§7παυσαμένου δὲ τοῦ Θέωνος, Εὔστροφον Ἀθηναῖον οἶμαι τὸν εἰπόντα εἶναι πρὸς ἡμᾶς
ὁρᾷς, ὡς ἀμύνει τῇ διαλεκτικῇ Θέων προθύμως, μονονοὺ τὴν λεοντῆν ἐπενδυσάμενος;
When Theon had finished speaking, I believe it was Eustrophus the Athenian who said to us, "Do you see how zealously Theon defends logic, almost as if he had put on the lion's skin?
οὕτως οὐδʼ ἡμᾶς τοὺς πάντα συλλήβδην πράγματα καὶ φύσεις καὶ ἀρχὰς θείων ὁμοῦ καὶ ἀνθρωπείων ἐν ἀριθμῷ τιθεμένους, καὶ πολὺ μάλιστα τῶν καλῶν καὶ τιμίων τοῦτον ἡγεμόνα ποιουμένους καὶ κύριον, εἰκὸς ἡσυχίαν ἄγειν, ἀλλʼ ἀπάρξασθαι τῷ θεῷ τῆς φίλης μαθηματικῆς, αὐτὸ μὲν ἐφʼ ἑαυτοῦ μήτε δυνάμει μήτε μορφῇ μήτε τῷ ῥήματι τὸ Ε τῶν ἄλλων στοιχείων διαφέρειν ἡγουμένους, ὡς δὲ μεγάλου πρὸς τὰ ὅλα καὶ κυρίου σημεῖον ἀριθμοῦ τετιμῆσθαι τῆς πεμπάδος, ἀφʼ οὗ τὸ ἀριθμεῖν οἱ σοφοὶ πεμπάζειν ὠνόμαζον. ταῦτα δὲ πρὸς ἡμᾶς ἔλεγεν οὐ παίζων ὁ Εὔστροφος, ἀλλʼ ἐπεὶ τηνικαῦτα προσεκείμην τοῖς μαθήμασιν ἐμπαθῶς, τάχα δʼ ἔμελλον εἰς πάντα τιμήσειν τὸ
μηδὲν ἄγαν
ἐν Ἀκαδημείᾳ γενόμενος.
Thus it is not reasonable for us either—who place all things together, both divine and human affairs, natures, and principles, under the category of number, and who, far more than anything beautiful and honorable, make this our leader and master—to keep silent, but rather we should dedicate to the god the first-fruits of our beloved mathematics, considering that, although E by itself in its own right differs from the other letters neither in power, nor in shape, nor in name, it has been honored as a symbol of the great and sovereign number five, which is prominent in relation to the whole, and from which the wise used to call counting 'counting by fives'." Eustrophus said these things to us not in jest; but since at that time I was passionately devoted to mathematics, and perhaps was soon going to honor 'nothing in excess' in everything once I had entered the Academy.
§8εἶπον οὖν κάλλιστα τὸν Εὔστροφον τῷ ἀριθμῷ λύειν τὴν ἀπορίαν;
Therefore I said, "Did Eustrophus not resolve the difficulty most beautifully by means of number?
ἐπεὶ γάρ
ἔφην
εἰς τὸ ἄρτιον νενεμημένου παντὸς ἀριθμοῦ καὶ τὸ περιττὸν τὸν ἡ μὲν μονὰς ἀμφοτέρων ἐπίκοινός ἐστι τῇ δυνάμει (διὸ καὶ προστιθεμένη τὸν μὲν περιττὸν ἀριθμὸν ἄρτιον ποιεῖ τὸν δʼ ἄρτιον περιττόν̓) ἀρχὴν δὲ τοῦ μὲν ἀρτίου τὰ δύο τοῦ δὲ περιττοῦ τὰ τρία ποιοῦνται, τὰ δὲ πέντε γεννᾶται τούτων πρὸς ἀλλήλους μιγνυμένων εἰκότως; ἔσχηκε τιμὴν πρῶτος ἐκ πρώτων ἀποτελούμενος, καὶ γάμος ἐπωνόμασται τῇ τοῦ ἀρτίου πρὸς τὸ θῆλυ περιττοῦ δʼ αὖ πρὸς τὸ ἄρρεν ὁμοιότητι·
For since," I said, "all number is divided into even and odd, and the monad is common to both in its power (for which reason, when added, it makes an odd number even, and an even number odd), while people make two the beginning of the even, and three the beginning of the odd, and five is generated when these are mixed with one another, it has reasonably obtained honor, being the first to be formed from the first principles; and it has been named 'marriage' because of the similarity of the even to the female, and again of the odd to the male.
ταῖς γὰρ εἰς ἴσα τομαῖς τῶν ἀριθμῶν, ὁ μὲν ἄρτιος πάντῃ διιστάμενος ὑπολείπει τινὰ δεκτικὴν ἀρχὴν οἷον ἐν ἑαυτῷ καὶ χώραν, ἐν δὲ τῷ περιττῷ ταὐτὸ παθόντι μέσον ἀεὶ περίεστι τῆς νεμήσεως γόνιμον·
For in the equal divisions of numbers, the even, when split completely apart, leaves some receptive principle, as it were, within itself and a space; but when the odd undergoes the same thing, a generative middle is always left over from the division.
ᾗ γονιμώτερός ἐστι τοῦ ἑτέρου, καὶ μιγνύμενος ἀεὶ κρατεῖ κρατεῖται δʼ οὐδέποτε γίγνεται γὰρ ἐξ ἀμφοῖν κατʼ οὐδεμίαν μῖξιν ἄρτιος ἀλλὰ κατὰ πάσας περιττός.
In this way it is more generative than the other, and when mixed, it always dominates and is never dominated; for from both, in no mixture does an even number arise, but in all mixtures an odd one.
ἔτι δὲ μᾶλλον αὐτὸς ἐπιβάλλων αὑτῷ καὶ συντιθέμενος δείκνυσι τὴν διαφορὰν ἑκάτερος· ἄρτιος μὲν γὰρ οὐδεὶς ἀρτίῳ συνελθὼν περισσὸν παρέσχεν οὐδʼ ἐξέβη τὸ οἰκεῖον ὑπʼ ἀσθενείας ἄγονος ὢν ἑτέρου καὶ ἀτελής περισσοὶ δὲ μιγνύμενοι περισσοῖς ἀρτίους πολλοὺς διὰ τὸ πάντῃ γόνιμον ἀποτελοῦσι.
And still more, when each is applied to itself and added together, it demonstrates the difference: for no even number, when combined with an even number, has ever produced an odd number, nor has it departed from its own nature due to weakness, being unproductive of another and incomplete; but odd numbers, when mixed with odd numbers, produce many even numbers because of their completely generative nature.
τὰς δʼ ἄλλας οὐκ ἄν τις ἐν καιρῷ νῦν ἐπεξίοι δυνάμεις καὶ διαφορὰς τῶν ἀριθμῶν.
But there is no fitting opportunity now to go through the other powers and differences of numbers.
ὡς οὖν ἄρρενός τε τοῦ πρώτου καὶ θήλεος ὁμιλίᾳ τὰ πέντε γιγνόμενα γάμον οἱ Πυθαγόρειοι προσεῖπον.
Since, therefore, five is generated by the intercourse of the first male and female, the Pythagoreans called it 'marriage'.
ἔστι δʼ καὶ φύσις λέλεκται τῷ περὶ αὑτὸν πολυπλασιασμῷ πάλιν εἰς ἑαυτὸν περαίνων.
And it is also called 'nature' because, in its multiplication by itself, it terminates again in itself.
ὡς γὰρ ἡ φύσις λαβοῦσα πυρὸν ἐν σπέρματι καὶ χεαμένη πολλὰ μὲν ἐν μέσῳ φύει σχήματα καὶ εἴδη, διʼ ὧν ἐπὶ τέλος ἐξάγει τὸ ἔργον, ἐπὶ πᾶσι δὲ πυρὸν ἀνέδειξεν ἀποδοῦσα τὴν ἀρχὴν ἐν τῷ τέλει τοῦ παντός· οὕτω τῶν λοιπῶν ἀριθμῶν, ὅταν αὑτοὺς πολλαπλασιάσωσιν, εἰς ἑτέρους τελευτώντων τῇ αὐξήσει, μόνος ὁ τῶν πέντε καὶ ἓξ γενόμενος τοσαυτάκις αὑτοὺς ἀναφέρουσι καὶ ἀνασῴζουσιν.
For just as nature, having received wheat in the seed and sown it, produces many intermediate forms and shapes in between, through which she leads the work to its end, and at the very last displays wheat, restoring the beginning in the end of the whole; so, while the other numbers, when they multiply themselves, end in other numbers in their increase, only the numbers generated from five and six bring back and preserve themselves just as many times.
ἑξάκις γὰρ τὰ ἓξ τριακονταὲξ, καὶ πεντάκις τὰ πέντε εἰκοσιπέντε γίγνεται.
For six times six is thirty-six, and five times five is twenty-five.
καὶ πάλιν ὁ μὲν τῶν ἓξ ἅπαξ τοῦτο ποιεῖ καὶ μοναχῶς αὐτὸς ἀφʼ ἑαυτοῦ τετράγωνος γιγνόμενος· τῇ δὲ πεμπάδι καὶ τοῦτο μὲν συμβέβηκε κατὰ πολυαπλασιασμόν, ἰδίως δὲ τὸ κατὰ σύνθεσιν ἢ ἑαυτὴν ἢ τὴν δεκάδα ποιεῖν παρὰ μέρος ἐπιβάλλουσαν αὑτῇ, καὶ τοῦτο γίγνεσθαι μέχρι παντὸς, ἀπομιμουμένου τοῦ ἀριθμοῦ τὴν τὰ ὅλα διακοσμοῦσαν ἀρχήν.
And again, the number six does this once and in a single way, becoming a square from itself; but to the number five, not only does this happen by multiplication, but its unique characteristic is that in addition, by adding to itself, it produces alternately either itself or ten, and this happens indefinitely, the number imitating the principle that orders the universe.
ὡς γὰρ ἐκείνην φυλάττουσαν ἐκ μὲν ἑαυτῆς τὸν κόσμον ἐκ δὲ τοῦ κόσμου πάλιν αὖ ἑαυτὴν ἀποτελεῖν
πυρὸς τʼ ἀνταμείβεσθαι πάντα
φησὶν ὁ Ἡράκλειτος
καὶ πῦρ ἁπάντων, ὅκωσπερ χρυσοῦ χρήματα καὶ χρημάτων χρυσός· οὕτως ἡ τῆς πεμπάδος πρὸς ἑαυτὴν σύνοδος οὐδὲν οὔτʼ ἀτελὲς οὔτʼ ἀλλότριον γεννᾶν πέφυκεν, ἀλλʼ ὡρισμένας ἔχει μεταβολάς· ἢ γὰρ αὑτὴν ἢ τὴν δεκάδα γεννᾷ, τουτέστιν ἢ τὸ οἰκεῖον ἢ τὸ τέλειον.
For just as that principle, preserving itself, produces the cosmos out of itself, and out of the cosmos again itself— 'all things are exchanged for fire, and fire for all things, just as goods for gold and gold for goods,' as Heraclitus says—so the coming together of five with itself is not natural to generate anything incomplete or foreign, but has defined changes: for it generates either itself or ten, that is, either what is its own or what is perfect."