# Synagoge 51

Pappus Alexandrinus, Synagoge, ΠΑΠΠΟΥ ΑΛΕΞΑΝΔΡΕΩΣ ΣΥΝΑΓΩΓΗΣ Γ.. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/pappus-alexandrinus/synagoge/51

51 κ΄. Ἡ τῇ ἁρμονικῇ ὑπεναντία μεσότης ἐκ τῆς ἀναλογίας οὕτως συνίσταται. ὅρων ἀνάλογον ὑποκειμένων τῶν Α Β Γ δύο μὲν τοῖς Α καὶ τρισὶ τοῖς Β καὶ ἑνὶ τῷ Γ ἴσος ἔστω ὁ ∠, δύο δὲ τοῖς Α καὶ δύο τοῖς Β καὶ ἑνὶ τῷ ὁ Ε, ἑνὶ δὲ τῷ Β καὶ ἑνὶ τῷ Γ ὁ Ζ· λέγω ὅτι οἱ ∠ Ε Ζ τὴν εἰρημένην ποιοῦσι μεσότητα. Πάλιν γὰρ ὁμοίως τοῖς προδεδειγμένοις ἔσται ὡς ὁ ∠ πρὸς τὸν Ζ, οὕτως δύο οἱ Α μετὰ τοῦ Β πρὸς τὸν Β. καὶ εἰσὶν δύο μὲν οἱ Α μετὰ τοῦ Β ὑπεροχὴ ᾗ ὑπερέχουσιν δύο οἱ Α καὶ δύο οἱ Β καὶ εἷς ὁ Γ ἑνὸς τοῦ Β καὶ ἑνὸς τοῦ Γ, τουτέστιν ἡ τῶν Ε Ζ ὑπεροχή. εἷς δὲ ὁ Β ὑπεροχὴ ᾗ ὑπιρέχουσι δύο οἱ A καὶ τρεῖς οἱ Β καὶ εἷς ὁ Γ δύο τῶν Α καὶ δύο τῶν Β καὶ ἑνὸς τοῦ Γ, τουτέστιν ἡ τῶν ∠ Ε ὑπεροχή· ὡς ἄρα ὁ Ζ πρὸς τὸν ∠, οὕτως ἡ τῶν ∠ Ε ὑπεροχὴ πρὸς τὴν τῶν Ε Ζ ὑπεροχήν, ὕπερ ἐστὶ κατὰ τὴν μεσότητα τὴν τῇ ἁρμονικῇ ὑπεναντίαν. δῆλον δʼ ὅτι καί, μονάδων ὑποτεθεισῶν τῶν Α Β Γ, εἴη ἂν ἐν ἐλαχίστοις ἀριθμοῖς ἡ μεσότης τοῖς Ϛ΄ ε΄ β΄.

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Pappus Alexandrinus, Synagoge, ed. Friedrich Hultsch. Text: The Greek Library, from the Perseus Digital Library and other open sources, CC BY-SA 4.0. Changes: apostrophes normalised and the text divided into reading sections by Interlinea.
License: CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Source: https://thegreeklibrary.pages.lisn.upsaclay.fr/works/tlg2032.tlg001.local001/.