1°. If , the invariant velocity, is infinite, as in classical science, it is found that the composition of velocities must be Euclidean; so we should be able to combine velocities graphically by the well-known rule of the parallelogram or of the Euclidean triangle. In this case, of course, velocities which lie in the same direction can be added up and subtracted like numbers. 2°. If , 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. [Pg 204] 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.