εἶπον οὖν κάλλιστα τὸν Εὔστροφον τῷ ἀριθμῷ λύειν τὴν ἀπορίαν; 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· 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· ᾗ γονιμώτερός ἐστι τοῦ ἑτέρου, καὶ μιγνύμενος ἀεὶ κρατεῖ κρατεῖται δʼ οὐδέποτε γίγνεται γὰρ ἐξ ἀμφοῖν κατʼ οὐδεμίαν μῖξιν ἄρτιος ἀλλὰ κατὰ πάσας περιττός. 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. τὰς δʼ ἄλλας οὐκ ἄν τις ἐν καιρῷ νῦν ἐπεξίοι δυνάμεις καὶ διαφορὰς τῶν ἀριθμῶν. ὡς οὖν ἄρρενός τε τοῦ πρώτου καὶ θήλεος ὁμιλίᾳ τὰ πέντε γιγνόμενα γάμον οἱ Πυθαγόρειοι προσεῖπον. ἔστι δʼ καὶ φύσις λέλεκται τῷ περὶ αὑτὸν πολυπλασιασμῷ πάλιν εἰς ἑαυτὸν περαίνων. 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. ἑξάκις γὰρ τὰ ἓξ τριακονταὲξ, καὶ πεντάκις τὰ πέντε εἰκοσιπέντε γίγνεται. 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. 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.
I answered therefore that Eustrophus has excellently solved the difficulty by number. For (said I) since all number is distributed into even and odd, unity is in efficacy common to them both,—for that being added to an even number, it makes it odd, and to an odd, it makes it even, two constituting the beginning of the even, and three of the odd. Now the number of five, composed of these two, is deservedly honored, as being the first compound made of the first simple numbers, and is called the marriage, for the resemblance of the odd with the male, and the even with the female. For in the divisions of the numbers into equal parts, the even, being wholly separated, leaves a certain capacious beginning and space in itself; but in the odd, suffering the same thing, there always remains a middle, of generative distribution, by which it is more fruitful than the other, and being mixed is always master, never mastered. For by the mixture of both, even and odd together, there is never produced an even number but always an odd. But which is more, either of them added to and compounded with itself shows the difference; for no even joined with another even ever produced an odd, or went forth of its proper nature, being through weakness unable to generate another and imperfect. But odd numbers mixed with odd do, through their being every way fruitful, produce many even ones. Time does not now permit us to set down the other powers and differences of numbers. Therefore have the Pythagoreans, through a certain resemblance, said that five is the marriage of the first male and the first female number. This also is it for which it is called Nature, by the multiplication of itself determining again into itself. For as Nature, taking a grain of wheat for seed and diffusing it, produces many forms and species between, by which she brings her work to an end, but at last she shows again a grain of wheat, restoring the beginning in the end of all; so, while the rest of the numbers, when they are multiplied into others, terminate by the increase only those of five and six, multiplied by themselves, bring back and preserve themselves. For six times six makes thirty-six, and five times five makes twenty-five. And again, six does this once, and only after one manner, to wit, when it is squared. But this indeed befalls five both by multiplication and by composition with itself, to which being added, it alternatively makes itself and ten; and this as far as all number can extend, this number imitating the beginning or first Cause which governs the universe. For as that first Cause, preserving the world by itself, does reciprocally perfect itself by the world, as Heraclitus says of fire, Fire turns to all things, and all things to fire; and as money is changed for gold, and gold for money; so the congress of five with itself is framed by Nature to produce nothing imperfect or strange, but has limited changes; for it either generates itself or ten, that is, either what is proper to itself, or what is perfect.
ἐὰν οὖν ἔρηταί τις, τί ταῦτα πρὸς τὸν Ἀπόλλωνα, φήσομεν οὐχὶ μόνον ἀλλὰ καὶ πρὸς τὸν Διόνυσον, ᾧ τῶν Δελφῶν οὐδὲν ἧττον ἢ τῷ Ἀπόλλωνι μέτεστιν. ἀκούομεν οὖν τῶν θεολόγων τὰ μὲν ἐν ποιήμασι τὰ δʼ ἄνευ μέτρου λεγόντων καὶ ὑμνούντων, ὡς ἄφθαρτος ὁ θεὸς καὶ ἀίδιος πεφυκώς, ὑπὸ δή τινος εἱμαρμένης γνώμης καὶ λόγου μεταβολαῖς ἑαυτοῦ χρώμενος, ἄλλοτε μὲν εἰς πῦρ ἀνῆψε τὴν φύσιν πάνθʼ ὁμοιώσας πᾶσιν, ἄλλοτε δὲ παντοδαπὸς ἔν τε μορφαῖς καὶ ἐν πάθεσι καὶ δυνάμεσι διαφόροις γιγνόμενος, ὡς γίγνεται νῦν, κόσμος ὀνομάζεται δὲ τῷ γνωριμωτάτῳ τῶν ὀνομάτων. κρυπτόμενοι δὲ τοὺς πολλοὺς οἱ σοφώτεροι τὴν μὲν εἰς πῦρ μεταβολὴν Ἀπόλλωνά τε τῇ μονώσει Φοῖβόν τε τῷ καθαρῷ καὶ ἀμιάντῳ καλοῦσι. τῆς δʼ εἰς πνεύματα καὶ ὓδωρ καὶ γῆν καὶ ἄστρα καὶ φυτῶν ζῴων τε γενέσεις τροπῆς αὐτοῦ καὶ διακοσμήσεως τὸ μὲν πάθημα καὶ τὴν μεταβολὴν διασπασμόν τινα καὶ διαμελισμὸν αἰνίττονται· Διόνυσον δὲ καὶ Ζαγρέα καὶ Νυκτέλιον καὶ Ἰσοδαίτην αὐτὸν ὀνομάζουσι, καὶ φθοράς τινας καὶ ἀφανισμοὺς οἱ τὰς ἀποβιώσεις καὶ παλιγγενεσίας, οἰκεῖα ταῖς εἰρημέναις μεταβολαῖς αἰνίγματα καὶ μυθεύματα περαίνουσι· καὶ ᾅδουσι τῷ μὲν διθυραμβικὰ μέλη παθῶν μεστὰ καὶ μεταβολῆς πλάνην τινὰ καὶ διαφόρησιν ἐχούσης· μιξοβόαν γὰρ Αἰσχύλος φησὶ πρέπει / διθύραμβον ὁμαρτεῖν / σύγκωμον Διονύσῳ; / τῷ δὲ παιᾶνα, τεταγμένην καὶ σώφρονα μοῦσαν. ἀγήρων τε τοῦτον ἀεὶ καὶ νέον ἐκεῖνον δὲ πολυειδῆ καὶ πολύμορφον ἐν γραφαῖς καὶ πλάσμασι δημιουργοῦσι· καὶ ὅλως τῷ μὲν ὁμοιότητα καὶ τάξιν καὶ σπουδὴν ἄκρατον, τῷ δὲ μεμιγμένην τινὰ παιδιᾷ καὶ ὕβρει καὶ σπουδῇ καὶ μανίᾳ προσφέροντες ἀνωμαλίαν, εὔιον ὀρσιγύναικα μαινομέναις Διόνυσον ἀνθέοντα τιμαῖς ἀνακαλοῦσιν, οὐ φαύλως ἑκατέρας μεταβολῆς τὸ οἰκεῖον λαμβάνοντες. ἐπεὶ δʼ οὐκ ἴσος τῶν περιόδων ἐν ταῖς μεταβολαῖς χρόνος, ἀλλὰ μείζων ὁ τῆς ἑτέρας ἣν κόρον· καλοῦσιν, ὁ δὲ τῆς χρησμοσύνης ἐλάττων, τὸ κατὰ λόγον τηροῦντες ἐνταῦθα τὸν μὲν ἄλλον ἐνιαυτὸν παιᾶνι χρῶνται περὶ τὰς θυσίας, ἀρχομένου δὲ χειμῶνος ἐπεγείραντες διθύραμβον τὸν δὲ παιᾶνα καταπαύσαντες, τρεῖς μῆνας ἀντʼ ἐκείνου τοῦτον κατακαλοῦνται τὸν θεὸν · ὅπερ τρία πρὸς ἕν, τοῦτο τὴν διακόσμησιν οἰόμενοι χρόνῳ πρὸς τὴν ἐκπύρωσιν εἶναι.
Now if any one shall say, What is all this to Apollo? we will answer, that it concerns not Apollo only, but Bacchus also, who has no less to do with Delphi than Apollo himself. For we have heard the divines, partly in verse partly in prose, saying and singing, that the God is of his own nature incorruptible and eternal, but yet, through a certain fatal decree and reason, suffers changes of himself, having sometimes his nature kindled into a fire, and making all things alike, and otherwhiles becoming various, in different shapes, passions, and powers, like unto the World, and is named by this best-known of names. But the wiser, concealing from the vulgar the change into fire, call him both Apollo from his unity and Phoebus from his purity and unpollutedness. But as for the passion and change of his conversion into winds, water, earth, stars, and the various kinds of plants and animals, and its order and disposition, this they obscurely represent as a certain distraction and dismembering; and they now call him Dionysus, Zagreus, Nyctelius, and Isodaetes, exhibiting and chanting forth certain corruptions, disparitions, deaths, and resurrections, which are all riddles and fables suited to the said mutations. To Dionysus or Bacchus they sing dithyrambic verses, full of passions and change, joined with a certain wandering and agitation backwards and forwards; for, as Aeschylus says, The dithyramb, whose sounds are dissonant, / ’Tis fit should wait on Bacchus. / But to Apollo they sing the well-ordered paean and a discreet song. And Apollo both in their sculptures and statues they always make to be young and never declining to old age; but Dionysus they represent in many shapes and forms. Lastly, to the one they attribute equality, order, and unmixed gravity; but to the other, a certain unequal mixture of sports, petulancy, gravity, and madness, surnaming him, Evius Bacchus, who to rage incites / Women on tops of mountains, and delights / In frantic worship. / Thus they not unfitly touch the property of both changes. Now because the time of the revolutions in these changes is not equal, but that of the one which they call Koros (that is, satiety) is longer, and that of the other named Chresmosyne (or want) shorter; observing in this the proportion, they all the rest of the year use in their sacrifices the paean; but at the beginning of winter, rousing up the dithyramb, and laying the paean to rest, they do for three months invocate Bacchus instead of Apollo, esteeming the creation of the world to be the same in proportion of time to the conflagration of it as three to one.
ἀλλὰ ταῦτα μὲν ἱκανοῦ καιροῦ μᾶλλον ἀπομεμήκυνται· δῆλον δʼ ὅτι συνοικειοῦσιν αὐτῷ τὴν πεμπάδα, νῦν μὲν αὐτὴν ἑαυτὴν ὡς τὸ πῦρ αὖθις δὲ τὴν δεκάδα ποιοῦσαν ἐξ ἑαυτῆς ὡς τὸν κόσμον. τῆς δὲ δὴ μάλιστα κεχαρισμένης τῷ θεῷ μουσικῆς οὐκ οἰόμεθα τούτῳ τῷ ἀριθμῷ· μετεῖναι; τὸ γὰρ πλεῖστον ὡς ἔνι εἰπεῖν ἔργον ἁρμονικῆς περὶ τὰς συμφωνίας ἐστίν. αὗται δʼ ὅτι πέντε καὶ οὐ πλείους, ὁ λόγος ἐξελέγχει τὸν ἐν χορδαῖς καὶ τρυπήμασι ταῦτα θηρᾶν ἀλόγως τῇ αἰσθήσει βουλόμενον. πᾶσαι γὰρ ἐν λόγοις τὴν γένεσιν ἀριθμῶν λαμβάνουσιν· καὶ λόγος ἐστὶ τῆς μὲν διὰ τεσσάρων ἐπίτριτος, τῆς δὲ διὰ πέντε ἡμιόλιος, διπλάσιος δὲ τῆς διὰ πασῶν, τῆς δὲ διὰ πασῶν καὶ διὰ πέντε τριπλάσιος, τῆς δὲ δὶς διὰ πασῶν τετραπλάσιος. ἣν δὲ ταύταις ἐπεισάγουσιν οἱ ἁρμονικοὶ διὰ πασῶν καὶ διὰ τεσσάρων ὀνομάζοντες, ἔξω μέτρου βαίνουσαν, οὐκ ἄξιόν ἐστι δέχεσθαι, τῆς ἀκοῆς τῷ ἀλόγῳ παρὰ τὸν λόγον ὥσπερ νόμον χαριζομένους. ἵνα τοίνυν ἀφῶ πέντε τετραχόρδων θέσεις, καὶ πέντε τοὺς πρώτους, εἴτε τόνους ἢ τρόπους; εἴθʼ ἁρμονίας χρὴ καλεῖν, ὧν ἐπιτάσει καὶ ὑφέσει τρεπομένων κατὰ τὸ μᾶλλον καὶ ἧττον αἱ λοιπαὶ βαρύτητές εἰσι καὶ ὀξύτητες ἆρʼ οὐχὶ πολλῶν, μᾶλλον δʼ ἀπείρων διαστημάτων ὄντων, τὰ μελῳδούμενα μόνα πέντʼ ἐστί, δίεσις καὶ ἡμιτόνιον καὶ τόνος καὶ τριημιτόνιον καὶ δίτονον, ἄλλο δʼ οὐδὲν οὔτε μικρότερον οὔτε μεῖζον ἐν φωναῖς χωρίον ὀξύτητι καὶ βαρύτητι περατούμενον μελῳδητόν ἐστι;
But these things have perhaps had more than sufficient time spent on them. This, however, is evident, that they properly attribute to this God the number of five, saying that it sometimes of itself produces itself like fire, and other whiles the number of ten, like the world. But do we think that this number is not also concerned with music, which is of all things most acceptable to this God For the chiefest operation of harmony is, as one may say, about symphonies. Now that these are five and no more, reason convinces even him who will by his senses without reasoning make trial either on strings or pipe-holes. For all these accords take their original in proportions of number; and the proportion of the symphony diatessaron is sesquitertial, of diapente sesquialter, of diapason duple, of diapason with diapente triple, and of disdiapason quadruple. But as for that which, transcending all measures, the musicians add to these, naming it diapason with diatessaron, it is not fit we should receive it, gratifying the unreasonable pleasure of the ear against proportion, which is as the law. I may therefore let pass the five positions of the tetrachords, and also the five first,—whether they are to be called tones, tropes, or harmonies,—according to the changes of which by rising or falling, either to more or less, the rest are bases or trebles. And, whereas there are many or rather infinite intervals, are not five of them only used in music, to wit, diesis, hemitonion, tons, triemitonion, and ditonon? Nor is there in the voice any other space, either greater or less, that, being distinguished by base or treble, comes into melody.
πολλὰ δʼ ἄλλα τοιαῦτʼ ἔφην ἐγὼ παρελθὼν τὸν Πλάτωνα προσάξομαι λέγοντα κόσμον ἕνα, ὡς εἴπερ εἰσὶ παρὰ τοῦτον ἕτεροι καὶ μὴ μόνος οὗτος εἷς, πέντε τοὺς πάντας ὄντας καὶ μὴ πλείονας. μὴν ἀλλὰ κἂν εἷς οὗτος μονογενής, ὡς οἴεται καὶ Ἀριστοτέλης, τρόπον τινὰ καὶ τοῦτον ἐκ πέντε συγκείμενον κόσμων καὶ συνηρμοσμένον εἶναι· ὧν ὁ μὲν ἐστι γῆς ὁ δʼ ὕδατος, τρίτος δὲ πυρὸς καὶ τέταρτος ἀέρος· τὸν δὲ πέμπτον οὐρανὸν οἱ δὲ φῶς οἱ δʼ αἰθέρα καλοῦσιν, οἱ δʼ αὐτὸ τοῦτο πέμπτην οὐσίαν, τὸ κύκλῳ περιφέρεσθαι μόνῃ τῶν σωμάτων κατὰ φύσιν ἐστίν, οὐκ ἐξ ἀνάγκης οὔτʼ ἄλλως συμβεβηκὸς δὴ καὶ τὰ πέντε καὶ κάλλιστα καὶ τελεώτατα σχήματα τῶν ἐν τῇ φύσει κατανοήσας, πυραμίδα καὶ κύβον καὶ ὀκτάεδρον καὶ εἰκοσάεδρον καὶ δωδεκάεδρον, ἕκαστον οἰκείως ἑκάστῳ προσένειμεν.
Passing by many other such like things, said I, I will produce only Plato; who says, that there is but one world, but that if this were not alone, so that there were others besides it, they would be in all five and no more. For indeed, though there is but this one only world, as Aristotle is also of opinion, yet this world is in some sort composed and assembled of five, of which one indeed is of earth, another of water, the third of fire, the fourth of air, and the fifth, being heaven, some call light and others the sky; and some also name this same the fifth essence, which alone of all, bodies is naturally carried about in a circle, and not of necessity or otherwise by accident. Wherefore Plato, knowing that, of the figures which are in Nature, there are five most excellent and perfect,—to wit, the pyramid, the cube, the octahedron, the icosahedron, and the dodecahedron,—has fitly accommodated each of them to each of these worlds or bodies.
εἰσὶ δʼ οἳ καὶ τὰς τῶν αἰσθήσεων δυνάμεις ἰσαρίθμους οὔσας τοῖς πρώτοις ἐκείνοις συνοικειοῦσι, τὴν μὲν ἁφὴν ὁρῶντες ἀντίτυπον οὖσαν καὶ γεώδη, τὴν δὲ γεῦσιν ὑγρότητι τῶν γευστῶν τὰς ποιότητας προσιεμένην. ἀὴρ δὲ πληγεὶς ἐν ἀκοῇ γίγνεται φωνὴ καὶ ψόφος. δυεῖν δὲ τῶν λοιπῶν ὀσμὴ μέν, ἣν ἡ ὄσφρσις εἴληχεν, ἀναθυμίασις οὖσα καὶ γεννωμένη θερμότητι πυρῶδές ἐστιν· αἰθέρι δὲ καὶ φωτὶ διὰ συγγένειαν διαλαμπούσης τῆς ὄψεως γίγνεται κρᾶσις ἀμφοῖν ὁμοιοπαθὴς καὶ σύμπηξις. ἄλλην δʼ οὔτε τὸ ζῷον αἴσθησιν οὔτε ὁ κόσμος ἔχει φύσιν ἁπλῆν καὶ ἄμικτον ἀλλὰ θαυμαστή τις, ὡς ἔοικε, διανομὴ γέγονε τῶν πέντε πρὸς τὰ πέντε καὶ σύλληξις.
There are some also who apply the faculties of the senses, being equal in number, to these five first bodies, seeing the touch to be firm and earthly, and the taste to perceive the qualities of savors by moisture. Now the air being struck upon is a voice and sound to the ear; and as for the other two,—the scent, which the smell has obtained for its object, being an exhalation and engendered by heat, is fiery; and the sight, which shines by reason of its affinity to the sky and light, has from them a temperature and complexion equally mingled of both. Now neither has any animal any other sense, nor the world any other nature simple and unmixed; but there has been made, as appears, a certain wonderful distribution and congruity of five to five.
Cite This Text