Hinton - The Fourth Dimension.pdf

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THE FOURTH DIMENSION
In this case all that we see of the plane about which
the turning takes place is the line AB.
But it is obvious that the axis plane may lie in our
space. A point near the plane determines with it a threedimensional
space. When it begins to rotate around the
plane it does not move anywhere in this three-dimensional
space, but moves out of it. A point can no more rotate
round a plane in three-dimensional space than a point
can move round a line in two-dimensional space.
We will now apply the second of the modes of representation
to this case of turning about a plane, building
up our analogy step by step from the turning in a plane
about a point and that in space about a line, and so on.
In order to reduce our considerations to those of the
greatest simplicity possible, let us realise how the plane
being would think of the motion by which a square is
turned round a line.
Let fig. 34, ABCD be a square on his plane, and represent
the two dimensions of his space by the axes Ax, Ay.
Now the motion in which the square is turned over
about the line AC involves the third dimension.
He cannot represent the motion of the whole square in
its turning, but he can represent the motions of parts of
it. Let the third axis perpendicular to the plane of the
paper be called the axis of z. Of the three axes, x, y, z,
the plane being can represent any two in his space. Let
him then drawn, in fig. 35, two axes, x and z. Here he has
in his plane a representation of what exists in the plane
which goes off perpendicularly to his space.
In this representation the square would not be shown,
for in the plane of xy simple the line AB of the square is
contained.
The plane being then would have before him, in fig. 35,
the representation of one line AB of his square and two
axes, x and z, at right angles. Now it would be obvious