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]
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.
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