Note 19 — Chapter V · AN ALTERNATIVE VIEW OF THE NON-EUCLIDEAN GEOMETRIES

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.