Rogers — The Logical Basis of Mathematics. 187 



the axioms and tlio preceding propositions ; others do not. Thus, for example, 

 Euc. I. 4 should be preceded by a series of axioms on congruence. Euclid's 

 proof of this (regarded as proof) is not intuitional, but materialistic ; and from 

 one point of view is either a contradiction or a 'petitio prrncipii. Moreover, 

 the method of superposition cannot be applied to the superposition of reflected 

 tetrahedra, i.e., those of opposite aspect. So to apply it to the third dimension 

 to prove equality of volumes, one must assert an intuition of the fourth 

 dimension ! 



The third stage in the generalization of geometry comprises analytic and 

 kindred geometries, which, though at first apparently Euclidean, provide a 

 weapon for the construction of non-Euclidean schemes. The Cartesian 

 method is essentially logical. At first it is used simply as a device for the 

 continued logical application of axioms once suggested by intuition ; but its 

 range goes much further. 



The fourth step in the generalizing process is represented by the earlier 

 non-Euclideans of the nineteenth century, such as Kiemann, who had not 

 clearly separated the question of existence from the question of logic. 

 Cayley's Theory of Distance forms an intermediate stage. 



The fifth step of generalization is the purely logical theory, which is now 

 in full swing, and is an explanatory development of the non-Euclidean 

 method. 



It is commonly urged that non-Eucliclean geometry has no objective 

 basis ; that it is merely fantastic, and has no connexion with Nature. This 

 objection, however, only applies to the earlier forms of non-Euclidean systems, 

 such as Riemann's investigation of the curvature of space. Here the terms 

 used imply, or are popularly thought to imply, that actual space may have 

 4 or n mutually perpendicular straight lines. This, however, is only pseudo- 

 logic, and is in fact self-contradictory. Likewise it is a logical contradiction 

 to say — using the terms in their ordinary sense — that two straight lines 

 may intersect twice, or that any two straight lines must meet. But w^hen 

 we define our whole system without reference to actual space, the contra- 

 diction disappears. It is probable, however, I think, that all non-Euclidean 

 systems have an actual meaning. The real question at issue is not, Is non- 

 Euclidean geometry actual, but Is it useful ? ISTow, projective geometry, in 

 which any two straight lines intersect, has been used to extend our knowledge 

 of actual space — e.g., by Cayley and Salmon. The whole doctrine of the 

 line at infinity where parallel lines meet, the circle at infinity, the / and J 

 points, is essentially non-Euclidean, and yet productive. The same applies 

 to spherical and all non-planar two-dimensional systems, where geodesies take 

 the place of straight lines, and may intersect any number of times. As regards 



