PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 297 



medium, in which if the length of lines be measured by the time 

 necessary for an sethereal disturbance to pass from one point to 

 another, the shortest line is curved and not straight. For such a 

 medium a straight line, spatially the longer, is in a temporal sense 

 the shorter. 



26. Space of n-dimensions as the boundary of {n + 1) dimensional 

 space. — In the generative scheme of § 4, space of n dimensions is 

 to space of (n+\) dimensions the analogue of a boundary, as 

 pointed out in § 24. The idea of space of one type being a locus 

 in space of another of higher dimensions, was conceived by Johann 

 Bolyai, and by Beltrami. It is shewn by Whitehead, 1 analytically, 

 that "euclidean space of n dimensions can be conceived as a limit- 

 surface of hyperbolic space of (n-\- 1) dimensions," and he remarks: 

 "There is an error, 2 popular even among mathematicians misled by 

 a useful technical phraseology, that euclidean space is in a special 

 sense flat? and that this flatness is exemplified by the possibility 

 of a euclidean space containing surfaces with the properties of 

 hyperbolic and elliptic spaces. But the text shews that the 

 relation to hyperbolic to euclidean space can be inverted. Thus 

 no theory of the flatness of euclidean space can be founded on it." 

 This dictum is based upon a theory of the relations between the 

 sides and angles of a "curvilinear' 1 '' triangle formed by great 

 circles on a "limit-surface." 4 Now it may be remarked that 

 "curvilinear" must be defined for such a statement to be intelligible. 

 Reverting to the illustration of refraction in last section, the 

 "shorter" line, i.e. the curved one, may be regarded as straight 

 in the geometry which measures its unit of length, and the actually 

 shorter line — the really straight one, may be regarded as curved. 

 And generally any line, curved or tortuous, may geometrically be 

 treated as straight and be made the basis of a geometry of. the form 

 of any other lines, straight or curved. More generally, a geometrical 

 figure of n dimensions may be treated as a "flat" for the develop- 



1 Universal Algebra i., p. 451. 



2 The italics are mine. 3 The word is Whitehead's. 



* For the theory of limit-surfaces see Whitehead op. cit., i., pp. 447-8. 



