Hinton - The Fourth Dimension.pdf

70
THE FOURTH DIMENSION
plane as the rotation of a number of sections of a body
about a number of lines in a plane, these rotations not
being inconsistent in a four-dimensional space as they are
in three-dimensional space.
We are not limited to any particular direction for the
lines in the plane about which we suppose the rotation
of the particular sections to take place. Let us draw
the section of the cube, fig. 36, through A, F, C, B, forming
a sloping plane. Now since the fourth dimension is at
right angles to every line in our space it is at right
angles to this section also. We can represent our space
by drawing an axis at right angles to the plane ACEG, our
space is then determined by the plane ACEG, and the perpendicular
axis. If we let this axis drop and suppose the
fourth axis, w, to take its place, we have a representation of
the space which runs off in the fourth dimension from the
plane ACEG. In this space we shall see simply the section
ACEG of the cube, and nothing else, for one cube does not
extend to any distance in the fourth dimension.
If, keeping this plane, we bring in the fourth dimension,
we shall have a space in which simply this section of
the cube exists and nothing else. This section can turn
about the line AF, and parallel sections can turn about
z
parallel lines. Thus in considering
the rotation about
a plane we can draw any
D F lines we like and consider
the rotation as taking place
G H
x in sections about them.
To bring out this point
C E
more clearly, let us take two
y
parallel lines, A and B, in
A B
the space of xyz, and let CD
Fig. 42.
and EF be two rods running
above and below the plane of xy, from these lines. If we