Note 26 — Extract from the Preface of Kepler’s Dioptrics. Augsburg, 1611.

Kepler, in his Mystery of the Universe, endeavoured to connect the orbits of the planets with the five regular solids, thus: If in a sphere (i.) a cube be inscribed, and in the cube a sphere (ii.); and in that sphere a tetrahedron, and in the tetrahedron a sphere (iii.); and in that sphere a dodecahedron, and in the dodecahedron a sphere (iv.); and in that sphere an icosahedron, and in the icosahedron a sphere (v.); and in that sphere an octahedron, and in the octahedron a sphere (vi.), the diameters of these six spheres will be proportional to the diameters of the orbits of Saturn, Jupiter, Mars, the Earth, Venus, and Mercury respectively; or, as Kepler expresses it, the common centre of these spheres represents the position of the Sun, and the six spheres represent the spheres of the planets.

By these considerations, however, Kepler was led to enunciate his third law, that the squares of the periodic times of planets are proportional to the cubes of their mean distances from the sun.—Kepler, Prodromus Dissertationum Mathematicarum continens Mysterium Cosmographicum, etc. (Tübingen, 1596.)