PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 245 



ceived as having an indefinite number of dimensions, but also as 

 possessing positive or zero curvature. 1 It is easy to see that the 

 introduction of such ideas into the theory of space, in no way 

 exhausts its possible complexity. Thus conceptually, there may 

 be tridimensional tortuous space, and even n-dimensional space of 

 manifold tortuosity, in exactly the same sense as there may be 

 "curved space," though the development of the geometries of these 

 has not so far been attempted. There is in fact no limit to the 

 complexity of conceptual space : To anyone, however, who is 

 sufficiently a Kantian 2 to still believe that the limitless infinity 

 of tridimensional homaloidal space 3 is the form in which the 

 intellect, by its original constitution, is forced to interpret all 

 natural phenomena, the interpretative argument of the non- 

 euclidean geometers appears to be wrongly founded. This is 

 assuredly so if the ordinary conception, at least when clearly 

 realised through attention to the matter, is not only the most 

 comprehensive, but is also the background against which as it 

 were, the lineaments of other geometries 4 are revealed, and without 

 which they would remain unintelligible. No argument can 

 logically be founded on analytic, i.e., algebraic geometry, taken by 

 itself, for it is evident that the quantitative expressions both in 

 arithmetic and algebra, are not in themselves a geometry, though 

 within certain limits, and subject to certain restrictions they are 

 susceptible of a geometrical interpretation. At best they cannot 



1 ' Space of positive curvature ' or ' synclastic space ' is now generally 

 known as 'elliptic space/ In the geometries of such space, Klein dis- 

 tinguishes between the 'polar' and ' antipodal ' forms, calling the former 

 f elliptic/ and the latter ' spherical geometry/ ' Space of negative curva- 

 ture/ or 'anticlastic space ' is similarly called 'hyperbolic space/ and its 

 geometry known as 'hyperbolic geometry/ or ' pseudospherical geometry/ 

 The limiting form between the two is 'parabolic space* or 'homaloidal 

 space/ Whitehead, however, evidently regards this last as an unsound 

 definition, see his Universal Algebra, Vol. i., p. 451 (footnote) 1898. 



2 Helmholtz imagined that Kant fell into error in the matter of the 

 ' axioms of intuition/ 



3 Space in which if two points be taken, one, and only one straight line 

 can be drawn j or if three points be taken, one and only one plane can 

 be drawn. 



4 Such as those of curved space, or those of space of manifold tortuosity. 



