Suppose the sectional plane to pass through Mala, cutting Cuspis, Dos, and Arctos each at unit from Corvus. To find the figure, the plane-being would have to take Moena, a section half-way between Moena and Murex, Murex, and an imaginary section half-a-unit beyond Murex (Fig. 24). He would produce Arctos and Cuspis to points half-a-unit from Ilex and Nugæ, and by joining these points, he would see that the line passes through the middle points of Callis and Far (a, b, Fig. 24). In the last square, the imaginary section, there would be the point m; for this is 1¹⁄₂ unit from Corvus measured along Dos produced. There would also be lines in the other two squares, the section and Murex, and to find these he would have to make many observations. He found the points a and b (Fig. 24) by drawing a line from r to s, r and s being each 1¹⁄₂ unit from Corvus, and simply seeing that it cut Callis and Far at the middle point of each. He might now imagine a cube Mala turned about Arctos, so that Alvus came into his plane; he might then produce Arctos and Dos until they were each unit long, and join their extremities, when he would see that Via and Bucina are each cut half-way. Again, by turning Syce into his plane, and producing Dos and Cuspis to points 1¹⁄₂ unit from Corvus and joining the points, he would see that Bolus and Cadus are cut half-way. He has now determined six points on Mala, through which the plane passes, and by referring them in pairs to Ilex, Olus, Cista, Crus, Nugæ, Sors, he would find that each was d (¹⁄₂)² from the next; so he would know that the figure is an equilateral hexagon. The angles he would not have got in this observation, and they might be a serious difficulty to him. It should be observed that a similar difficulty does not come to us in our observation of the sections of a tessaract: for, if the angles of each side of a solid figure are determined, the solid angles are also determined. There is another, and in some respects a better, way by which he might have found the sides of this figure. If he had noticed his plane-space much, he would have found out that, if a line be drawn to cut two other lines which meet, the ratio of the parts of the two lines cut off by the first line, on the side of the angle, is the same for those lines, and any other two that are parallel to them. Thus, if a b and a c (Fig. 25) meet, making an angle at a, and b c crosses them, and also crosses a′ b′ and a′ c′, these last two being parallel to a b and a c, then a b ∶ a c ∷ a′ b′ ∶ a′ c′. [Illustration: Fig. 25] If the plane-being knew this, he would rightly assume that if three lines meet, making a solid angle, and a plane passes through them, the ratio of the parts between the plane and the angle is the same for those three lines, and for any other three parallel to them.