Hinton - The Fourth Dimension.pdf

THE HIGHER WORLD 67
to him that, by a turning such as he knows, by a rotation
about a point, the line AB can turn round A, and occupying
all the intermediate positions, such as AB1, come,
after half a revolution to lie as Ax produced through A.
Again, just as he can represent the vertical plane
through AB, so he can represent the vertical plane
through A’B’, fig. 34, and in a like manner can see that
the line A’B’ can turn about the point A’ till it lies in the
opposite direction from that which it runs at first.
Now these two turnings are not inconsistent. In his
plane, if AB is turned about A, and A’B’ about A’, the consiststency
of the square would be destroyed, it would be an
impossible motion for a rigid body to perform. But in
the turning which he studies portion by portion there
is nothing inconsistent. Each line in the square can turn
in this way, hence he would realise the turning of the
whole square as the sum of a number of turnings of
isolated parts. Such turnings, if they took place in his
plane, would be inconsistent, but by virtue of a third
dimension they are consistent, and the result of them all
is that the square turns about the line AC and lies in a
position in which it is the mirror image of what it was in
its first position. Thus he can realise a turning about a
line by relinquishing one of his axes, and representing his
body part by part.
Let us apply this method to the turning of a cube so as
to become the mirror image of itself. In our space we can
construct three independent axis, x, y, z, shown in fig. 36.
Suppose that there is a fourth axis, w, at right angles to
each and every one of them. We cannot, keeping all
three axes, x, y, z, represent w in our space; but if we
relinquish one of our three axes we can let the fourth axis
take its place, and we can represent what lies in the
square, determined by the two axes we retain and the
fourth axis.