# Synagoge 78-2

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

Canonical: https://interlinea.aeterna-institute.org/library/greek/pappus-alexandrinus/synagoge/78-2

78 μθ΄ (Ϛ΄). Ἐὰν εὐθεῖα γραμμὴ ἄκρον καὶ μέσον λόγον τμηθῇ, τὸ ἀπὸ τῆς ὅλης πρὸς τὸ πεντάκις ἀπὸ τῆς ἐλάσσονος μείζονα λόγον ἔχει ἤπερ δ΄ πρὸς γ΄. Εὐθεῖα γὰρ ἡ ΑΒ ἄκρον καὶ μέσον λόγον τετμήσθω κατὰ τὸ Γ, καὶ ἔστω ἔλασσον αὐτῆς τμῆμα τὸ ΓΒ· λέγω ὅτι τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ πεντάκις ἀπὸ ΓΒ μείζονα λόγον ἔχει ἤπερ δ΄ πρὸς γ΄. Κείσθω τῇ ΓΒ ἴση Γ∠· γίνεται ἄρα, διὰ τὸ ἄκρον καὶ μέσον λόγον τετμῆσθαι τὴν ΑΒ εὐθεῖαν, τὸ ἀπὸ ΑΒ καὶ τὸ ἀπὸ ΓΒ ἴσα τῷ τρὶς ἀπὸ ΑΓ, ὡς ἔστιν στοιχείοις δʼ τοῦ τρισκαιδεκάτου θεωρήματι, τουτέστιν τῳ τρὶς ὑπὸ ΑΒΓ. ἀλλὰ τὸ τρὶς ὑπὸ ΑΒΓ ἴσον ἐστὶν τῷ τρὶς ὑπὸ ΑΓΒ καὶ τῷ τρὶς ἀπὸ ΓΒ (ἐπεὶ καὶ τὸ ἅπαξ ὑπὸ ΑΒΓ ἴσον ἐστὶν τῷ ὑπὸ ΑΓΒ καὶ τῷ ἀπὸ ΒΓ, ὥς ἔστι στοιχείοις τὸ γ΄ θεώρημα τοῦ β΄)· τὰ ἄρα ἀπὸ ΑΒΓ ἴσα ἐστὶν τῷ τρὶς ὑπὸ ΑΓΒ καὶ τῷ τρὶς ἀπὸ ΓΒ· κοινοῦ ἀφαιρεθέντος τοῦ ἀπὸ ΒΓ λοιπὸν ἄρα τὸ ἀπὸ ΑΒ ἴσον ἐστὶ τῷ τρὶς ὑπὸ ΑΓΒ καὶ τῷ δὶς ἀπὸ ΒΓ, τουτέστιν τῷ τρὶς ὑπὸ ΑΓ∠ καὶ τῷ δὶς ἀπὸ Γ∠. ἀλλὰ τὸ τρὶς ὑπὸ ΑΓ∠ ἴσον ἐστὶν τῷ τρὶς ἀπὸ Α∠Γ καὶ τῷ τρὶς ἀπὸ Γ∠ (ἐπεὶ καὶ τὸ ἅπαξ ὑπὸ ΑΓ∠ ἴσον τῷ ὑπὸ Α∠Γ καὶ τῷ ἀπὸ ∠Γ διὰ τὸ αὐτὸ γ΄ τοῦ β’ στοιχείων θεώρημα)· τὸ ἄρα ἀπὸ ΑΒ ἴσον ἐστὶν τῷ τρὶς ὑπὸ Α∠Γ καὶ τῷ πεντάκις ἀπὸ Γ∠, τουτέστιν τῷ πεντάκις ἀπὸ ΓΒ.

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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/.