1878.] The British Association, 507 
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 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 threefold ; the direction of the axis twofold ; 
the distance of the plane of section from the vertex one- 
fold ; and the direction of that plane onefold if the conic be 
defined, twofold if it be not defined. Hence, space as a 
plenum of definite conics will be sevenfold, as a plenum of 
conics in general eightfold. 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 repre- 
sentation, 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 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 dimen- 
sion of space, as represented in a pidture, has its origin in 
the painter’s mind, and is due to his skill, but has no real 
existence upon the canvas which is the groundwork of his 
art. Or, again, turning to literature, when in legendary 
tales, or in works of fidtion, things past and future are pic- 
tured as present, has not the poetic fancy correlated time 
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 im- 
material 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 experi- 
mental fadt ? 
The third method proposed for special remark is that 
which has been termed Non-Euclidean Geometry; and the 
train of reasoning which has led to it may be described in 
general terms as follows Some of the properties of space 
which on account of their simplicity, theoretical as well as 
practical, have, in constructing the ordinary system of 
geometry, been considered as fundamental, are now seen to 
be particular cases of more general properties. Thus a 
