Περὶ τῆς Νικομάχου ἀριθμητικῆς εἰσαγωγῆς
(156)
urn:cts:greekLit:tlg2023.tlg004.1st1K-grc1:156
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ἁρμονικὴν
ὡς ἀεὶ ἀκρότητα· τὰ γὰρ ἑκατέρας ἰδιώματα ἐφ’ ἑαυτῆς ἀναμίξει. ἦν μὲν γὰρ τῆς ἁρμονικῆς ἴδιον τὸ τὸν μέσον ὅρον ὑπερέχειν τε καὶ ὑπερέχεσθαι μέρει αὐτῶν τῶν ἄκρων ποιότητι τῷ αὐτῷ, εἰ καὶ μὴ ποσότητι, οὐδέποτε δὲ τοῦ μέσου· τῆς δὲ ἀριθμητικῆς ἀνάπαλιν οὐκέτι τῶν ἄκρων, ἀλλὰ τοῦ μέσου καὶ ποσότητι τῷ αὐτῷ. ἐπὶ δὲ τῆς γεωμετρικῆς ὁ μέσος ὅρος ᾧ ὑπερέχει καὶ ὑπερέχεται μέρει, ἐκεῖνο οὔτε μόνων τῶν ἄκρων ἐστὶν οὔτε μόνου τοῦ μέσου, ἀλλὰ καὶ μέσου καὶ ἄκρων· τοῦ μὲν γὰρ ἑτέρου τῶν ἄκρων ὑπερέξει αὑτοῦ μέρει, ὑπερσχεθήσεται δὲ ὑπὸ θατέρου τοῦ ἐκείνου μέρει· τὸ δὲ αὐτὸ ἔσται ποιότητι, εἰ καὶ μὴ ποσότητι τὸ μέρος, ὡς ἐπὶ τῆς ἁρμονικῆς. πολλάκις δὲ καὶ πλείοσι μέρεσιν ὑπερέξει τε καὶ ὑπερσχεθήσεται, ἐπὶ ποιότητι πάλιν τοῖς αὐτοῖς, ὥστε καὶ κοινόν τι ἕξει πρὸς τὴν ἁρμονικὴν τὸ μόνον ποιότητι ταὐτὸν εἶναι τὸ μέρος, μηκέτι δὲ ποσότητι, καὶ κατὰ τοῦτο οὐκ ἔσται αὐτῇ ὑπεναντία ἡ ἁρμονική. πάλιν ἐν μὲν τῇ ἀριθμητικῇ κατὰ μὲν τοὺς μείζονας ὅρους οἱ ἐλάττονες λόγοι ἐφαίνοντο, κατὰ δὲ τοὺς ἐλάττονας
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