296 G. H. KNIBBS. 



and hyperbolic space have no existence, as representing a possibility 

 of "actual" space. 1 



25. Elliptic and hyperbolic space merely a specialised region in 

 a homaloid. — The confidence which mathematicians have felt as to 

 the possible existence of types of space other than homaloidal, has 

 arisen from the inherent consistency of "analytic or symbolic 

 geometry." In the introduction to this paper, it was pointed out 

 that a region of space may be specialised, as for example, by being 

 referred to curvilinear axes, varied in intensity, regarded as 

 seolotropic, or supposed to be otherwise specially constituted. Any 

 specialisation of space whatever that can be represented by symbols, 

 has its appropriate analytical geometry, or scheme of symbolic 

 operation, in which geometrical interpretation is indifferent until 

 necessitated in applying the final results. The belief that algebras, 

 as such, can establish results which reveal the true nature of 

 factual" tridimensional space, or reach results that, though funda- 

 mentally affecting the concepts or intuitions of space, cannot 

 immediately be apprehended by pure geometry, is based upon a 

 complete misconception of the nature of the "actuality" of space. 

 The function of geometries, either analytic, metric, or projective, 

 is to reveal the properties, not of space, but of geometrical figures 

 either in pure space, or in any specialised space which can be clearly 

 conceived, and interpreted into and from the symbols. 



In pure space, any geometrical figure whatever, [e.g. point, 

 straight or curved line, plane or curved surface, or solid of any 

 form) may exist, because conceptually they are spatial, while space 

 of (3 + m) dimensions is purely abstract in regard to every one of 

 the m dimensions, and is spatially not representable, though by 

 analogies, such space is schematically representable. Elliptic and 

 hyperbolic geometries refer therefore to figures in space or to space 

 specially constituted, as for example, space filled with a refracting 



1 Stringham affirms "there is yet no theory of knowledge that can tell 

 us which of these three diverging paths we must take" i.e. interpret space 

 in terms of parabolic, elliptic or hyperbolic geometry, vide " On the funda- 

 mental differential equations of Geometry"! Journal B. A. A. Sc. 1899, 

 p. 647. With this dictum we of course, join issue. 



