It might approach the latter at first, but
would eventually diverge. The two lines AB and CD, starting
parallel, would eventually, perhaps at distances greater than that
of the fixed stars, gradually diverge from each other. This system
does not admit of being shown by analogy so easily as the other,
but an idea of it may be had by supposing that the surface of
"flat-land," instead of being spherical, is saddle-shaped.
Apparently straight parallel lines drawn upon it would then
diverge, as supposed by Bolyai. We cannot, however, imagine such a
surface extended indefinitely without losing its properties. The
analogy is not so clearly marked as in the other case.
To explain hypergeometry proper we must first set forth what a
fourth dimension of space means, and show how natural the way is
by which it may be approached. We continue our analogy from "flat-
land" In this supposed land let us make a cross--two straight
lines intersecting at right angles. The inhabitants of this land
understand the cross perfectly, and conceive of it just as we do.
But let us ask them to draw a third line, intersecting in the same
point, and perpendicular to both the other lines. They would at
once pronounce this absurd and impossible.
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