lers 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-land" people these lines would both be perfectly
straight, because the only curvature would be in the direction
downward, which they could never either perceive or discover. The lines
would also correspond to the definition of straight lines, because any
portion of either contained between two of its points would be the
shortest distance between those points. And yet, if these people should
extend their measures far enough, they would find any two parallel
lines to meet in two points in opposite directions. For all small
spaces the axioms of their geometry would apparently hold good, but
when they came to spaces as immense as the semi-diameter of the earth,
they would find the seemingly absurd result that two parallel lines
would, in the course of thousands of miles, come together. Another
result yet more astonishing would be that, going ahead far enough in a
straight line, they would find that although they had been going
forward all the time in what seemed to them the same direction, they
would at the end of 25,000 miles find themselves once more at their
starting-point.

One form of the modern non-Euclidian geometry assumes that a similar
theorem is true for the space in which our universe is contained.
Although two straight lines, when continued indefinitely, do not appear
to converge even at the immense distances which separate us from the
fixed stars, it is possible that there may be a point at which they
would eventually meet without either line having deviated from its
primitive direction as we understand the case. It would follow that, if
we could start out from the earth and fly through space in a perfectly
straight line with a velocity perhap