and not
, as at first sight we might be inclined to believe.
[19] An equivalent form of the criticisms discussed in the text consists in assuming that as Euclidean geometry is associated with the number zero, it must be logically antecedent to the non-Euclidean varieties since these are associated with non-vanishing numbers. But here, again, it is only because we take “curvature” as fundamental that Euclidean geometry is connected with zero. Should we choose to take “radius of curvature,” Euclidean geometry would be associated with infinity, and the non-Euclidean varieties with finite numbers. Hence, the problem would now resolve itself into determining whether “curvature” or “radius of curvature” was the more fundamental; and, in point of fact, “curvature” is a more complex conception than “radius of curvature.”
Then again, we might characterise the various geometries by means of the parallel postulate. In this case, Riemann’s geometry would be associated with the number zero, Euclidean geometry with the number one, and Lobatchewski’s geometry with infinity. Still another method of presentation would be to approach the problem through projective geometry. Then we should find that Lobatchewski’s geometry was associated with a circle, and Euclid’s with the intersection of the line at infinity with two imaginary lines, yielding the “circular points at infinity,” whereas Riemann’s geometry would be connected with a circle of imaginary radius. With this method of representation, zero would never enter into our discussions. And no one would maintain that the concept of imaginary points at infinity (the circular points) was logically antecedent to a real circle of finite radius. In short, we see that there are a large number of different methods of representing the various geometries; and, according to the method selected, the number zero may be associated with Euclideanism or with non-Euclideanism, or, again, with neither. All that we are justified in saying is that Euclidean geometry is the easiest of the geometries, but not necessarily the most fundamental.
[20] It is to be noted that had the source of light been placed at the centre of the sphere instead of at the north pole, all the great circles on the sphere would have cast straight lines for shadows on the tangent plane. Still, the shadows on the plane would, as before, have yielded Riemann’s geometry (of the elliptical variety). We see then, once again, that absolute shape, size and straightness escape us in every case, and we cannot even say that Reimann’s straight lines are curved with respect to Euclidean ones. All that is relevant is the mutual behaviour of bodies, their laws of disposition.
[21] For the sake of completeness we may mention that connectivity can be studied by the same procedure as that by which dimensionality was investigated. Thus, a space was considered to be two-dimensional when it was possible to intercept the path of continuous transfer between any two points by tracing a continuous line throughout the space. If we apply this test to the surface of a doughnut, it will be seen that a continuous line, such as a circumference enclosing part of the doughnut, like a ring, would be insufficient to divide the doughnut into two separate parts which could not be connected by a route of continuous transfer over the surface. This property would differentiate the surface of a doughnut from that of a sphere or an ellipsoid, and yet the surface of the doughnut would still be two-dimensional, but its connectivity would be different owing to the hole passing through its substance.
[22] An instant of time is of course a mere abstraction, but so is a mathematical point in space.
[23] We are not referring solely to the tides of the sea, but also to the solid tides generated in the earth’s substance.
[24] The problem is of course much more complex than would appear from the present analysis. For the slowing down of the earth’s rate of rotation, owing to tidal action, would also result in causing a retreat of the moon, producing thereby a variation in the period of its revolution round the earth.