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 A_x_, A_y_ 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 A_z_ be the axis, which stands perpendicular to his plane at A. He can draw in his plane two lines to represent the two axes, A_x_ and A_z_. Let Fig. 2 be this drawing. Here the _z_ axis has taken the place of the _y_ axis, and the plane of A_x_ A_z_ is represented in his plane. In this figure all that exists of the square ABCD will be the line AB. [Illustration: 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 AB_{1} and after half a revolution will lie on A_x_ 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. In this turning AB and A´B´ 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 began to make this turning. He could see it half way round when it came to lie on the opposite side of the line AC. But in intermediate portions he could not see it, for it runs out of the plane. Coming now to the question of a four-dimensional body, let us conceive of it as a series of cubic sections, the first in our space, the rest at intervals, stretching away from our space in the unknown direction. We must not think of a four-dimensional body as formed by moving a three-dimensional body in any direction which we can see.