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of Edinburgh, Session 1879 - 80 . 
allowed by most of those who have considered the question of 
axioms in what I believe to be by far the most useful and 
effective way, viz., by examining and pushing the conclusions to 
be drawn from them to the utmost; and by investigating what 
change on these conclusions would be induced by varying one or 
more of the axioms themselves. 
The question might also be approached from the side of experience. 
I take, for the sake of illustration, an instance which brings me at 
once to my subject. We have, by generalisation from experience, 
ideas more or less refined according to our individual physical 
education of a geometrical straight line, and of a geometrical point. 
Let us think, then, of two straight lines intersecting at a point, and 
let us ask ourselves, Can two such lines intersect again 1 Our first 
impulse is to answer no ; but due consideration will show us that, in 
point of fact, experience does not settle the question. All we can say 
is that no one starting from the point of intersection of two straight 
lines has ever followed them by physical (say optical) observation to 
a second intersection. But then we must admit that, on our usual 
assumption that space is of infinite extent, and straight lines of 
infinite length, the distance through which any one has so followed 
them is, after all, relatively speaking, but an infinitely little way. Our 
assertion, therefore, that two straight lines never intersect again is 
merely an assumption, accordant, no doubt, with our limited experi- 
ence, but otherwise unfounded, and certainly not of necessity involved 
in our idea of straightness, though we may superadd it thereto if we 
please. I recommend those who doubt this statement to begin by 
defining a straight line by a single geometrical property, which is not 
verbally equivalent to the assertion in question, and to attempt to 
prove it. 
It may be well to remark here that the discussion of the properties 
of tridimensional space in reality divides itself into two parts : — first, 
what may the properties of space be conceived to be 'l conceive being 
understood in the sense above explained ; second, what are the pro- 
perties of space as we know, or think we know, them ? The former 
question is a purely mathematical one ; the latter is one in the main 
for the physicist or the mental philosopher, and the function of the 
mathematician in connection with it is to make clear what the question 
exactly is, and what alternatives are open for us. What the bearing 
