After the judgment of the thoughts, an exact discernment of how the thoughts happen to be, whether they are good or otherwise; but imagination is the indication of the thought that has appeared. But that which appears is not in every case also true; for sometimes, without judgment, the true is presented to the mind as false, and the false is received as true through a lack of discernment. And these divisions are powers of the mind which it puts forth according to its own measure. An explanation by the same author of the mathematical existence or generation of the soul in Plato's Timaeus. You have asked me in part about the Platonic diagram concerning the soul, about which you must learn the hidden thought of the philosopher. But it is necessary first to state the entire mathematical scope of the argument, and then thus to explain to you the meaning of what has been said. For Plato, then, says thus concerning the demiurge: "he took one portion from the whole, and after this he took away a portion double this, and a third, one and a half times the second and three times the first, and a fourth double the second, and a fifth three times the third, and a sixth eight times the first, and a seventh twenty-seven times the first. After this he went on to fill up both the double and the triple intervals, cutting off yet more portions from the original mixture and placing them in the intervals, so that in each interval there were two means, the one exceeding and being exceeded by the extremes by the same fraction of the extremes, the other exceeding and being exceeded by the same number. And when from these links intervals of one and a half and one and a third and one and an eighth had been formed in the previous intervals, he filled up all the intervals of one and a third with the interval of one and an eighth, leaving over of each a fraction, the terms of the interval of this fraction left over being in the numerical ratio of two hundred and fifty-six to two hundred and forty-three. And indeed the mixture from which he was cutting these things he had now consumed entirely in this way." So then, this is the verbal sketch of Plato. So that we may proceed in order, let us first take, among the numbers from the monad, the ratios called primary by Plato. Let the monad, then, be set out; and double this, the dyad; then the triad, one and a half times the dyad and three times the monad; then the tetrad, four times the monad; then the ennead, three times the triad; then the ogdoad, eight times the monad; and on top of all, a seventh portion, being twenty-seven times the monad. And since Plato exhorts us to connect the double and triple intervals with harmonic and arithmetic means, but it is impossible to find these means between the monad and the dyad, some first number must be taken, which being the smallest will have both a half and a third. Let six, therefore, be taken, and let this be the portion taken by the demiurge from the whole; and after the six, the twelve, being double the six; then the eighteen, one and a half times the twelve and three times the six; then the twenty-four, being double the second term, the twelve; then the fifty-four, being three times the third term, I mean, the eighteen; then the forty-eight, being eight times the first term, I mean, the six; then the one hundred and sixty-second, being twenty-seven times the first term, the six. Therefore, between the six and twelve will fall the harmonic mean, the eight, and the arithmetic, the nine; and between the twelve and the double twenty-four, the harmonic mean is the sixteen, and the arithmetic, the eighteen; and between the third double, of the twenty-four and forty-eight, the harmonic mean is 32, and the arithmetic, 36; and in the triples, between the six and 18, the harmonic mean is 9, and the arithmetic, 12. And let these terms be set out in order: 6, 8, 9, 12, 16, 24, 32, 36, 40 50, 54, 81, 108, 162. But if it were possible in 5 these
μετὰ τὴν κρίσιν τῶν νοημάτων ἀκριβὴς διάγνωσις ὡς ἂν τύχῃ ὄντα τὰ νοήματα, εἴτε καλῶς εἴτε καὶ ἄλλως ἔχοντα· φαντασία δὲ ἡ τοῦ φανέντος νοήματος ἔνδειξις. οὐ πάντως δὲ τὸ φαινόμενον καὶ ἀληθές· ἀκρίτως γὰρ ἔστιν ὅτε παραπέμπεται τῇ διανοίᾳ τὸ ἀληθὲς ὡς ψευδὲς καὶ τὸ ψεῦδος ὡς ἀληθὲς παραδεχομένῃ δι' ἔλλειψιν διακρίσεως. αὗται δὲ αἱ διαιρέσεις δυνάμεις εἰσὶ τοῦ νοὸς ἃς κατὰ τὸ αὐτοῦ μέτρον προβάλλεται. Τοῦ αὐτοῦ ἐξήγησις τῆς ἐν τῷ Τιμαίῳ τοῦ Πλάτωνος μαθηματικῆς περὶ ψυχῆς ὑπάρξεως ἢ γεννήσεωσ Ἐκ μέρους ἠρώτηκάς με τὸ Πλατωνικὸν περὶ τῆς ψυχῆς διάγραμμα, περὶ οὗ δέῃ μαθεῖν τὴν κεκρυμμένην τοῦ φιλοσόφου διάνοιαν. δεῖ δὲ πρῶτον πᾶσαν ἐρεῖν τὴν μαθηματικὴν τοῦ λόγου περιοχήν, εἶθ' οὕτως σοι τῶν εἰρημένων τὸν νοῦν ἐξηγήσασθαι. λέγει γοῦν περὶ τοῦ δημιουργοῦ ὁ Πλάτων οὑτωσί· «μίαν ἀφεῖλεν ἀπὸ παντὸς μοῖραν, μετὰ δὲ ταύτην ἀφῄρει διπλασίαν ταύτης, τὴν δὲ αὖ τρίτην ἡμιολίαν μὲν τῆς δευτέρας, τριπλασίαν δὲ τῆς πρώτης, τετάρτην δὲ τῆς δευτέρας διπλῆν, πέμπτην δὲ τριπλῆν τῆς τρίτης, τὴν δὲ ἕκτην τῆς πρώτης ὀκταπλασίαν, ἑβδόμην δ' ἑπτακαιεικοσαπλασίαν τῆς πρώτης. μετὰ δὲ ταῦτα συνεπλήρου 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 διαστάσεσι, τῷ τοῦ ἐπογδόου διαστήματι τὰ ἐπίτριτα πάντα συνεπληροῦτο, λείπων αὐτῶν ἑκάστου μόριον, τῆς δὲ τοῦ μορίου ταύτης διαστάσεως λειφθείσης ἀριθμοῦ πρὸς ἀριθμὸν ἐχούσης τοὺς ὅρους ἓξ καὶ πεντήκοντα καὶ διακοσίων πρὸς τρία καὶ τετταράκοντα καὶ διακόσια. καὶ δὴ τὸ μιχθὲν ἐξ οὗ ταῦτα κατέτεμνεν οὕτως ἤδη πάντα κατανηλώκει.» Ἡ μὲν οὖν τοῦ Πλάτωνος ἐπὶ λέξεων ὑποτύπωσις αὕτη. ἵνα οὖν ἐν τάξει προΐωμεν, λάβωμεν πρῶτον ἐν τοῖς ἀπὸ μονάδος ἀριθμοῖς τοὺς λεγομένους πρώτους ὑπὸ τοῦ Πλάτωνος λόγους. ἐκκείσθω οὖν μονάς· καὶ ταύτης διπλασία δυάς· εἶτα τριάς, ἡμιολία μὲν τῆς δυάδος, τριπλασία δὲ τῆς μονάδος· εἶτα τετράς, τετραπλασία τῆς μονάδος· εἶτα ἐννεάς, τριπλασία τῆς τριάδος· εἶτα ὀγδοάς, ὀκταπλασία τῆς μονάδος· ἐπὶ πᾶσι δὲ ἑβδόμη {καὶ εἰκοστὴ} μοῖρα ἑπτακαιεικοσαπλασία οὖσα τῆς μονάδος. ἐπειδὴ δὲ παρακελεύεται ἡμῖν ὁ Πλάτων τὰ διπλάσια καὶ τριπλάσια διαστήματα ταῖς ἁρμονικαῖς καὶ ἀριθμητικαῖς μεσότησι συνδεῖν, μονάδος δὲ καὶ δυάδος μεταξὺ ταύτας τὰς μεσότητας εὑρεῖν ἀδύνατον, ληπτέον τινὰ πρῶτον ἀριθμόν, ὃς ἐλάχιστος ὢν ἕξει καὶ ἥμισυ καὶ τρίτον. εἰλήφθω οὖν ὁ ἕξ, καὶ ἔστω οὗτος ἡ ἀφῃρημένη παρὰ τοῦ δημιουργοῦ ἀπὸ παντὸς μοῖρα· μετὰ δὲ τὸν ἓξ ὁ δώδεκα, διπλάσιος ὢν τοῦ ἕξ· εἶτα ὁ ὀκτωκαίδεκα, ἡμιόλιος μὲν τοῦ δώδεκα, τριπλάσιος δὲ τοῦ ἕξ· εἶτα ὁ εἰκοσιτέτταρα, διπλάσιος ὢν τοῦ δευτέρου ὅρου τοῦ δώδεκα· εἶτα ὁ πεντηκοντατέτταρα, τριπλάσιος ὢν τοῦ τρίτου ὅρου, φημὶ δὴ τοῦ ὀκτωκαίδεκα· εἶτα ὁ τετταρακονταοκτώ, ὀκταπλάσιος ὢν τοῦ πρώτου ὅρου, φημὶ δὴ τοῦ ἕξ· εἶτα ὁ ἑκατοστὸς ἑξηκοστὸς δεύτερος, ἑπτακαιεικοσαπλάσιος ὢν τοῦ πρώτου ὅρου τοῦ ἕξ. τοιγαροῦν μεταξὺ μὲν τοῦ ἓξ καὶ δώδεκα ἐμπεσοῦνται ἁρμονικὴ μὲν μεσότης ὁ ὀκτώ, ἀριθμητικὴ δὲ ὁ ἐννέα· μεταξὺ δὲ τοῦ δώδεκα καὶ τοῦ εἰκοστοῦ τετάρτου διπλασίου ἁρμονικὴ μὲν μεσότης ὁ δεκαέξ, ἀριθμητικὴ δὲ ὁ ὀκτωκαίδεκα· μεταξὺ δὲ τοῦ τρίτου διπλασίου τοῦ εἰκοσιτέτταρα καὶ τετταρακονταοκτὼ ἁρμονικὴ μὲν μεσότης ὁ ˉλˉβ, ἀριθμητικὴ δὲ ὁ ˉλˉϛ· ἐν δὲ τοῖς τριπλασίοις μεταξὺ τοῦ ἓξ καὶ ˉιˉη ἁρμονικὴ μὲν μεσότης ὁ ˉθ, ἀριθμητικὴ δὲ ὁ ˉιˉβ. καὶ κείσθωσαν ἐφεξῆς ὅροι οὗτοι· ˉϛ, ˉη, ˉθ, ˉιˉβ, ˉιˉϛ, ˉκˉδ, ˉλˉβ, ˉλˉϛ, ˉμˉν, ˉνˉδ, ˉπˉα, ˉρˉη, ˉρˉξˉβ. ἀλλ' εἰ μὲν ἦν δυνατὸν ἐν 5 τούτοις τοῖς