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 dwellers in "flatland" could construct a plane geometry
which would be exactly like ours in being based on the axioms of
Euclid. Two parallel straight lines would never meet, though
continued indefinitely.

But suppose that the surface on which these beings live, instead
of being an infinitely extended plane, is really the surface of an
immense globe, like the earth on which we live. It needs no
knowledge of geometry, but only an examination of any globular
object--an apple, for example--to show that if we draw a line as
straight as possible on a sphere, and parallel to it draw a small
piece of a second line, and continue this in as straight a line as
we can, the two lines will meet when we proceed in either
direction one-quarter of the way around the sphere. For our "flat-