Note 39 — Introductory material

On the whole, then, a modification of Sir R. Ball's view, which is practically a generalized statement of Beltrami's method, seems the most tenable. He imagines what, with Grassmann, he calls a Content, i.e. a perfectly general three-dimensional manifold, and then correlates its elements, one by one, with points in Euclidean space. Thus every element of the Content acquires, as its coordinates, the ordinary Euclidean coordinates of the corresponding point in Euclidean space. By means of this correlation, our calculations, though they refer to the Content, are carried on, as in Beltrami's Saggio, in ordinary Euclidean space. Thus the confusion disappears, but with it, the supposed Euclidean interpretation also disappears. Sir R. Ball's Content, if it is to be a space at all, must be a space radically different from Euclid's[55]; to speak, as Klein does, of ordinary planes with hyperbolic or elliptic measures of distance, is either to incur a contradiction, or to forego any metrical meaning of distance. Instead of ordinary planes, we have surfaces like Beltrami's, of constant measure of curvature; instead of Euclid's space, we have hyperbolic or spherical space. At the same time, it remains true that we can, by Klein's method, give a Euclidean meaning to every symbolic proposition in non-Euclidean Geometry. For by substituting, for distance, the logarithm above alluded to, we obtain, from the non-Euclidean result, a result which follows from the ordinary Euclidean axioms. This correspondence removes, once for all, the possibility of a lurking contradiction in Metageometry, since, to a proposition in the one, corresponds one and only one proposition in the other, and contradictory results in one system, therefore, would correspond to contradictory results in the other. Hence Metageometry cannot lead to contradictions, unless Euclidean Geometry, at the same moment, leads to corresponding contradictions. Thus the Euclidean plane with hyperbolic or elliptic measure of distance, though either contradictory or not metrical as an independent notion, has, as a help in the interpretation of non-Euclidean results, a very high degree of utility.