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 of the square on the side of the octahedron.
Proof
Take two perpendicular diameters in a circle; through the centre
erect a perpendicular axis equal in length to the diameter. The
four endpoints in the circle and two endpoints on the axis form the
six vertices of the octahedron. Diameter-squared / side-squared =2.