# Synagoge 3.68

Pappus Alexandrinus, Synagoge, Book 3. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/pappus-alexandrinus/synagoge/3.68

68 μδ΄ Ἑτέρως δὲ τὴν ἀνάλυσιν τοῦ τρίχα τεμεῖν τὴν γωνίαν ἢ περιφέρειαν ἐξέθεντό τινες ἄνευ τῆς νεύσεως. ἔστω δὲ ἐπὶ περιφερείας ὁ λόγος· οὐδὲν γὰρ διαφέρει γωνίαν ἢ περιφέρειαν τεμεῖν. Γεγονέτω δή, καὶ τῆς ΑΒΓ περιφερείας τρίτον ἀπειλήφθω μέρος ἡ ΒΓ, καὶ ἐπεζεύχθωσαν αἱ ΑΒ ΒΓ ΑΓ· διπλασίων ἄρα ἡ ὑπὸ ΑΓΒ τῆς ὑπὸ ΒΑΓ. τετμήσθω δίχα ἡ ὑπὸ ΑΓβ τῇ Γ∠, καὶ κάθετοι αἱ ∠Ε ΖΒ· ἴση ἄρα ἡ Α∠ τῇ ∠Γ, ὥστε καὶ ἡ ΑΕ τῇ ΕΓ δοθὲν ἄρα τὸ Ε. ἐπεὶ οὖν ἐστιν ὡς ἡ ΑΓ πρὸς ΓΒ, οὕτως ἡ Α∠ πρὸς ∠Β, τουτέστιν ἡ ΑΕ πρὸς ΕΖ, καὶ ἐναλλὰξ ἄρα ἐστὶν ὡς ἡ ΓΑ πρὸς ΑΕ, ἡ ΒΓ πρὸς ΕΖ. διπλῆ δὲ ἡ ΓΑ τῇ ΑΕ· διπλῆ ἄρα καὶ ἡ ΒΓ τῆς ΕΖ· τετραπλάσιον ἔρα τὸ ἀπὸ ΒΓ, τουτέστιν τὰ ἀπὸ τῶν ΒΖΓ, τοῦ ἀπὸ τῆς ΕΖ. ἐπεὶ οὖν δύο δοθέντα ἐστὶν τὰ Ε Γ, καὶ ὀρθὴ ἡ ΒΖ, καὶ λόγος ἐστὶν τοῦ ἀπὸ ΕΖ πρὸς τὰ ἀπὸ τῶν ΒΖΓ, τὸ Β ἄρα πρὸς ὑπερβολῇ. ἀλλὰ καὶ πρὸς θέσει περιφερείᾳ· δοθὲν ἄρα τὸ Β. καὶ ἡ σύνθεσις φανερά.

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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/.