This line of reasoning would be correct in the present case, but it is important to note that this conception of an inside and an outside is in no wise essential to non-Euclidean geometry. Had we adopted the more rational method of presenting three-dimensional non-Euclidean space as due to the peculiar behaviour of our measuring rods, the conception of an inside and an outside would have been meaningless. We must understand that this notion arose merely as a result of our particular choice of a mode of presentation, and in no wise constitutes an intrinsically necessary condition. If these rather delicate points have been understood, no harm can be done by discussing non-Euclidean geometries as the geometries obtained by applying Euclidean rods on curved surfaces. In many respects, indeed, this method of presentation is helpful; for accustomed as we are in everyday life to effect measurements with Euclidean rods, we are able to visualise more easily a series of abstract investigations when this familiar procedure is followed. Let us now proceed to a more thorough study of the preceding method. We have said that non-Euclidean results are obtained when we confine ourselves to Euclidean rods while conducting our measurements on curved surfaces. In the case of Riemannian geometry of two dimensions, the required surface is that of a sphere embedded in three-dimensional Euclidean space. In the case of Lobatchewski’s geometry, it is a saddle-shaped surface known as a pseudosphere (subject to the limitations mentioned previously). We will confine ourselves to the study of Riemann’s geometry for the time being. Such a study in two dimensions permits easy visualisation, owing to the fact that since the earth is spherical in shape it is precisely this type of geometry which we obtain when we conduct measurements on its surface with rigid Euclidean measuring rods (assuming the surface to be perfectly smooth, like that of the ocean on a calm day). In fact, we shall see that the postulates of Riemann’s geometry are immediately verified on the sphere. To begin with, the shortest distance, or, better still, the most direct distance, between two points on the earth’s surface, when Euclidean rigid rods or inextensible tape measures are employed as a [Pg 65] means of measurement, is an arc of a great circle. Great circles, such as meridians on the surface of our planet, constitute therefore the straight lines or geodesics of Riemann’s geometry; and when discussing Riemann’s geometry we may call them straight lines.