Z X Y 0 . ¹⁄₂ . ¹⁄₂. This set of figures will be expressed Z X Y Z X Y Z X Y 0 . ¹⁄₂ . ¹⁄₂ 0 . 1 . 1 0 . 1¹⁄₂ . 1¹⁄₂ It will be seen that these sections are parallel to each other; and that in each figure Cuspis and Dos are cut at equal distances from Corvus. We may express the whole set thus:-- Z X Y O . I . I it being understood that where Roman figures are used, the numbers do not refer to the length of unit cut off any given line from Corvus, but to the proportion between the lengths. Thus Z X Y O . I . II means that Arctos is not cut at all, and that Cuspis and Dos are cut, Dos being cut twice as far from Corvus as is Cuspis. These figures will also be rectangles. Take the third case. Suppose Arctos, Cuspis, and Dos are each cut half-way. This figure is an equilateral triangle, whose sides are the diagonal of a half-unit squared. The figure Z X Y 1 . 1 . 1 is also an equilateral triangle, and the figure Z X Y 1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂ is an equilateral hexagon. It is easy for us to see what these shapes are, and also, what the figures of any other set would be, as Z X Y I . II . II or Z X Y I . II . III but we must learn them as a two-dimensional being would, so that we may see how to learn the three-dimensional sections of a tessaract. It is evident that the resulting figures are the same whether we fix the cube, and then turn the sectional plane to the required position, or whether we fix the sectional plane, and then turn the cube. Thus, in the first case we might have fixed the plane, and then so placed the cube that one plane side coincided with the sectional plane, and then have drawn the cube half-way through, in a direction at right angles to the plane, when we should have seen the square first mentioned. In the second case (Z X Y) (O . I . I) we might have put the cube with Arctos coinciding with the plane and with Cuspis and Dos equally inclined to it, and then have drawn the cube through the plane at right angles to it until the lines (Cuspis and Dos) were cut at the required distances from Corvus. In the third case we might have put the cube with only Corvus coinciding with the plane and with Cuspis, Dos, and Arctos equally inclined to it (for any of the shapes in the set Z X Y) I . I . I) and then have drawn it through as before. The resulting figures are exactly the same as those we got before; but this way is the best to use, as it would probably be easier for a two-dimensional being to think of a cube passing through his space than to imagine his whole space turned round, with regard to the cube. We have already seen (p. 117) how a two-dimensional being would observe the sections of a cube when it is put with one plane side coinciding with his space, and is then drawn partly through.