WAUGURAL ADDRESS BEFORE THE BRITISH ASSOCIATION. 507 



upon the degree of complexity of the subject of 

 our choice will depend the internal structure or 

 manifoldness of space. 



Thus, beginning with the simplest case, a 

 point may have any singly infinite multitude of 

 positions in a line, which gives a onefold system 

 of points in a line. The line may revolve in a 

 plane about any one of its points, giving a two- 

 fold system of points in a plane ; and the plane 

 may revolve about any one of the lines, giving a 

 threefold system of points in space. 



Suppose, however, that we take a straight 

 line as our element, and 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 twofold sys- 

 tem, and it therefore follows that the system of 

 lines is fourfold ; in other words, space, regard- 

 ed as a plenum of lines, is fourfold. The same 

 result follows from the consideration that the 

 lines in a plane, and the planes through a point, 

 are each twofold. 



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 fourfold. And, generally, 

 space as a plenum of surfaces has a manifoldness 

 equal to the number of constants required to de- 

 termine the surface. Although it would be be- 

 yond our present purpose to attempt to pursue 

 the-subject further, it should not pass unnoticed 

 that the identity in the fourfold 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 prin- 

 ciples established by Sqphus 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 threefold system, 

 and, as the planes in space form a threefold sys- 

 tem, it follows that space as a plenum of circles 

 is sixfold. 



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 threefold ; the 

 directions of the plane, like that of a pencil of 

 lines perpendicular thereto, twofold ; and the ra- 

 dius of the sphere onefold — sixfold in all. In the 



latter case, the position of the vertex is three- 

 fold, the direction of the axis twofold, and the 

 distance of the plane of section onefold — sixfold 

 in all, as before. Hence space as a plenum of 

 circles is sixfold. 



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 na- 

 ture of the conic be defined, the plane of section 

 will be inclined at a fixed angle to the axis, oth- 

 erwise it will be free to take any inclination what- 

 ever. This being so, the position of the vertex 

 will be threefold, the direction of the axis two- 

 fold, the distance of the plane of section from the 

 vertex onefold, and the direction of that plane 

 onefold if the conic be defined, twofold if it be 

 not defined. Hence, space as a plenum of defi- 

 nite conies will be sevenfold, as a plenum of con- 

 ies in general eightfold. And so on for curves 

 of higher degrees. 



This is, in fact, the whole story and mystery 

 of manifold space. If 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 tangi- 

 ble 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 life, I might remind you that perspective 

 in drawing is itself a method not altogether dis- 

 similar to that of which I have been speaking, 

 and that the third dimension of space, as repre- 

 sented in a picture, has its origin in the painter's 

 mind, and is due to his skill, but has no real ex- 

 istence upon the canvas which is the ground- 

 work of his art. Or, again, turning to literature, 

 when, in legendary tales or in works of fiction, 

 things past and future are pictured as present, 

 has not the poetic fancy brought time into corre- 

 lation with the three dimensions of space, and 

 brought all alike to a common focus ? Or, once 

 more, when space already filled with material 

 substances is mentally peopled with immaterial 

 beings, may not the imagination be regarded as 

 having added a new element to the capacity of 

 space, a fourth dimension of which there is no 

 evidence in experimental fact ? 



The third method proposed for special re- 

 mark is that which has been termed non-Euclid- 

 ean geometry, and the train of reasoning which 



