Hence the geometry must be non-Euclidean in a field of gravitation.
10. Since the tracks of particles in a gravitational field are evidently governed by some law, the possible geometries must be limited to certain types.
11. The limitation concerns the absolute structure of the world, and must be independent of the choice of mesh-system. This narrows down the possible discriminating characters. Practically the only reasonable suggestion is that the world must (in empty space) be "curved no higher than the first degree"; and this is taken as the law of gravitation.
12. The simplest type of hummock with this limited curvature has been investigated. It has a kind of infinite chimney at the summit, which we must suppose cut out and filled up with a region where this law is not obeyed, i.e. with a particle of matter.
13. The tracks of the geodesics on the hummock are such as to give a very close accordance with the tracks computed by Newton's law of gravitation. The slight differences from the Newtonian law have been experimentally verified by the motion of Mercury and the deflection of light.
14. The hummock might more properly be described as a ridge extending linearly. Since the interval-length along it is real or time-like, the ridge can be taken as a time-direction. Matter has thus a continued existence in time. Further, in order to conform with the law, a small ridge must always follow a geodesic in the general field of space-time, confirming the conclusion arrived at under (8).
15. The laws of conservation of energy and momentum in mechanics can be deduced from this law of world-curvature.
16. Certain phenomena such as the FitzGerald contraction and the variation of mass with velocity, which were formerly thought to depend on the behaviour of electrical forces concerned, are now seen to be general consequences of the relativity of knowledge. That is to say, length and mass being the relations of some absolute thing to the observer's mesh-system, we can foretell how these relations will be altered when referred to another mesh-system.
