32 CONCRETE REPRESENTATIONS OF 



that it is only a representation. A vast deal of misconception 

 has grown around it. The following points have been most 

 generally misunderstood : 



(1) There is an essential difference between Riemann's 

 geometry and the geometry on the surface of a sphere. 1 The 

 former is a true metrical geometry of two dimensions, and is 

 no more dependent upon three dimensions 2 than ordinary 

 geometry is on the ' fourth dimension.' The geometry on 

 the surface of a sphere, on the other hand, is a body of doctrine 

 forming a part of ordinary geometry of three dimensions. 



(2) The fact that there is in ordinary space only one 

 uniform real surface other than the plane has led certain 

 critics 3 to reject Hyperbolic and Elliptic geometries as false 

 and absurd, while they admit Spherical geometry only as a 

 branch of ordinary geometry of three dimensions. This view 

 is not so common now since the investigations of Pasch, Hilbert, 

 and others on geometries defined by systems of axioms have 

 become better known. 



(3) The term ' curvature,' especially when extended to 

 space of three dimensions, has given rise to much confusion, 

 and has led to the notion that non- Euclidean geometry of 



1 Cf. P. Mansion, ' Sur la non-identite du plan riemannien et de la sphere euclidienne,' 

 Bruxclles, Ann. Soc. sclent., 20 B (1896), a reply to Lechalas in the same volume. See 

 also B. Russell, ' Geometry, non-Euclidean,' Encycl. Brit. (10th ed.), p. 669d. 



2 This statement must not be confused with the result that plane projective geometry, 

 which is free from metrical considerations, and in which the Euclidean and non-Euclidean 

 hypotheses are not distinguished, cannot be established completely without using space 

 of three dimensions. The theorem of Desargues relating to perspective triangles, which 

 is proved easily by projection in space of three dimensions, is incapable of deduction 

 from the axioms of plane projective geometry alone. Thus there are two-dimensional 

 but not three-dimensional non-Desarguesian geometries. In the same way the theorem 

 of Pascal for a conic, or, in the special form, the theorem of Pappus, when the conic 

 reduces to two straight lines, from which Desargues' theorem can be deduced, is in- 

 capable of deduction from the axioms of plane projective geometry alone. In this sense 

 plane geometry is dependent upon three dimensions ; but it is only necessary to make 

 some additional assumption, Pascal's theorem or an equivalent, in order to construct 

 plane geometry without reference to three dimensions. 



3 Cf., e.g., E. T. Dixon, The Foundations ofQeometry (Cambridge, Bell, 1891), p. 140. 



