Hinton - The Fourth Dimension.pdf

THE HIGHER WORLD 63
dimensions. His limitation to a plane must be the result
of physical conditions.
We will therefore think of him as of a figure cut out of
paper placed on a smooth plane. Sliding over this plane,
and coming into contact with other figures equally thin
as he in the third dimension, he will apprehend them only
by their edges. To him they will be completely bounded
by lines. A “solid” body will be to him a two-dimensional
extent, the interior of which can only be reached by
penetrating through the bounding lines.
Now such a plane being can think of our threedimensional
existence in two ways.
First, he can think of it as a series of sections, each like
the solid he knows of extending in a direction unknown
to him, which stretches transverse to his tangible
universe, which lies in a direction at right angles to every
motion which he made.
Secondly, relinquishing the attempt to think of the
three-dimensional solid body in its entirety he can regard
it as consisting of a number of plane sections, each of them
in itself exactly like the two-dimensional bodies he knows,
but extending away from his two-dimensional space.
A square lying in his space he regards as a solid
bounded by four lines, each of which lies in his space.
A square standing at right angles to his plane appears
to him as simply a line in his plane, for all of it except
the line stretches in the third dimension.
He can think of a three-dimensional body as consisting
of a number of such sections, each of which starts from a
line in his space.
Now, since in his world he can make any drawing or
model which involves only two dimensions, he can represent
each such upright section as it actually is, and can represent
a turning from a known into the unknown dimension
as a turning from one to another of his known dimensions.