40 



triply infinite ; hence space as a plenum of spheres is 

 four-fold. And generally, space as a plenum of surfaces 

 has a mani-foldness equal to the number of constants 

 required to determine the surface. Although it would 

 be beyond our present purpose to attempt to pursue 

 the subject further, it should not pass unnoticed that 

 the identity in the four-fold character of space, as 

 derived on the one hand from a system of straight 

 lines, and on the other from a system of spheres, is 

 intimately connected with the principles established by 

 Sophus Lie in his researches on the correlation of these 

 figures. 



If we take a circle as our element we can around any 

 point in a plane as a centre draw a singly infinite system 

 of circles ; but the number of such centres in a plane 

 is doubly infinite ; hence the circles in a plane form a 

 three-fold system, and as the planes in space form a 

 three-fold system, it follows that space as a plenum of 

 circles is six-fold. 



Again, if we take a circle as our element, we may 

 regard it as a section either of a sjDhere, or of a right 

 cone (given except in position) by a plane perpendicular 

 to the axis. In the former case the position of the 

 centre is three-fold; the directions of the plane, like 

 that of a pencil of lines perpendicular thereto, two-fold ; 

 and the radius of the sphere one-fold ; six-fold in all. 

 In the latter case, the position of the vertex is three- 

 fold ; the direction of the axis two-fold ; and the 

 distance of the plane of section one-fold; six-fold in 

 all, as before. Hence space as a plenum of circles is 

 six-fold. 



Similarly, if we take a conic as our element we may 

 regard it as a section of a right cone (given except in 

 position) by a plane. If the nature of the conic be 



