There is another way in which a plane being can think about three-dimensional movements; and, as it affords the type by which we can most conveniently think about four-dimensional movements, it will be no loss of time to consider it in detail.
Fig. 1 (129).
We can represent the plane being and his object by figures cut out of paper, which slip on a smooth surface. The thickness of these bodies must be taken as so minute that their extension in the third dimension escapes the observation of the plane being, and he thinks about them as if they were mathematical plane figures in a plane instead of being material bodies capable of moving on a plane surface. Let Ax, Ay be two axes and ABCD a square. As far as movements in the plane are concerned, the square can rotate about a point A, for example. It cannot rotate about a side, such as AC.
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 neglects one of his axes and represents 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 lines to represent the two axes, Ax and Az. Let Fig. 2 be this drawing. Here the z axis has taken the place of the y axis, and the plane of Ax Az is represented in his plane. In this figure all that exists of the square ABCD will be the line AB.
Fig. 2 (130).
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 AB in this diagram. It is simply a case of plane turning around the point A. The line AB occupies intermediate portions like AB1 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, A´B´, 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 A, so A´C´ around A.