s 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 of the fourth dimension, he
would reply to us as we replied to the "flat-land" people: "The problem
is absurd and impossible if you confine your line to space as you
understand it. But for me there is a fourth dimension in space. Draw
your line through that dimension, and the problem will be solved. This
is perfectly simple to me; it is impossible to you solely because your
conceptions do not admit of more than three dimensions."

Supposing the inhabitants of "flat-land" to be intellectual beings as
we are, it would be interesting to them to be told what dwellers of
space in three dimensions could do. Let us pursue the analogy by
showing what dwellers in four dimensions might do. Place a dweller of
"flat-land" inside a circle drawn on his plane, and ask him to step
outside of it without breaking through it. He would go all around, and,
finding every inch of it closed, he would say it was impossible from
the very nature of the conditions. "But," we would reply, "that is
because of your limited conceptions. We can step over it."

"Step over it!" he would exclaim. "I do not know what that means. I can
pass around anything if there is a way open, but I cannot imagine what
you mean by stepping over it."

But we should simply step over the line and reappear on the other side.
So, if we confine a being able to move in a fourth dimension in the
walls of a dungeon of wh