100 ARCHIMEDES 
symmetrically, from the corners). We have seen that, accord 
ing to Heron, two of the semi-regular solids had already been 
discovered by Plato, and this would doubtless be his method. 
The methods (A) and (B) applied to the five regular solids 
give the following out of the 13 semi-regular solids. We 
obtain (1) from the tetrahedron, P 1 by cutting off angles 
so as to leave hexagons in the faces ; (2) from the cube, P 2 by 
leaving squares, and P 4 by leaving octagons, in the faces; 
(3) from the octahedron, P 2 by leaving triangles, and P 3 by 
leaving hexagons, in the faces; (4) from the icosahedron, 
Bj by leaving triangles, and P s by leaving hexagons, in the 
faces; (5) from the dodecahedron, P 7 by leaving pentagons, 
and P 9 by leaving decagons in the faces. 
Of the remaining six, four are obtained by cutting off all 
the edges symmetrically and equally by planes parallel to the 
edges, and then cutting off' angles. Take first the cube. 
(1) Cut off from each four parallel edges portions which leave 
an octagon as the section of the figure perpendicular to the 
edges; then cut off equilateral triangles from the corners 
(see Fig. 1); this gives P 5 containing 8 equilateral triangles 
and 18 .squares. (P 5 is also obtained by bisecting all the 
edges of P 2 and cutting off corners.) (2) Cut off from the 
edges of the cube a smaller portion so as to leave in each 
face a square such that the octagon described in it has its 
side equal to the breadth of the section in which each edge is 
cut; then cut off hexagons from each angle (see Fig. 2); this 
Fig. 1. Fig. 2. 
gives 6 octagons in the faces, 12 squares under the edges and 
8 hexagons at the corners; that is, we have P 6 . An exactly