Hinton - The Fourth Dimension.pdf

THE HIGHER WORLD 65
If now he turns the square about the point A in the
plane of zy, each parallel section turns with the square
he moves. In each of the sections there is a point of
rest, that vertically over A. Hence he would conclude
that in the turning of a three-dimensional body there is
one line which is at rest. That is, a three-dimensional
turning is a turning about a line.
In a similar way let us regard ourselves as limited to a
three-dimensional world by a physical condition. Let us
imagine that there is a direction at right angles to every
direction in which we can move, and that we are prevented
from passing in this direction by a vast solid, that
against which in every movement we make we slip as
the plane being slips against his plane sheet.
We can then consider a four-dimensional body as consisting
of a series of sections, each parallel to our space,
and each a little further off than the preceding on the
unknown dimension.
Take the simplest four-dimensional body—one which
z
begins as a cube, fig. 36, in our
space, and consists of sections, each
F H
a cube like fig. 36, lying away from
our space. If we turn the cube
y
B D
which is its base in our space
G about a line, if, e.g., in fig. 36 we
E
turn the cube about the line AB,
A C x not only it but each of the parallel
Fig. 36.
cubes moves about a line. The
cube we see moves about the line AB, the cube beyond it
about a line parallel to AB and so on. Hence the whole
four-dimensional body moves about a plane, for the
assemblage of these lines is our way of thinking about the
plane which, starting from the line as in our space, runs
off in the unknown direction.