§pr## ΕΥΚΛΕΙΔΟΥ
Κατατομὴ κανόνος.
## EUCLID'S Division of the Monochord.
Εἰ ἡσυχία εἴη καὶ ἀκινησία, σιωπὴ ἂν εἴη· σιωπῆς δὲ οὔσης καὶ μηδενὸς κινουμένου οὐδὲν ἂν ἀκούοιτο·
If there were quiet and immobility, there would be silence; and if there were silence and nothing were moving, nothing would be heard.
εἰ ἄρα μέλλει τι ἀκουσθήσεσθαι, πληγὴν καὶ κίνησιν πρότερον δεῖ γενέσθαι.
If, therefore, anything is to be heard, there must first be impact and movement.
ὥστε ἐπειδὴ πάντες οἱ φθόγγοι γίνονται πληγῆς τινος γινομένης, πληγὴν δὲ ἀμήχανον γενέσθαι μὴ οὐχὶ κινήσεως πρότερον γενομένης, — τῶν δὲ κινήσεων αἱ μὲν πυκνότεραί εἰσιν, αἱ δὲ ἀραιότεραι, καὶ αἱ μὲν πυκνότεραι ὀξυτέρους ποιοῦσι τοὺς φθόγγους, αἱ δὲ ἀραιότεραι βαρυτέρους, — ἀναγκαῖον τοὺς μὲν ὀξυτέρους εἶναι, ἐπείπερ ἐκ πυκνοτέρων καὶ πλειόνων σύγκεινται κινήσεων, τοὺς δὲ βαρυτέρους ἐπείπερ ἐξ ἀραιοτέρων καὶ ἐλασσόνων σύγκεινται κινήσεων.
Since, then, all sounds occur when some impact occurs, and it is impossible for an impact to occur unless movement has occurred beforehand — and of movements, some are closer together and others are more spaced out, and the closer ones make the sounds higher, while the more spaced out ones make them lower — it is necessary that the higher sounds, since they are composed of closer and more numerous movements, are indeed such, and the lower sounds, since they are composed of more spaced out and fewer movements, are indeed such.
ὥστε τοὺς μὲν ὀξυτέρους τοῦ δέοντος ἀνιεμένους ἀφαιρέσει κινήσεως τυγχάνειν τοῦ δέοντος, τοὺς δὲ βαρυτέρους ἐπιτεινομένους προσθέσει κινήσεως τυγχάνειν τοῦ δέοντος.
Consequently, the higher sounds, being relaxed beyond what is proper by the subtraction of movement, reach what is proper, and the lower sounds, being tensed by the addition of movement, reach what is proper.
διόπερ ἐκ μορίων τοὺς φθόγγους συγκεῖσθαι φατέον, ἐπειδὴ προσθέσει καὶ ἀφαιρέσει τυγχάνουσι τοῦ δέοντος.
Therefore, we must say that sounds are composed of parts, since they reach what is proper by addition and subtraction.
πάντα δὲ τὰ ἐκ μορίων συγκείμενα ἀριθμοῦ λόγῳ λέγεται πρὸς ἄλληλα, ὥστε καὶ τοὺς φθόγγους ἀναγκαῖον ἐν ἀριθμοῦ λόγῳ λέγεσθαι πρὸς ἀλλήλους·
And all things composed of parts are spoken of in relation to one another in a ratio of number, so that it is also necessary that sounds be spoken of in relation to one another in a ratio of number.
τῶν δὲ ἀριθμῶν οἱ μὲν (p. 24 Meib) ἐν πολλαπλασίῳ λόγῳ λέγονται, οἱ δὲ ἐν ἐπιμορίῳ, οἱ δὲ ἐν ἐπιμερεῖ, ὥστε καὶ τοὺς φθόγγους ἀναγκαῖον ἐν τοῖς τοιούτοις λόγοις λέγεσθαι πρὸς ἀλλήλους.
Of numbers, some are spoken of in a multiple ratio, others in a superparticular ratio, and others in a superpartient ratio; so that it is also necessary that sounds be spoken of in relation to one another in such ratios.
τούτων δὲ οἱ μὲν πολλαπλάσιοι καὶ ἐπιμόριοι ἑνὶ ὀνόματι λέγονται πρὸς ἀλλήλους.
Of these, those that are multiple and superparticular are spoken of in relation to one another by a single name.
Γινώσκομεν δὲ καὶ τῶν φθόγγων τοὺς μὲν συμφώνους ὄντας, τοὺς δὲ διαφώνους, καὶ τοὺς μὲν συμφώνους μίαν κρᾶσιν τὴν ἐξ ἀμφοῖν ποιοῦντας, τοὺς δὲ διαφώνους οὔ.
We also know that of sounds, some are consonant and others are dissonant, and that the consonant sounds make a single mixture out of both, while the dissonant sounds do not.
τούτων οὕτως ἐχόντων εἰκὸς τοὺς συμφώνους φθόγγους, ἐπειδὴ μίαν τὴν ἐξ ἀμφοῖν ποιοῦνται κρᾶσιν τῆς φωνῆς, εἶναι τῶν ἐν ἑνὶ ὀνόματι πρὸς ἀλλήλους λεγομένων ἀριθμῶν, ἤτοι πολλαπλασίους ὄντας ἢ ἐπιμορίους.
This being so, it is reasonable that the consonant sounds, since they make a single mixture of the voice out of both, are among those numbers that are spoken of in relation to one another by a single name, being either multiple or superparticular.
§1Ἐὰν διάστημα πολλαπλάσιον δὶς συντεθὲν ποιῇ τι διάστημα, καὶ αὐτὸ πολλαπλάσιον ἔσται.
If a multiple interval, when combined twice, makes a certain interval, this itself will also be multiple.
Ἔστω διάστημα τὸ βγ, καὶ ἔστω πολλαπλάσιος ὁ β τοῦ γ, καὶ γεγενήσθω ὡς ὁ γ πρὸς τὸν β ὁ β πρὸς τὸν δ.
Let there be an interval βγ, and let β be a multiple of γ, and let it be made that as γ is to β, so β is to δ.
φημὶ δὴ τὸν δ τοῦ γ πολλαπλάσιον εἶναι.
I say indeed that δ is a multiple of γ.
ἐπεὶ γὰρ ὁ β τοῦ πολλαπλάσιός ἐστι, μετρεῖ ἄρα ὁ τὸν β.
For since β is a multiple of [γ], [γ] therefore measures β.
ἦν δὲ καὶ ὡς ὁ (p. 25) πρὸς (η) (δ) (β) τὸν β ὁ β πρὸς τὸν δ, ὥστε μετρεῖ ὁ γ καὶ τὸν δ.
But as γ was to β, so β was to δ, so that γ measures δ as well.
πολλαπλάσιος ἄρα ἐστὶν ὁ δ τοῦ γ.
Therefore, δ is a multiple of γ.
§2Ἐὰν διάστημα δὶς συντεθὲν τὸ ὅλον ποιῇ πολλαπλάσιον, καὶ αὐτὸ ἔσται πολλαπλάσιον. γ β δ σ ιβ κδ
Ἔστω διάστημα τὸ βγ καὶ γεγενήσθω ὡς ὁ πρὸς τὸν β οὕτως ὁ β πρὸς τὸν δ, καὶ ἔστω ὁ δ τοῦ γ πολλαπλάσιος.
If an interval, when combined twice, makes the whole multiple, this itself will also be multiple. γ β δ σ ιβ κδ Let there be an interval βγ, and let it be made that as [γ] is to β, so β is to δ, and let δ be a multiple of γ.
φημὶ καὶ τὸν β τοῦ εἶναι πολλαπλάσιον.
I say that β is also a multiple of [γ].
ἐπεὶ γὰρ ὁ δ τοῦ πολλαπλάσιός ἐστι, μετρεῖ ἄρα ὁ τὸν δ.
For since δ is a multiple of [γ], [γ] therefore measures δ.
ἐμάθομεν δὲ ὅτι, ἐὰν ὦσιν ἀριθμοὶ ἀνάλογον ὁποσοιοῦν, ὁ δὲ πρῶτος τὸν ἔσχατον μετρῇ, καὶ τοὺς μεταξὺ μετρήσει. μετρεῖ ἄρα ὁ γ τὸν β, πολλαπλάσιος ἄρα ὁ β τοῦ γ.
And we have learned that if there are any number of proportional numbers, and the first measures the last, it will also measure those in between. γ therefore measures β; β is therefore a multiple of γ.
§3Ἐπιμορίου διαστήματος οὐδεὶς μέσος, οὔτε εἷς οὔτε πλείους ἀνάλογον ἐμπεσεῖται ἀριθμός.
Between a superparticular interval, no mean proportional number, neither one nor more, will fall.
(p. 26. )
Ἔστω γὰρ ἐπιμόριον διάστημα τὸ βγ· ἐλάχιστοι δὲ ἐν τῷ αὐτῷ λόγῳ τοῖς βγ ἔστωσαν οἱ δζ θ.
For let there be a superparticular interval βγ, and let δζ and θ be the least numbers in the same ratio as βγ.
οὗτοι οὖν ὑπὸ μονάδος μόνης μετροῦνται κοινοῦ μέτρου.
These therefore are measured by a unit alone as a common measure.
ἄφελε ἶσον τῷ θ τὸν ηζ.
Subtract ηζ equal to θ.
καὶ ἐπεὶ ἐπιμόριός ἐστιν ὁ δζ τοῦ θ, ἡ ὑπεροχὴ ὁ δη κοινὸν μέτρον τοῦ τε δζ καὶ τοῦ θ ἐστι· μονὰς ἄρα ὁ δη·
And since δζ is superparticular of θ, the excess δη is a common measure of both δζ and θ. δη is therefore a unit.
οὐκ ἄρα ἐμπεσεῖται εἰς τοὺς δζ θ μέσος οὐδείς.
Therefore, no mean will fall between δζ and θ.
ἔσται γὰρ ὁ ἐμπίπτων τοῦ δζ ἐλάττων, τοῦ δὲ θ μείζων, ὥστε τὴν μονάδα διαιρεῖσθαι, ὅπερ ἀδύνατον.
For that which falls between would be less than δζ and greater than θ, so that the unit would be divided, which is impossible.
οὐκ ἄρα ἐμπεσεῖται εἰς τοὺς δζ θ τις.
Therefore, no number will fall between δζ and θ.
ὅσοι δὲ εἰς τοὺς ἐλαχίστους μέσοι ἀνάλογον ἐμπίπτουσι, τοσοῦτοι καὶ εἰς τοὺς τὸν αὐτὸν λόγον ἔχοντας ἀνάλογον μπεσοῦνται.
And as many mean proportionals as fall between the least numbers, so many will also fall between those having the same ratio.
οὐδεὶς δὲ εἰς τοὺς δζ θ ἐμπεσεῖται, οὐδὲ εἰς τοὺς β ἐμπεσεῖται.
Since none falls between δζ and θ, none will fall between β [and γ] either.