Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

NON-EUCLIDEAN GEOMETRIES 357
from the third line is finite, and all perpendiculars to this
line meet at that point.
One more: In parabolic geometry each summit angle
of an isosceles birectangular quadrilateral is a right angle;
in hyperbolic geometry, it is acute; and in elliptic geometry it is obtuse. So it is seen that neither of the nonEuclidean geometries contains rectangles among its
figures.
It would be easy to prolong the list of such differences; the theorems already stated, which are not difficult to prove, are sufficient, however, for illustration and
they ought, I think, to challenge the curiosity of any intellectual student.
There are certain questions which you are doubtless
bursting to ask, for in a discussion of this subject thoughtful beginners always ask them.
One of the questions is this: Can we be quite certain that neither of the non-Euclidean geometries involves
an inner contradiction? In other words, can we be certain with respect to each of them that the propositions
constituting it are compatible with one another? The
answer is, yes. The propositions constituting a geometry
consist, as you know, of its postulates and of the propositions logically deducible therefrom; and so, if the postulates are mutually compatible, the whole geometry is selfconsistent. The question thus reduces to this: Can we
be certain with respect to each of the non-Euclidean
geometries that its postulates are mutually compatible?
Now, in respect of the Euclidean postulates, we saw in
Lecture VI, you will remember, that we can be as certain of their compatibility as it is possible to be of any
reasoned proposition, and that is what I mean by "quite
certain" and it is what you mean. Well, it can be shown