§ 88. The remaining propositions relate to circles and spheres. Of the sphere only one property is proved, viz.:—
Prop. 18. Spheres have to one another the triplicate ratio of that which their diameters have. The mensuration of the sphere, like that of the circle, the cylinder and the cone, had not been settled in the time of Euclid. It was done by Archimedes.
Book XIII.
§ 89. The 13th and last book of Euclid’s Elements is devoted to the regular solids (see [Polyhedron]). It is shown that there are five of them, viz.:—
1. The regular tetrahedron, with 4 triangular faces and 4 vertices;
2. The cube, with 8 vertices and 6 square faces;
3. The octahedron, with 6 vertices and 8 triangular faces;
4. The dodecahedron, with 12 pentagonal faces, 3 at each of the 20 vertices;
5. The icosahedron, with 20 triangular faces, 5 at each of the 12 vertices.
It is shown how to inscribe these solids in a given sphere, and how to determine the lengths of their edges.