It is obvious that the unknown direction, the direction in which the white line runs at first, is quite distinct from any direction which the plane creature knows. The white line may come in towards him, or running down. If he is looking at a square, which is the face of a cube (looking at it by a line), then any one of the bounding lines remaining unmoved, another face of the cube may come in, any one of the faces, namely, which have the white line in them. And the white line comes sometimes in one of the space directions he knows, sometimes in another. Now this turning which leaves a line unchanged is something quite unlike any turning he knows in the plane. In the plane a figure turns round a point. The square can turn round the null point in his plane, and the red and yellow lines change places, only of course, as with every rotation of lines at right angles, if red goes where yellow went, yellow comes in negative of red’s old direction. This turning, as the plane creature conceives it, we should call turning about an axis perpendicular to the plane. What he calls turning about the null point we call turning about the white line as it stands out from his plane. There is no such thing as turning about a point, there is always an axis, and really much more turns than the plane being is aware of. Taking now a different point of view, let us suppose the cubes to be presented to the plane being by being passed transverse to his plane. Let us suppose the sheet of matter over which the plane being and all objects in his world slide, to be of such a nature that objects can pass through it without breaking it. Let us suppose it to be of the same nature as the film of a soap bubble, so that it closes around objects pushed through it, and, however the object alters its shape as it passes through it, let us suppose this film to run up to the contour of the object in every part, maintaining its plane surface unbroken. Then we can push a cube or any object through the film and the plane being who slips about in the film will know the contour of the cube just and exactly where the film meets it. [Illustration: Fig. 90.] Fig. 90 represents a cube passing through a plane film. The plane being now comes into contact with a very thin slice of the cube somewhere between the left and right hand faces. This very thin slice he thinks of as having no thickness, and consequently his idea of it is what we call a section. It is bounded by him by pink lines front and back, coming from the part of the pink face he is in contact with, and above and below, by light yellow lines. Its corners are not null-coloured points, but white points, and its interior is ochre, the colour of the interior of the cube.