In entering on this enquiry we must make a plan of procedure. The method which I shall adopt is to trace out the steps of reasoning by which a being confined to movement in a two-dimensional world could arrive at a conception of our turning and rotation, and then to apply an analogous process to the consideration of the higher movements. The plane being must be imagined as no abstract figure, but as a real body possessing all three dimensions. His limitation to a plane must be the result of physical conditions. We will therefore think of him as of a figure cut out of paper placed on a smooth plane. Sliding over this plane, and coming into contact with other figures equally thin as he in the third dimension, he will apprehend them only by their edges. To him they will be completely bounded by lines. A “solid” body will be to him a two-dimensional extent, the interior of which can only be reached by penetrating through the bounding lines. Now such a plane being can think of our three-dimensional existence in two ways. First, he can think of it as a series of sections, each like the solid he knows of extending in a direction unknown to him, which stretches transverse to his tangible universe, which lies in a direction at right angles to every motion which he made. Secondly, relinquishing the attempt to think of the three-dimensional solid body in its entirety he can regard it as consisting of a number of plane sections, each of them in itself exactly like the two-dimensional bodies he knows, but extending away from his two-dimensional space. A square lying in his space he regards as a solid bounded by four lines, each of which lies in his space. A square standing at right angles to his plane appears to him as simply a line in his plane, for all of it except the line stretches in the third dimension. He can think of a three-dimensional body as consisting of a number of such sections, each of which starts from a line in his space. Now, since in his world he can make any drawing or model which involves only two dimensions, he can represent each such upright section as it actually is, and can represent a turning from a known into the unknown dimension as a turning from one to another of his known dimensions. To see the whole he must relinquish part of that which he has, and take the whole portion by portion. Consider now a plane being in front of a square, fig. 34. The square can turn about any point in the plane—say the point A. But it cannot turn about a line, as AB. For, in order to turn about the line AB, the square must leave the plane and move in the third dimension. This motion is out of his range of observation, and is therefore, except for a process of reasoning, inconceivable to him. [Illustration: Fig. 34.] Rotation will therefore be to him rotation about a point. Rotation about a line will be inconceivable to him.