In the case we are dealing with he knows that from Ilex to the point on Arctos produced where the plane cuts, it is half-a-unit; and as the Z, X, and Y lines are cut equally from Corvus, he would conclude that the X and Y lines are cut the same distance from Ilex as the Z line, that is half-a-unit. He knows that the X line is cut at 1¹⁄₂ units from Corvus; that is, half-a-unit from Nugæ: so he would conclude that the Z and Y lines are cut half-a-unit from Nugæ. He would also see that the Z and X lines from Cista are cut at half-a-unit. He has now six points on the cube, the middle points of Callis, Via, Bucina, Cadus, Bolus, and Far. Now, looking at his square sections, he would see on Moena a line going from middle of Far to middle of Callis, that is, a line d (¹⁄₂)² long. On the section he would see a line from middle of Via to middle of Bolus d (1)² long, and on Murex he would see a line from middle of Cadus to middle of Bucina, d (¹⁄₂)² long. Of these three lines a b, c d, e f, (Fig. 24)--a b and e f are sides, and c d is a section of the required figure. He can find the distances between a and c by reference to Ilex, between b and d by reference to Nugæ, between c and e by reference to Olus, and between d and f by reference to Crus; and he will find that these distances are each d (¹⁄₂)². Thus, he would know that the figure is an equilateral hexagon with its sides d (¹⁄₂)² long, of which two of the opposite points (c and d) are d (1)² apart, and the only figure fulfilling all these conditions is an equilateral and equiangular hexagon. Enough has been said about sections of a cube, to show how a plane-being would find the shapes in any set as in Z X Y I . II . II or Z X Y I . I . II. He would always have to bear in mind that the ratio of the lengths of the Z, X, and Y lines is the same from Corvus to the sectional plane as from any other point to the sectional plane. Thus, if he were taking a section where the plane cuts Arctos and Cuspis at one unit from Corvus and Dos at one-and-a-half, that is where the ratio of Z and of X to Y is as two to three, he would see that Dos itself is not cut at all; but from Cista to the point on Dos produced is half-a-unit; therefore from Cista, the Z and X lines will be cut at ²⁄₃ of ¹⁄₂ unit from Cista. It is impossible in writing to show how to make the various sections of a tessaract; and even if it were not so, it would be unadvisable; for the value of doing it is not in seeing the shapes themselves, so much as in the concentration of the mind on the tessaract involved in the process of finding them out.