help for it, if the longest track can be a spiral like that known to be described by the earth. Non-Euclidean geometry is necessary. In Euclidean geometry the shortest track is always a straight line; and the slight modification of Euclidean geometry described in Chapter iii is found to give a straight line as the longest track. The status of non-Euclidean geometry has already been thrashed out in the Prologue; and there seems to be no reason whatever for preferring Euclid's geometry unless observations decide in its favour. Equation (1), p. 53, is the expression of the Euclidean (or semi-Euclidean) geometry we have hitherto adopted; we shall have to modify it, if we adopt non-Euclidean geometry.
But the point arises that the geometry arrived at in Chapter iii was not arbitrary. It was the synthesis of measures made with clocks and scales, by observers with all kinds of uniform motion relative to one another; we cannot modify it arbitrarily to fit the behaviour of moving particles like the earth. Now, if the worst came to the worst, and we could not reconcile a geometry based on measures with clocks and scales and a geometry based on the natural tracks of moving particles—if we had to select one or the other and keep to it—I think we ought to prefer to use the geometry based on the tracks of moving particles. The free motion of a particle is an example of the simplest possible kind of phenomenon; it is unanalysable; whereas, what the readings of any kind of clock record, what the extension of a material rod denotes, may evidently be complicated phenomena involving the secrets of molecular constitution. Each geometry would be right in its own sphere; but the geometry of moving particles would be the more fundamental study. But it turns out that there is probably no need to make the choice; clocks, scales, moving particles, light-pulses, give the same geometry. This might perhaps be expected since a clock must comprise moving particles of some kind.
A formula, such as equation (1), based on experiment can only be verified to a certain degree of approximation. Within certain limits it will be possible to introduce modifications. Now it turns out that the free motion of a particle is a much more sensitive way of exploring space-time, than any practicable measures with scales and clocks. If then we employ our accurate
