22 report— 1878. 



conceive space as filled with such lines. This will be the case if we take 

 two planes, e.g., two parallel planes, and join every point in one with 

 every point in the other. Now the points in a plane form a two-fold 

 system, and it therefore follows that the system of lines is fonr-f old ; in 

 other words, space regarded as a plenum of lines is four-fold. The same 

 result follows from the consideration that the lines in a plane, and the 

 planes through a point, are each two-fold. 



Again, if we take a sphere as our element we can through any point 

 as a centre draw a singly infinite number of spheres, but the number of 

 such centres is 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 sphere, 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 defined, the plane of section will be inclined at a 

 fixed angle to the axis ; otherwise it will be free to take any inclination 

 whatever. This being so, the position of the vertex will be three-fold ; 

 the direction of the axis two-fold ; the distance of the plane of section 

 from the vertex one-fold ; and the direction of that plane one-fold if the 

 conic be defined, two-fold if it be not defined. Hence, space as a plenum 

 of definite conies will be seven-fold, as a plenum of conies in general, 

 eight-fold. And so on for curves of higher degrees. 



This is in fact the whole story and mystery of manifold space. It is 

 not seriously regarded as a reality in the same sense as ordinary space ; 



