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 curve and the curve which intersect at this point; and these two numbers will be called the Gaussian numbers or co-ordinates of the point. The necessity of generalising Cartesian co-ordinates by introducing Gaussian ones arises from the fact that Cartesian mesh-systems of equal squares can be traced only on a plane and could never be drawn on a [Pg 85] curved surface, like that of a sphere, for example. Hence, were we to ignore the use of Gaussian mesh-systems, it would be impossible for us to localise points on a curved surface, by means of a mesh-system applied on the surface. The nearest approach to a network of squares on the surface of a sphere would be a network of meridians and parallels, and such a network is not one of equal Euclidean squares; it is a curvilinear or Gaussian mesh-system tapering to points at the North and South Poles. As a matter of fact, we also make use of Gaussian co-ordinates in everyday life. Such is the case when we state that the position of a ship is so many degrees of latitude and so many degrees of longitude; our mesh-system, being one of meridians and parallels, is a Gaussian one, and the latitude and longitude of the ship constitute its Gaussian co-ordinates. Having determined how the positions of points may be defined on any surface in terms of some mesh-system, let us now see how it is proposed to express the distance between two points, as measured over the surface with rigid Euclidean rods. Here we must proceed with the utmost caution. Let us recall exactly what is involved. Any definite line is one along which remains constant while varies continuously, and inversely any definite line is one along which remains constant while varies continuously. In much the same way, on the earth’s surface a latitude line or parallel is one along which the latitude remains constant and the longitude varies, whereas a longitude line or meridian is one along which the reverse is true. Fig. II