PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 289 



Fig. 17 denotes the saddle-shaped surface of (variable) negative 

 curvature, 1 the centre of curvature of BOB' being and that of 

 BCS, 0', CO and CO' being opposed directions on the one line. 2 

 The system of meridians on the surface, cutting one another at 

 right angles (like RAS, RDS, RQS, and PAQ, PA'Q, PSQ, Fig. 

 16) are RR', TT', CB, AD, UU', SS', and RCS, GH, IJ, etc. If 

 AD' is equidistant from CB, the shortest distance between those 

 points is nearer to CB than the equidistant line, and similarly if 

 D'F is equidistant from IJ, the geodesic is nearer AS : the fine 

 dotted lines indicate their positions. A triangle ABC has the 

 sum of its angles less than two right angles, or a quadrilateral 

 BCAD', less than four right angles, the amount of the defect being 

 expressed by formula (24a). 3 As in the previous example parallel 

 and equidistant geodesies cannot exist on a surface of constant 

 negative curvature. 



By analogy the two classes of surfaces suggest the possible 

 existence of types of ro-dimensional space, in which a system of 

 'straight' (!) lines 4 cutting another 'straight' line at right angles, 

 should either converge and meet (elliptic space) or diverge and 

 therefore never meet (hyperbolic space). It will be observed that 

 the doctrine is obviously true for surfaces (not plane): provided 

 they are essentially three-dimensional; but not otherwise. It may 

 be inferred, therefore, that in a (supposititious) space of four 

 dimensions, a three-dimensional space could be represented, which 

 should have the properties indicated. 5 



22. Symmetrical elliptic and hyperbolic space of two-dimensions. 

 — Since the sphere is a surface of uniform positive curvature, 

 absolutely symmetrical in all respects, and unbounded, elliptical 

 2-dimensional space can be perfectly represented thereupon, and 



1 The inner side of a ring or tore, is a good example of such a surface. 



2 Thus the product of the radii is a negative quantity. 



3 That is, the excess is negative. 



4 These lines may start out from any point in all directions, maintain- 

 ing relations of symmetry, from any point in 3-dimensional space. 



5 It will be seen the inference is of extremely doubtful validity, or 

 rather that the space is not what we ordinarily conceive as " space." 



S-Dec. 4, 1901. 



