41 



defined, tlie 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 ; it is a mode 

 of representation, or a method which, having served its 

 purpose, vanishes from the scene. Like a rainbow, if we 

 try to grasp it, it eludes our very touch; but, like a 

 rainbow, it arises out of real conditions of known and 

 tangible quantities, and if rightly apprehended it is a 

 true and valuable expression of natural laws, and serves 

 a definite purpose in the science of which it forms part. 



Again, if we seek a counterpart of this in common Illustrations, 

 life, I might remind you that perspective in drawing is 

 itself a method not altogether dissimilar to that of which 

 I have been speaking ; and that the third dimension of 

 space, as represented in a picture, has its origin in the 

 painter's mind, and is due to his skill, but has no real 

 existence upon the canvass which is the groundwork of 

 his art. Or again, turning to literature, when in legen- 

 "dary tales, or in works of fiction, things past and future 

 are pictured as present, has not the poetic fancy 

 brought time into correlation with the three dimen- 

 sions of space, and brought all alike to a common 

 focus? Or once more, when space already filled with 

 material substances is mentally peopled with imma- 



