 The treatment of the regular polyhedra constitutes one of the two principal pillars of HM, Book 5 (chapters 1–2), where the third law is formulated (see Caspar in KGW 6, Nachbericht, p. 497, and Engl. trans. p. xxxiii). Following his approach from MC and complementing it with occasional references to his EAC, Kepler makes use again of the Platonic solids to determine the number of the planets and their distances from the Sun. Meanwhile, he has learned that the application of his old polyhedral hypothesis has limits. As he tells us in a footnote in the second edition of his MC from 1621, he was earlier convinced of the possibility of explaining the eccentricities of the planetary orbits by values derived “a priori” from this hypothesis (MC, Engl. trans., p. 189). Now, with access to the observational data of Tycho, Kepler had to exclude this explanation and look for another. And this is one of the most significant achievements of his basic harmonies (which in turn are derived from the regular polygons), constituting the second great pillar of Book 5.