WE HAVE mentioned in a general way the significance of congruence and of spatial distance. It now remains for us to find a means of defining these concepts in a rigorous mathematical form. We remember that the equality or congruence of two spatial distances between two point pairs and was an indeterminate concept, depending essentially on the behaviour of our measuring rods. An alternative presentation of non-Euclideanism (in the case of two-dimensional geometry) was then found to be afforded by assuming that the distance between points could in all cases be determined by measurements with rigid Euclidean rods; but that, whereas in the case of Euclidean geometry all the points should be considered as existing in the same plane, in the case of non-Euclidean geometry it would be as though the points were situated on a suitably curved surface. Thus, in the case of the earth, the non-Euclidean distance between two points, say New York and Paris, would be given by the Euclidean length of a great circle extending between these points, hence by a curved line following the contour of the earth’s surface. On the other hand, the Euclidean distance between these two same points would be given by the Euclidean length of the straight line joining them and passing, of course, through the earth’s interior. We now propose to investigate the mathematical expression of distance in two dimensions; we will assume that we are discussing it from the standpoint of Euclidean measurements conducted on surfaces. Let us first consider the case where the surface is an unlimited plane. If we wish to define the position of a point of the plane, we must refer it to some system of reference. Three centuries ago Descartes devised a method whereby this result could be accomplished. He considered two families of Euclidean straight lines which we may call horizontals and verticals, respectively. The lines of these two intersecting families are equally spaced (Euclideanly speaking), so that they form a mesh-system or network of equal Euclidean squares. The scientific name for a mesh-system is co-ordinate system, but the appellation “mesh-system” introduced by Eddington has the advantage of giving a more graphic picture of what is involved. The type of mesh-system constituted by horizontals and verticals introduced by Descartes is called a Cartesian co-ordinate system.