PKINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 



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a at oo 



Fig. 21 



Eig. 23 



24. Is elliptic? and hyperbolic space of n dimensions representable 

 in space 0/(11+ 1) dimensions? — Consider the circumference of a 

 circle, and the surface of a sphere : these are geometrical figures 

 without terms. As line and surface they are respectively 1- and 

 2-dimensional figures, but as boundaries they are respectively 2- 

 and 3-dimensional ■ and we have seen that if the radius of the 

 circle or sphere be infinite, they are pseudo- or finitely homa- 

 loidal, see § 12. Observing further that the boundary of a sur- 

 face is a line, and that of a solid is a surface, and remembering 

 that a 4-dimensional quantum is (conceptually) generated by the 

 motion of a solid into the fourth dimension, see § 4, we conclude 

 that space of n dimensions, is, conceptually, the boundary of space 

 of (n + 1) dimensions. Consistently with the developments of 

 §§ 7, 9, and 12, we may define an w-dimensional homaloid as the 

 boundary of an (n+l) dimensional space, whose (n+l)th axis 

 only, is infinitesimally curved, positively or negatively: conse- 

 quently the question arises whether the ^-dimensional space will 

 be elliptic or hyperbolic according as the curvature, if finite, is 

 positive or negative, as represented in Fig. 22. In the tridimen- 

 sional or ordinary homaloid the triangles ABC are, by hypothesis, 

 plane — the curvature not existing on their tridimensional repre- 



