# Synagoge 52

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

Canonical: https://interlinea.aeterna-institute.org/library/greek/pappus-alexandrinus/synagoge/52

52 Ἡ πέμπτη μεσότης ἐκ τῆς ἀναλογίας οὕτως συνίσταται. ἐκκείσθωσαν ἀνάλογον ὅροι τρεῖς οἱ Α Β Γ, καὶ ἑνὶ μὲν τῷ Α καὶ τρισὶ τοῖς Β καὶ ἑνὶ τῷ Γ ἴσος ἔστω ὁ ∠, ἑνὶ δὲ τῷ Α καὶ δύο τοῖς, B καὶ ἑνὶ τῷ Γ ὁ E, ἑνὶ δὲ τῷ B καὶ ἑνὶ τῷ Γ ὁ Ζ· λέγω ὅτι οἱ ∠ Ε Ζ κατὰ τὴν πέμπτην εἰσὶ μεσότητα. Ἐπεὶ γὰρ διὰ τὴν ἀναλογίαν ἐστὶν ὡς ὁ Α μετὰ τοῦ Β πρὸς τὸν Β, οὕτως ὁ Β μετὰ τοῦ Γ πρὸς τὸν Γ, ἔσται καὶ συναμφότερος ὁ ἡγούμενος ὁ Α Β μετὰ συναμφοτέρου τοῦ Β Γ πρὸς τὸν ἑπόμενον συναμφότερον τὸν Β Γ, τουτέστιν ὡς ὁ Ε πρὸς τὸν Ζ, οὕτως συναμφότερος ὁ Α Β πρὸς τὸν Β. καὶ ἔστι συναμφότερος μὲν ὁ Α Β ἡ ὑπεροχὴ ᾗ ὑπερέχει εἷς ὁ Α καὶ δύο οἱ Β καὶ εἷς ὁ Γ ἑνὸς τοῦ Β καὶ ἑνὸς τοῦ Γ, τουτέστιν ἡ τῶν Ε Ζ ὑπεροχή, ὁ δὲ Β ᾗ ὑπερέχει εἷς ὁ Α καὶ τρεῖς οἱ Β καὶ εἷς ὁ Γ ἑνὸς τοῦ Α καὶ δύο τῶν Β καὶ ἑνὸς τοῦ Γ, τουτέστιν ἡ τῶν ∠ Ε ὑπεροχή· ὡς ἄρα ὁ Ζ πρὸς τὸν E, οὕτως ἡ τῶν ∠ E ὑπεροχὴ πρὸς τὴν τῶν Ε Ζ ὑπεροχήν, ὅπερ τῇ πέμπτῃ συμβέβηκεν μεσότητι. καὶ λέγοιτʼ ἂν ἐν ἐλαχίστοις ἀριθμοῖς τοῖς ε΄ δ΄ β΄, μονάδων ὑποτεθεισῶν τῶν Α Β Γ. .

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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/.