PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 317 



as to exhibit the features of elliptic, parabolic and hyperbolic 

 geometry, 1 and is moreover tortuous. 



These surfaces have been designed merely to illustrate how 

 easily, and in what an illimitable number of ways space can be 

 specialised, and that for such space special geometries may be 

 created. If the intensity of the surfaces in these figures be also 

 varied the geometries may be still further complicated. 



The constants of space of positive curvature may, as an abstract 

 question, be of any magnitude whatever, consequently if the view 

 that space may be positively curved were correct, it is a possibility 

 that ordinary space is the locus of an infinity of unbounded curved 

 "spaces." 3 It is equally possible also, that it consists of an infinity 

 of "spaces" which are of the complex type roughly sketched or of 

 more complex type still. The over-sublety of the conception 

 would sufficiently argue its inutility as a foundation for geometry, 

 even if it could be shewn to be consistent. 



38. Pangeometry. — Non-euclidean- and pan-geometry have 

 generally been regarded as practically identical. Much of what 

 has been assigned these names is, as I have endeavoured to shew, 

 really a geometry, not of space, but of specialised regions of space, 

 analogous in some features to the geometry of curved surfaces. 

 The true homaloid or continuum of three dimensions, i.e. space in 

 the vulgar sense, is the foundation element of all spatial concep- 

 tions; and its dimensionless point, its straight line without breadth 

 and thickness, its plane surface without thickness, its isotropy and 

 homogeneity, are the fundamental forms which render any 

 geometry intelligible, and which make possible a consistent study 

 of its various specialisations. The bizarre idea that space may be 

 discovered to be anything different from a homaloid, is really self- 



1 G, G', G" are successive positions of the generating surface AB, AC, 

 AD, AE, AF, etc., join corresponding points on the generatrix. Helui- 

 holtz's sphere-dweller, if on this surface, would conclude that parallel 

 lines may be parallel, diverge or converge. If he supposed to perceive 

 intensity as he looked along a line, the straight lines would be unique in 

 that respect. 



2 That is spatial regions, each "unbounded." 



