self-evident. Yet their vital principle is not so much that of
being self-evident as being, from the nature of the case,
incapable of demonstration. Our edifice must have some support to
rest upon, and we take these axioms as its foundation. One example
of such a geometric axiom is that only one straight line can be
drawn between two fixed points; in other words, two straight lines
can never intersect in more than a single point. The axiom with
which we are at present concerned is commonly known as the 11th of
Euclid, and may be set forth in the following way: We have given a
straight line, A B, and a point, P, with another line, C D,
passing through it and capable of being turned around on P. Euclid
assumes that this line C D will have one position in which it will
be parallel to A B, that is, a position such that if the two lines
are produced without end, they will never meet. His axiom is that
only one such line can be drawn through P. That is to say, if we
make the slightest possible change in the direction of the line C
D, it will intersect the other line, either in one direction or
the other.

The new geometry grew out of the feeling that this proposition
ought to be proved rather than taken as an axiom; in fact, that it
could in some way be derived from the other axioms. Many
demonstrations of it were attempted, but it was always found, on
critical examination, that the proposition itself, or its
equivalent, had slyly worked itself in as part of the base of the
reasoning, so that the very thing to be proved was really taken
for granted.

[Illustration with caption: FIG. I]