156 Proceedings of the Royal Irish Acadermj. 
To give vividness to our conceptions in abstract geometry, we imagine 
lines and surfaces, as we see and know them in our Euclidian space ; 
but the results obtained by mathematical operations have no relation 
to, and are completely independent of, any definitions of length, and 
still remain true, if space possessed properties altogether different from 
those which it has been found to have from experience. That is, of 
course, provided these results are stated in the most general form in 
accordance with the ideas of modern geometry. 
From the algebraic point of view there is such a thing as a pure, 
perfectly neutral space. Such a space would not be cognisable by our 
senses, for it does not possess any attributes — it does not involve the 
existence of length, of area, or volume. In order to realize these ideas, 
the mathematician finds it necessary to suppose the existence of some 
absolute surface or curve, which, for the ordinary Euclidian space is a 
curve of the second order, and in the elliptic space is a surface of the 
same order. 
The Euclidian geometry forms the connecting-link between algebra 
and modern geometry on the one hand, and physics on the other. 
Physical mathematics may inform us that there is something peculiarly 
appropriate in having the law of attraction in nature equal to that of 
the inverse square of the distance, but still we do not perceive any a 
priori necessity for such a law. On the other hand, it may be conceded 
that we have the idea of necessary truth in arithmetic or algebra, but 
it does not seem so easy to assign the exact boundary ; for, according 
to Eiemann, the belief that our space is Euclidian is founded on 
experienee, while the more usual view seems to be that it is a 
necessary truth. 
