He shows how to inscribe the five regular polyhedra within it.

If we project both polyhedra orthogonally on a plane perpendicular to the axis of the paraboloid, we obtain two figures which are reciprocal, except that corresponding lines are orthogonal instead of parallel.

Incidentally Pappus describes the thirteen other polyhedra bounded by equilateral and equiangular but not similar polygons, discovered by Archimedes, and finds, by a method recalling that of Archimedes, the surface and volume of a sphere.

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To describe a circle touching three given circles, which he discovered in 1805, his generalization of Euler's theorem on polyhedra in 1811, and in several other elegant problems. More important is his memoir on wave-propagation which obtained the Grand Prix of the Institut in 1816.

The interrelations of the polyhedra enumerated above are considerably elucidated by the introduction of the following terms: (1) Correspondence.

Of lost works by Archimedes we can identify the following: (I) investigations on polyhedra mentioned by Pappus; (2) Archai, Principles, a book addressed to Zeuxippus and dealing with the naming of numbers on the system explained in the Sand Reckoner; (3) Peri zygon, On balances or levers; (4) Kentrobarika, On centres of gravity; (5) Katoptrika, an optical work from which Theon of Alexandria quotes a remark about refraction; (6) Ephodion, a Method, mentioned by Suidas; (7) Peri sphairopeoia, On Sphere-making, in which Archimedes explained the construction of the sphere which he made to imitate the motions of the sun, the moon and the five planets in the heavens.

The general theory of polyhedra properly belongs to combinatorial analysis.

Mensuration of the Platonic Solids.-The mensuration of the regular polyhedra is readily investigated by the methods of elementary geometry, the property that these solids may be inscribed in and circumscribed to concentric spheres being especially useful.

Stevinus was the first to show how to model regular and semiregular polyhedra by delineating their frames in a plane.