This argument is radically incorrect, and arises from too loose an understanding of non-Euclidean geometry. In the first place, to assume that the concept of curvature presupposes the concept of straightness is no more inevitable than to assume that the concept of straightness presupposes that of curvature. For if it is true to say that that which is curved is that which is not straight, it is equally true to say that that which is straight is that which is not curved; so that in the absence of curvature, straightness would in turn be meaningless. The question of deciding, for example, whether a circle or a straight line is the more fundamental is strictly a matter of opinion. The Greeks held circular motion to be the noblest of all motions; and between the designations “noblest” and “most fundamental,” the distinction is exceedingly slight. And even this is not all, for thus far we have been arguing as though Euclidean space were truly flat, and non-Euclidean space truly curved. But we must remember that this curvature of which we speak is not to be construed as representing anything absolute; it arises solely from the particular type of representation we have agreed to select. Thus, in the presentation of non-Euclidean geometry which we have just been discussing, we started from three-dimensional Euclidean space as a basis and considered curved surfaces embedded in this space. This form of presentation was equivalent, as we know, to stating that Euclidean measurements must be adhered to on the curved surface. But we might have proceeded otherwise. We might have started with three-dimensional non-Euclidean space and considered the geometry of curved surfaces (of a sphere, for example) embedded in this space. This would have been equivalent to applying non-Euclidean rods on our curved surface. Under suitable conditions the geometry of our curved surface (or sphere) would then become Euclidean. Euclideanism itself would thus be linked with curvature, while non-Euclideanism, which would here be the type of geometry obtained on the plane embedded in our three-dimensional non-Euclidean space, would accordingly be linked with flatness. In short, we must not allow the word “curvature” to mislead us into forming a false impression about non-Euclidean geometry. In many respects Riemann’s choice of the word “curvature” has proved unfortunate and the more scientific appellation non-Euclideanism [Pg 63] should be adhered to.[19]