Now if we use i, j, k, for the three space directions, i left to right, j from near away, k from below up; then, using the colour names for the axes, we have that first of all white runs i, yellow runs j, red runs k; then after the first turning round the k axis, white runs negative j, yellow runs i, red runs k; thus we have the table:—
| i | j | k | |
| 1st position | white | yellow | red |
| 2nd position | yellow | white— | red |
| 3rd position | red | yellow | white— |
Here white with a negative sign after it in the column under j means that white runs in the negative sense of the j direction.
We may express the fact in the following way:— In the plane there is room for two axes while the body has three. Therefore in the plane we can represent any two. If we want to keep the axis that goes in the unknown dimension always running in the positive sense, then the axis which originally ran in the unknown dimension (the white axis) must come in in the negative sense of that axis which goes out of the plane into the unknown dimension.
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.
