circle in space, drawn out into a spiral by continuous displacement in time. Any other course would have had shorter interval-length.
In this way the study of fields of force is reduced to a study of geometry. To a certain extent this is a retrograde step; we adopt Kepler's description of the sun's gravitational field instead of Newton's. The field of force is completely described if the tracks through space and time of particles projected in every possible way are prescribed. But we go back in order to go forward in a new direction. To express this unmanageable mass of detail in a unified way, a world-geometry is found in which the tracks of greatest length are the actual tracks of the particles. It only remains to express the laws of this geometry in a concise form. The change from a mechanical to a geometrical theory of fields of force is not so fundamental a change as might be supposed. If we are now reducing mechanics to a branch of natural geometry, we have to remember that natural geometry is equally a branch of mechanics, since it is concerned with the behaviour of material measuring-appliances.
Reference has been made to weird geometry. There is no
