NON-EUCLIDEAN GEOMETRY 33 



three dimensions necessarily implies space of four dimensions. 1 

 The truth is that Beltr ami's representation, as he himself 

 expressly states, breaks down when we pass to three dimen- 

 sions, and it is necessary, in order to obtain an analogous 

 representation, to introduce space of four dimensions. The 

 geometry, however, is a true geometry of three dimensions, 

 having its own axioms or assumptions, one of which is that 

 there exists no point outside its space. The term ' curvature ' 

 is therefore without meaning. The constant K 2 which occurs 

 in the Cayley-Klein formula, and which corresponds to the 

 measure of curvature of the surface upon which the geometry 

 may be represented, has been called on this account the 

 measure of curvature of the space, but as this is so mislead- 

 ing the term is now generally replaced by ' space-constant.' 

 When it is finite it gives a natural unit of length like the natural 

 angular unit. In Elliptic geometry it may be replaced by the 

 length of the complete straight line ; in Hyperbolic geometry 

 where K 2 is negative iK can be constructed as follows : 2 

 Take two lines OA, OB at right angles, and draw A'B' so that 

 A'B r || OB and B'A' || OA ; then draw an arc OL of a limit- 

 curve through perpendicular to OA and B'A' ; the arc 

 OL=iK. Another natural unit based upon K is the area of 

 the maximum triangle, which has all its angles zero, the 

 limit being TrK 2 . 



(4) Confusion has also existed with regard to the compari- 

 son of spaces with different space-constants. As there can be 

 no comparison between one line and another unless they are 

 in the same space, it appears clear that it is meaningless to 



1 For example, S. Newcomb, ' Elementary Theorems relating to the Geometry of a 

 Space of three Dimensions and of uniform positive Curvature in the Fourth Dimension,' 

 J. Math., Berlin, 83 (1877). Clifford attempted, playfully no doubt, but with a certain 

 seriousness, to explain physical phenomena by periodic variations in the curvature of 

 space (Common-sense of the Exact Sciences, chap, iv., 19). Helmholtz also, by his 

 popularisation of the results of Beltrami and Riemann, did a good deal to promulgate 

 this view especially among philosophers. Cf. Russell, loc. cit. 



2 See Engel, Leipzig, Ber. Ges. Wiss., 50 (1898), p. 190. 



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