PRINCIPLE OF CONTINUITY IN THE THEORY OF SPACE. 279 



B, becoming identical therewith simultaneously. That and D 



are conjugate, in respect of OB as unit length, is obvious, since if 



the angle A'OA" = ODA"= 9, and if the diameter of the circle be 



a, we shall have 



OD = a cosec 6, OC = a sin 6 • 



hence, as before, denoting these quantities by x and £ respectively 



sc£ = OD . OC e= a cosec 6 a sin Q = a? (20) 



the same equation as before, when a is unity. 



Continuity as between plus and minus infinity may, in these 

 examples, be variously represented and interpreted. Two illustra- 

 tions will suffice. In Fig. 9, suppose the point C to move from 

 A, ( = - 1 from 0) in the positive direction, the conjugate point D' 

 will move from A in the negative direction. When C is at the 

 origin O, D' may be conceived to be at the antipodal point 1 0', as 

 C" continues to move towards B, D" moves towards B from the 

 antipodal point. The formal advantage of this is, that the double 

 infinite discontinuity at the moment of crossing is apparently 

 resolved. 2 For a discrete point geometry, the points on the range 

 being separated by zero distances of the first order, 00' need be 

 an infinity also of the first order only. 



In obtaining D' or D" by construction, let the line ANM and 

 the point N be fixed, then as C moves positively, the intersecting 

 diagonals move continuously counterclockwise from the line NM 

 to NB. Since the line NL o is parallel to AB it will 'meet it at 

 ± oo,' hence the intersection by NL' may, by a, fiction, be considered 

 as beyond infinity? On an infinite spherical surface the line D'NL' 

 will intersect the line AOBO' at the point antipodal to D', that is 

 at infinity therefrom, hence that intersection is ignored; and the 

 second intersection, viz. at D' (or the nearer intersection in the 



1 C in the figure is the centre of the infinite sphere, whose diametral 

 plane is D"00'D'. 



2 If C" continues, D" moves towards O, reaching it when C ' ' reaches 

 O' and so on. 



3 The intersecting lines from N through B, D" etc., moving clockwise, 

 reach infinity when the moving point reaches O, moving from B, conse- 

 quently the more divergent line may be conceived to intersect at a point 

 still further away. This 'further away' however is not of the nature of 

 a higher degree of infinity, i.e. it is not oo 11 . 



