The book continues Euclid 's comparison of regular solids inscribed in spheres , with the chief result being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes , the ratio being The spurious Book XV was probably written , at least in part , by Isidore of Miletus .
The chief result is that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes . The spurious Book XV was probably written , at least in part , by Isidore of Miletus .