t 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. 1]

This suggested another course of inquiry. If this axiom of parallels
does not follow from the other axioms, then from these latter we may
construct a system of geometry in which the axiom of parallels shall
not be true. This was done by Lobatchewsky and Bolyai, the one a
Russian the other a Hungarian geometer, about 1830.

To show how a result which looks absurd, and is really inconceivable by
us, can be treated as possible in geometry, we must have recourse to
analogy. Suppose a world consisting of a boundless flat plane to be
inhabited by reasoning beings who can move about at pleasure on the
plane, but are not able to turn their heads up or down, or even to see
or think of such terms as above them and below them, and things around
them can be pushed or pulled about in any direction, but cannot be
lifted up. People and things can pass around each other, but cannot
step over anything. These dwel