If an equilateral triangle is inscribed in a circle, then the square on the side of the triangle is triple the square on the radius of the circle.
Proposition 13.
To construct a pyramid, to comprehend it in a given sphere; and to prove that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid.
Lemma for XIII.13.
Proposition 14.
To construct an octahedron and comprehend it in a sphere, as in the preceding case; and to prove that the square on the diameter of the sphere is double the square on the side of the octahedron.
Proposition 15.
To construct a cube and comprehend it in a sphere, like the pyramid; and to prove that the square on the diameter of the sphere is triple the square on the side of the cube.
Proposition 16.
To construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the icosahedron is the irrational straight line called minor.
Corollary. The square on the diameter of the sphere is five times the square on the radius of the circle from which the icosahedron has been described, and the diameter of the sphere is composed of the side of the hexagon and two of the sides of the decagon inscribed in the same circle.
Proposition 17.
To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the dodecahedron is the irrational straight line called apotome.
Corollary. When the side of the cube is cut in extreme and mean ratio, the greater segment is the side of the dodecahedron.
Proposition 18.
To set out the sides of the five figures and compare them
with one another.
