Euclid's Elements
Occasionally in ancient times, writings were attributed to celebrated authors but were not written by them. It is in this way that the apocryphal books XIV and XV of the Elements were sometimes included in the collection. The spurious Book XIV was probably written by Hypsicles on the basis of a treatise by Apollonius of Perga. The book continues Euclid's comparison of regular solids inscribed in spheres. 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.