To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the side of the dodecahedron is the irrational straight line called apotome.
Proof
The dodecahedron has twelve regular pentagonal faces; the side is
the apotome formed when the cube-edge is cut in extreme and mean
ratio (XIII.6). Inscribe by placing pentagonal faces on the six
square faces of the inscribed cube (XIII.15).