The fourth dimension

198 THE FOURTH DIMENSION

would be smaller; if it were to pass farther, we should have a different figure, the outlines of which can be determined in a similar manner.

The preceding method is open to the objection that it depends rather on our inferring what must be, than our seeing what is. Let us therefore consider our sectional space as consisting of a number of planes, each very close to the last, and observe what is to be found in each plane.

The corresponding method in the case of two dimensions is as follows :—The plane being can see that line of the sectional plane through null y, null w, null rv, which lies in Null-y, eee the orange plane. Let him = now suppose the cube and the section plane to pass half way through his plane. Replacing the red and yellow axes are lines parallel to them, sections of the pink and light yellow faces.

Where will the section plane cut these parallels to the red and yellow axes?

Let him suppose the cube, in the position of the drawing, fig. 124, turned so that the pink face lies against his plane. He can see the line from the null r point to the null wh point, and can see (compare fig. 119) that it cuts aB a parallel to his red axis, drawn at a point half way along the white line, in a point B, half way up. . I shall speak of the axis as having the length of an edge of the cube. Similarly, by letting the cube turn so that the light yellow square swings against his plane, he can see (compare fig. 119) that a parallel to his yellow axis drawn from a point half-way along the white axis, is cut at half its length by the trace of the section plane in the light yellow face.

Null : N ullwh, Fig. 124,