, the invariant velocity, is no longer infinite but finite, and if it is a real number, as in Einstein’s theory, the rules according to which velocities combine are found to be those of Lobatchewskian geometry. If, therefore, in Einstein’s theory, we wish to compound velocities, we must operate no longer on a Euclidean but on a Lobatchewskian triangle. We might, for example, trace our triangles on the surface of a pseudosphere, since the geometry of this surface is, as we know, Lobatchewskian. In this case velocities lying in the same direction will no longer add up like the numbers of arithmetic.
3°. In a similar way we should find that if the invariant velocity
, while finite, happened to be an imaginary number, the composition of velocities would follow the rules of Riemann’s geometry. In this case we saw that four-dimensional space-time would be Euclidean and not semi-Euclidean.[66] We merely mention this third type of universe for motives of symmetry. Obviously it does not correspond to the world we live in, hence we will not refer to it in future, confining ourselves to the classical world and to that of relativity.
When we are called upon to decide which of the two alternatives is correct (space-time or separate space and time), which one corresponds to reality, a priori speculations are futile, since both solutions are conceivable and satisfy the facts of crude observation. Our only recourse is then to appeal to experiment, and by experiment we mean observations that are more reliable than our crude perceptions unaided by ultra-precise instruments. It is only thanks to such experiments that we may succeed in ascertaining whether or not a finite invariant velocity is demanded by the world-structure. Should our experiments point to the existence of a finite invariant velocity, our problem would be settled in favour of a world of space-time as against one of separate space and time.
Now, experiment suggests that the propagation of light in vacuo furnishes us with a concrete example of this critical velocity. It follows that a measure of the speed of light propagation will yield us the precise numerical value of the critical velocity. For this reason, the propagation of light assumes a position of vast theoretical importance. Needless to say, this importance of light is due solely to its physical behaviour in vacuo, solely to the fact that it moves with the invariant speed, not to the fact that because of some accidental circumstance it happens to be visible to the human eye. For light passing through glass or water is also visible, but its speed, being reduced, is no longer invariant. Hence light propagation through matter can no longer serve to define the invariant velocity directly. Besides we can get away from light as a visible propagation by considering an invisible electromagnetic propagation in its stead.
If we wish, we can do better still and obviate electromagnetic waves entirely. For when account is taken of the negative experiments in electrodynamics, we realise that the invariant velocity is none other than the constant
which enters into Maxwell’s equations, whence any experiment capable of measuring