PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 315 



spatial relationship. 1 With rigorous conceptions this expression 

 however, denotes a uniquely straight line, for each element must be 

 regarded as straight and at right angles, not on a spherical surface 

 but absolutely (i.e. they must not be merely infinitesimal portions 

 of curves) : consequently the notion of an n-sheeted surface 

 (assumed to be plane, but not really so) or of an w-ply extended 

 magnitude of any other kind, throws no real light upon the con- 

 stitution of what is popularly meant by space. A little consider- 

 ation will shew that it is just because we do "assume with Euclid 

 not merely an existence of lines independent of position but of 

 bodies also," 2 that geometry itself and mathematical thought has 

 any validity. Once that basis is departed from, nothing is certain 

 or valid ; and not only do the conclusions of geometry fail, but 

 the constructions and conclusions of analytical geometry fail also. 

 Their logical basis is not a whit more assured than that of pure 

 geometry: the certainty or uncertainty is of the same type. All 

 ratiocination on the subject is unmeaning, and the apodictic cer- 

 tainty of the conclusions disappears, unless it is true that, in the 

 words of Kant, "space is no mere empirical concept derived from 

 external experience," nor "a determination produced by phe- 

 nomena": "it is rather the "condition of their possibility," or, "a 

 representation a priori which necessarily precedes" them. 3 The 

 fact that schemes have been discovered by means of which analy- 

 tical functions may be readily represented, though of great 

 importance in the development of mathematical science, in reality 

 establishes nothing of moment with regard to the foundations of 

 geometry. When geometrical meaning is attached to algebraic 

 or other symbols, the validity obviously depends upon the funda- 

 mental geometrical ideas assigned to them not to any consequences 

 that flow therefrom. 



1 The next degree of simplicity according to Riemann is where the line 

 element may be expressed as the fourth root of a quantic differential 

 expression. The difficulty of the theory cannot be logically avoided by 

 refusing to recognise the difference between 8x, 8y and 8z as parts of 

 straight lines or parts of curves. 



2 Eiemann's treatise, in., § 1. 



3 See Kritik d. rein. Vernunft: transscend. Aesthetik l er Abseh. §2, 1.2. 



