The fourth dimension

66 THE FOURTH DIMENSION

Tn 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 round 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 Aw Ay.

Now the motion by 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 @, y, 2, the plane being can represent any two in his space. Let him then draw, 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 wz simply the line 48 of the square is contained.

The plane being then would have before him, in fig. 35, the representation of one line 4B of his square and two axes, # and z, at right angles. Now it would be obvious