Our next problem is to determine the nature of the geometry of this mysterious continuum. The mere fact that it has been possible to write the expression of this square of the interval with squared magnitudes, such as , , etc., without having recourse to terms such as , , etc., would of itself imply that the continuum was flat and free from all trace of non-Euclideanism or curvature. Furthermore, it implies that the mesh-system to which we refer this distance is one of straight lines, and is not curvilinear (see Chapter VII). The only flat continuum we have studied so far is the Euclidean one, but in the present case we see that though flat, our continuum is not strictly Euclidean owing to the minus sign preceding ; or, again, owing to the fact that , instead of , as in a four-dimensional Euclidean space. Yet, because of the constancy in the values of the ’s throughout the mesh-system, hence because of the flatness or absence of non-Euclideanism or curvature of the continuum, the analogy with a Euclidean continuum is very great, and for this reason it is called semi-Euclidean.[64] We say, therefore, that space-time is a four-dimensional semi-Euclidean continuum, and that it is differentiated from a truly Euclidean four-dimensional one solely because it has one imaginary dimension and three positive ones in place of four positive ones. In this continuum, time represents the so-called imaginary dimension, and the three dimensions of space represent the three positive ones; though it must be remembered that all we wish to imply by this statement is that there exists a difference between the dimensions of space and that of time. We should be equally justified in calling time a positive dimension, and the three dimensions of space imaginary dimensions. [Pg 197] Now we have also seen that the mathematical form of the expression which gives the square of the distance depends not only on the geometry of the continuum, but also on the system of co-ordinates we may have adopted. In the present case the form of the mathematical expression tells us that our co-ordinate systems or mesh-systems are necessarily Cartesian, that is, are constituted by mesh-systems of four-dimensional cubes. Hence we see that all our different Galilean mesh-systems, when taken as frames of reference, correspond to differently orientated Cartesian mesh-systems in four-dimensional space-time.[65] In short, space-time must be regarded as the fundamental continuum of the universe. Of itself it is neither space nor time. Any Galilean observer splits it up into four directions by means of his particular Cartesian mesh-system; one of these directions corresponds to his time measurements, while the three others correspond to his space measurements.