Hinton - The Fourth Dimension.pdf

222
THE FOURTH DIMENSION
diagrams. Now, in fig. 14 let the zw rotation take place,
the x axis will turn toward the point w of the w axis, and
the point p will move in a circle about the point x.
Thus in fig. 13 the point p moves in a circle parallel to
the xy plane: in fig. 14 it moves in a circle parallel to the
zw plane, indicated by the arrow.
Now, suppose both of these independent rotations compounded,
the point p will move in a circle, but this circle
will coincide with neither of the circles in which either
one of the rotations will take it. The circle the point p
will move in will depend on its position on the surface of
the four sphere.
In this double rotation, possible in four-dimensional
space, there is a kind of movement totally unlike any
with which we are familiar in three-dimensional space.
It is a requisite preliminary to the discussion of the
behaviour of the small particles of matter, with a view to
determining whether they show the characteristics of fourdimensional
movements, to become familiar with the main
characteristics of this double rotation. And here I must
rely on a formal and logical assent rather than on the
intuitive apprehension, which can only be obtained by a
more detailed study.
In the first place this double rotation consists in two
varieties or kinds, which we will call the A and B kinds.
Consider four axes, x, y, z, w. The rotation of x to y can
be accompanied with the rotation of z to w. Call this
the A kind.
But also the rotation of x to y can be accompanied by
the rotation, of not z to w, but w to z. Call this the
B kind.
They differ in only one of the component rotations. One
is not the negative of the other. It is the semi-negative.
The opposite of an x to y, z to w rotation would be y to x,
w to z. The semi-negative is x to y and w to z.