When he had mingled them with the essence and out of three made one, he again divided this whole into as many portions as was fitting, each portion being a compound of the same, the other, and the essence. And he proceeded to divide after this manner:-First of all, he took away one part of the whole [1], and then he separated a second part which was double the first [2], and then he took away a third part which was half as much again as the second and three times as much as the first [3], and then he took a fourth part which was twice as much as the second [4], and a fifth part which was three times the third [9], and a sixth part which was eight times the first [8], and a seventh part which was twenty-seven times the first [27]. After this he filled up the double intervals [i.e. between 1, 2, 4, 8] and the triple [i.e. between 1, 3, 9, 27] cutting off yet other portions from the mixture and placing them in the intervals, so that in each interval there were two kinds of means, the one exceeding and exceeded by equal parts of its extremes [as for example 1, 4/3, 2, in which the mean 4/3 is one-third of 1 more than 1, and one-third of 2 less than 2], the other being that kind of mean which exceeds and is exceeded by an equal number. Where there were intervals of 3/2 and of 4/3 and of 9/8, made by the connecting terms in the former intervals, he filled up all the intervals of 4/3 with the interval of 9/8, leaving a fraction over; and the interval which this fraction expressed was in the ratio of 256 to 243. And thus the whole mixture out of which he cut these portions was all exhausted by him. This entire compound he divided lengthways into two parts, which he joined to one another at the centre like the letter X, and bent them into a circular form, connecting them with themselves and each other at the point opposite to their original meeting-point; and, comprehending them in a uniform revolution upon the same axis, he made the one the outer and the other the inner circle. Now the motion of the outer circle he called the motion of the same, and the motion of the inner circle the motion of the other or diverse. The motion of the same he carried round by the side to the right, and the motion of the diverse diagonally to the left. And he gave dominion to the motion of the same and like, for that he left single and undivided; but the inner motion he divided in six places and made seven unequal circles having their intervals in ratios of two-and three, three of each, and bade the orbits proceed in a direction opposite to one another; and three [Sun, Mercury, Venus] he made to move with equal swiftness, and the remaining four [Moon, Saturn, Mars, Jupiter] to move with unequal swiftness to the three and to one another, but in due proportion.

[35b] μειγνὺς δὲ μετὰ τῆς οὐσίας καὶ ἐκ τριῶν ποιησάμενος ἕν͵ πάλιν ὅλον τοῦτο μοίρας ὅσας προσῆκεν διένειμεν͵ ἑκάστην δὲ ἔκ τε ταὐτοῦ καὶ θατέρου καὶ τῆς οὐσίας μεμειγμένην. ἤρχετο δὲ διαιρεῖν ὧδε. μίαν ἀφεῖλεν τὸ πρῶτον ἀπὸ παντὸς μοῖραν͵ μετὰ δὲ ταύτην ἀφῄρει διπλασίαν ταύτης͵ τὴν δ΄ αὖ τρίτην ἡμιολίαν μὲν τῆς δευτέρας͵ τριπλασίαν δὲ τῆς πρώτης͵ τετάρτην δὲ τῆς δευτέρας διπλῆν͵ πέμπτην δὲ τριπλῆν τῆς τρίτης͵ [35c] τὴν δ΄ ἕκτην τῆς πρώτης ὀκταπλασίαν͵ ἑβδόμην δ΄ ἑπτακαιεικοσιπλασίαν τῆς πρώτης· [36a] μετὰ δὲ ταῦτα συνεπληροῦτο τά τε διπλάσια καὶ τριπλάσια διαστήματα͵ μοίρας ἔτι ἐκεῖθεν ἀποτέμνων καὶ τιθεὶς εἰς τὸ μεταξὺ τούτων͵ ὥστε ἐν ἑκάστῳ διαστήματι δύο εἶναι μεσότητας͵ τὴν μὲν ταὐτῷ μέρει τῶν ἄκρων αὐτῶν ὑπερέχουσαν καὶ ὑπερεχομένην͵ τὴν δὲ ἴσῳ μὲν κατ΄ ἀριθμὸν ὑπερέχουσαν͵ ἴσῳ δὲ ὑπερεχομένην. ἡμιολίων δὲ διαστάσεων καὶ ἐπιτρίτων καὶ ἐπογδόων γενομένων ἐκ τούτων τῶν δεσμῶν ἐν ταῖς πρόσθεν διαστάσεσιν͵ [36b] τῷ τοῦ ἐπογδόου διαστήματι τὰ ἐπίτριτα πάντα συνεπληροῦτο͵ λείπων αὐτῶν ἑκάστου μόριον͵ τῆς τοῦ μορίου ταύτης διαστάσεως λειφθείσης ἀριθμοῦ πρὸς ἀριθμὸν ἐχούσης τοὺς ὅρους ἓξ καὶ πεντήκοντα καὶ διακοσίων πρὸς τρία καὶ τετταράκοντα καὶ διακόσια. καὶ δὴ καὶ τὸ μειχθέν͵ ἐξ οὗ ταῦτα κατέτεμνεν͵ οὕτως ἤδη πᾶν κατανηλώκει. ταύτην οὖν τὴν σύστασιν πᾶσαν διπλῆν κατὰ μῆκος σχίσας͵ μέσην πρὸς μέσην ἑκατέραν ἀλλήλαις οἷον χεῖ προσβαλὼν κατέκαμψεν εἰς ἓν κύκλῳ͵ [36c] συνάψας αὑταῖς τε καὶ ἀλλήλαις ἐν τῷ καταντικρὺ τῆς προσβολῆς καὶ τῇ κατὰ ταὐτὰ ἐν ταὐτῷ περιαγομένῃ κινήσει πέριξ αὐτὰς ἔλαβεν͵ καὶ τὸν μὲν ἔξω͵ τὸν δ΄ ἐντὸς ἐποιεῖτο τῶν κύκλων. τὴν μὲν οὖν ἔξω φορὰν ἐπεφήμισεν εἶναι τῆς ταὐτοῦ φύσεως͵ τὴν δ΄ ἐντὸς τῆς θατέρου. τὴν μὲν δὴ ταὐτοῦ κατὰ πλευρὰν ἐπὶ δεξιὰ περιήγαγεν͵ τὴν δὲ θατέρου κατὰ διάμετρον ἐπ΄ ἀριστερά͵ κράτος δ΄ ἔδωκεν τῇ ταὐτοῦ καὶ ὁμοίου περιφορᾷ· [36d] μίαν γὰρ αὐτὴν ἄσχιστον εἴασεν͵ τὴν δ΄ ἐντὸς σχίσας ἑξαχῇ ἑπτὰ κύκλους ἀνίσους κατὰ τὴν τοῦ διπλασίου καὶ τριπλασίου διάστασιν ἑκάστην͵ οὐσῶν ἑκατέρων τριῶν͵ κατὰ τἀναντία μὲν ἀλλήλοις προσέταξεν ἰέναι τοὺς κύκλους͵ τάχει δὲ τρεῖς μὲν ὁμοίως͵ τοὺς δὲ τέτταρας ἀλλήλοις καὶ τοῖς τρισὶν ἀνομοίως͵ ἐν λόγῳ δὲ φερομένους.