In terms of our non-Euclidean measurements the same plane would be curved. We see, therefore, that the non-vanishing of the Gaussian curvature does not necessarily represent curvature in the usual visualising sense. It represents more truly a relationship between the surface and the behaviour of our measuring rods; in other words, it represents non-Euclideanism, and the word “curvature” is apt to be misleading. In a general way, therefore, we may state that the type of geometry of our two-dimensional space from place to place is defined by the value of the Gaussian curvature from point to point, hence by the law of -distribution throughout the space; and that when the space is Euclidean, the -distribution is always such that the Gaussian curvature vanishes at all points, regardless of the particular mesh-system selected. Now all these discoveries of Gauss relating to a two-dimensional space were extended by Riemann to spaces of any number of dimensions. Riemann found that for spaces of more than two dimensions the results became very much more complex. In the case of a space of dimensions we of course require Gaussian co-ordinates to define the position of a point in our mesh-system, which now becomes -dimensional. As before, we can conceive of Cartesian and Gaussian mesh-systems, [Pg 96] the former being a generalisation to dimensions of our network of Euclidean squares. As for the invariant mathematical expression of the square of a distance between two points, it now contains a greater number of terms. In the case of three-dimensional space we have no longer three separate quantities at every point; this number is increased to six. In the case of a four-dimensional space it is increased to ten, and in the case of an -dimensional space to . Owing to the importance of a four-dimensional extension in the theory of relativity, we shall write out the expression of the square of the distance for a four-dimensional space. If we call , , , the four differences between the four Gaussian co-ordinates of our two infinitely close points and and designate the ten numbers at every point by , , etc., we have or, more concisely, where we give to and all whole values from 1 to 4, permuting them in all possible ways. Accordingly, in what follows, we shall refer to the ’s as the ’s.