The fourth dimension

RECAPITULATION AND EXTENSION 209

But if the plane being is aware of the existence of a third dimension he can study the movements possible in the ample space, taking his figure portion by portion.

His plane can only hold two axes. But, since it can hold two, he is able to represent a turning into the third dimension if he neglect one of his axes and represent the third axis as lying in his plane. He can make a drawing in his plane of what stands up perpendicularly from his_

plane. Let az be the axis, which

stands perpendicular to his plane at

A. He can draw in his plane two

8, lines to represent the two axes, Ax

and az. Let Fig. 2 be this draw-

ing. Here the 2 axis has taken

A @ * the place of the y axis, and the

Fig. 2 (130). plane of Aw az is represented in his

plane. In this figure all that exists of the square ABCD will be the line as.

The square extends from this line in the y direction, but more of that direction is represented in Fig. 2. The plane being can study the turning of the line ap in this diagram. It is simply a case of plane turning around the point A. The line aB occupies intermediate portions like AB, and after half a revolution will lie on ax produced through a.

Now, in the same way, the plane being can take another point, a’, and another line, 4’s’, in his square. He can make the drawing of the two directions at A’, one along a’B’, the other perpendicular to his plane. He will obtain a figure precisely similar to Fig. 2, and will see that, as AB can turn around 4, so 4’c’ around A.

In this turning aB and a’s’ would not interfere with each other, as they would if they moved in the plane around the separate points a and a’.

Hence the plane being would conclude that a rotation round a line was possible. He could see his square as it