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 movable both round the sun and upon its own axis; and I showed that the differences of their orbits corresponded to the five regular Pythagorean figures, which had been already distributed by their author among the elements of the world, though the attempt was admirable rather than happy or legitimate, and for which figures' sake Euclid wrote the whole of his Geometry. Now, in that Mystery you may find a sort of combination of Astronomy and Euclid's Geometry, and through this combination a most thorough completion and finishing of them both; and this was the reason why I waited with intense longing to see what sort of an argument Galileo would produce in favour of the Pythagorean system of the universe. After this explanation, Galileo's letter about this argument was as follows:—

"Illustrissimo e Reverendissimo Signore mio colendissimo.

"E tempo che io deciferi a Vostra Signoria Illustrissima e Reverendissima e per lei al Sig. Keplero le lettere trasposte le quali alcune settimane sono le inviai; è tempo dico, giacchè sono interamente chiaro della verità del fatto, sicchè non ci resta un minimo scrupolo, o dubbio. Sapranno dunque come circa a tre mesi fa vedendosi Venere vespertina la cominciai ad osservar diligentemente coll' occhiale, per