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.
If to each vertical and to each horizontal of the mesh-system we assign consecutive whole numbers from zero on indefinitely, we see that the point of the plane which happens to coincide with the intersection of some particular horizontal and some particular vertical is defined by the numbers which represent the two lines, respectively. In this way every point of intersection is defined by two numbers, and these numbers are called the Cartesian co-ordinates of the point. Of course, by this method we are unable to define the positions of points which do not happen to coincide with the corners of our squares. But there is nothing to prevent us from assuming that between the verticals and horizontals we have mentioned there lie an indefinite number of other similar lines, to which intermediary fractional numbers will be assigned. Henceforth every point of the plane can be regarded as defined by the intersection of some particular horizontal and some particular vertical.
To what extent is it permissible to say that points on the plane have been defined by this method? If we disregard the existence of the co-ordinate system, nothing has been defined, but if we consider the co-ordinate system as given, then every point of the plane can be considered as defined unambiguously. In short, the points are defined not in the abstract, but in relation to the co-ordinate system. There is nothing mysterious about this method of defining the positions of points. Thus, in everyday life, when we agree to meet a friend at the corner of Fifth Avenue and Forty-second Street, we are inadvertently locating our point of meeting in terms of the Cartesian co-ordinate system defined by the avenues and streets. In the present case the co-ordinate system is not strictly Cartesian, since the streets and avenues may not enclose perfectly equal Euclideanly square blocks, but the general principle involved is the same. Needless to say, the definition of our point of meeting would convey no significance were the avenues and streets non-existent. Hence once again we see that it is only relative to the co-ordinate system that points can be defined.
Now, the essential characteristic of the Cartesian procedure is its use of a system of reference represented by separate families of intersecting lines. The fact that the lines we have considered are mutually perpendicular straight lines forming a network of Euclidean squares is of no particular importance. It would be just as feasible, in place of our horizontals and verticals, to select two families of intersecting curves, which we might call the
and
curves. Of course, our mesh system would now be curvilinear and the spaces enclosed by the meshes would no longer be Euclidean squares, nor even necessarily equal in area. This generalization of Descartes’ method was introduced by Gauss, and for this reason curvilinear mesh-systems are also called Gaussian mesh-systems. As before, every point will be defined by the numbers designating the two curves of either family, that is, by the numbers designating the