"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. It is equally absurd
and impossible to us if we require the third line to be drawn on
the paper. But we should reply, "If you allow us to leave the
paper or flat surface, then we can solve the problem by simply
drawing the third line through the paper perpendicular to its
surface."

[Illustration with caption: FIG. 2]

Now, to pursue the analogy, suppose that, after we have drawn
three mutually perpendicular lines, some being from another sphere
proposes to us the drawing of a fourth line through the same
point, perpendicular to all three of the lines already there. We
should answer him in the same way that the inhabitants of "flat-
land" answered us: "The problem is impossible. You cannot draw any
such line in space as we understand it." If our visitor conceived