PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 287 



euclidean space, which latter, being the lower limiting form of 

 hyperbolic space is called parabolic, or homaloidal [i.e. "even" or 

 "flat" space 1 ). This also is infinite. Space in which two straight 

 lines intersect twice, is defined as elliptic space; its upper limit 

 being parabolic, i.e. infinite space, and its lower a point! 2 Elliptic 

 space has been divided into the single-, or the polar form, in which 

 every line returns into itself (and two intersecting straight lines 

 intersect really in the one point only, the second intersection being 

 coincident with the original one); and the double or antipodal 

 form, in which the second intersection is the antipodes of the first. 

 Elliptic space is presumed to be finite in volume, 3 if its "constant" 

 were not finite, it would be undistinguishable from parabolic space. 

 Since a line cuts any plane only in one point, in the polar form of 

 elliptic space, a single plane cannot divide it: two planes however 

 can. In the antipodal form, since a straight line cuts it in two 

 points, a plane does divide the space. Hyperbolic space is also 

 divided by a plane. If any three points are taken in elliptic or 

 hyperbolic space, and joined by "straight " lines, to form a triangle 

 of area A, then its angles will be greater than 180° by the amount 



"? < 24 > 



that is numerically greater in elliptic space since p is then positive, 

 and numerically less in hyperbolic space, since p is then negative, 



1 Not surface. 



2 This latter limit to the conception is never insisted on : all that is 

 xirged (Clifford, Chrystal and many others) is that its volume is finite 

 though unbounded, in the sense that the surface of a sphere is unbounded. 

 It is left to the reader to satisfy himself whether a three-dimensional 

 figure, i.e. a volume, can be unbounded and finite. The scale of the figure 

 is of no moment : if the conception has validity it may be a microscopic 

 quantum, an yet unbounded space. It is hardly necessary to add, that it 

 is not a volume with an unbounded surface, but an unbounded volume. 



3 The greatest distance of two points is 8, an absolute linear constant 

 characterising the space: it is the distance one would have to travel on 

 the straight line to return to the point of starting. In single elliptic 

 space we should return inverted, as in passing along a tape rejoined after 

 putting a 'half -twist' in it, and would have to traverse the distance 2S 

 to become erect again ! See Klein, Math. Annal. vi.; Chrystal, Proc. 

 B.. S. Edinb., x., 655. The volume of single elliptic space is 



F=7r> 3 = S 3 /7i- 

 p being the radius of curvature. 



