Euclid's Elements
Apocrypha

It was not uncommon in ancient times to attribute to celebrated authors works that were not written by them. It is by these means 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. 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 10 3 (5 − 5) = 5 + 5 6. {\displaystyle {\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}={\sqrt {\frac {5+{\sqrt {5}}}{6}}}.}

The spurious Book XV was probably written, at least in part, by Isidore of Miletus. This book covers topics such as counting the number of edges and solid angles in the regular solids, and finding the measure of dihedral angles of faces that meet at an edge.
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