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 perhaps millions of times that of light, we
might at length find ourselves approaching the earth from a
direction the opposite of that in which we started. Our straight-