ADDRESS. 9 
IT am aware, a subject of philosophical discussion or enquiry. As regards 
the older metaphysical writers this would be quite accounted for by saying 
that they knew nothing, and were not bound to know anything, about it ; 
but at present, and, considering the prominent position which the notion 
occupies—say even that the conclusion were that the notion belongs to 
mere technical mathematics, or has reference to nonentities in regard to 
which no science is possible, still it seems to me that (as a subject of 
philosophical discussion) the notion ought not to be thus ignored; it 
should at least be shown that there is a right to ignore it. 
Although in logical order I should perhaps now speak of the notion 
just referred to, it will be convenient to speak first of some other quasi- 
geometrical notions; those of more-than-three-dimensional space, and of 
non-Huclidian two- and three-dimensional space, and also of the general- 
ised notion of distance. It is in connection with these that Riemann 
considered that our notion of space is founded on experience, or rather 
that it is only by experience that we know that our space is Euclidian 
space. ‘ 
It is well known that Euclid’s twelfth axiom, even in Playfair’s form of 
it, has been considered as needing demonstration ; and that Lobatschewsky 
constructed a perfectly consistent theory, wherein this axiom was assumed 
not to held good, or say a system of non-Euclidian plane geometry. 
There is a like system of non-Euclidian solid geometry. My own view is 
that Euclid’s twelfth axiom in Playfair’s form of it does not need 
demonstration, but is part of our notion of space, of the physical space of 
our experience—the space, that is, which we become acquainted with by 
experience, but which is the representation lying at the foundation of all 
external experience. Riemann’s view before referred to may I think be 
said to be that, having in intellectu a more general notion of space (in fact 
a notion of non-Huclidian space), we learn by experience that space (the 
physical space of our experience) is, if not exactly, at least to the highest 
degree of approximation, Euclidian space. 
But suppose the physical space of our experience to be thus 
only approximately Euclidian space, what is the consequence which 
follows ? Not that the propositions of geometry are only approximately 
true, but that they remain absolutely true in regard to that Euclidian 
Space which has been so long regarded as being the physical space of our 
experience. 
It is interesting to consider two different ways in which, without 
any modification at all of our notion of space, we can arrive at a system 
of non-Enclidian (plane or two-dimensional) geometry; and the doing so 
will, I think, throw some light on the whole question. 
First, imagine the earth a perfectly smooth sphere; understand by 
a plane the surface of the earth, and by a line the apparently straight 
line (in fact an are of a great circle) drawn on the surface; what ex- 
perience would in the first instance teach would be Euclidian geometry; 
there would be intersecting lines which produced a few miles or so would 
