# Synagoge 85-2

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

Canonical: https://interlinea.aeterna-institute.org/library/greek/pappus-alexandrinus/synagoge/85-2

85 νβ΄ (θ΄). Ἔστω ἡμικύκλιον τὸ ΑΒΓ, οὗ κέντρον τὸ Ε, καὶ τριπλῆ ἡ ΑΓ τῆς Γ∠, καὶ τῇ ΑΓ πρὸς ὀρθὰς ἡ ∠Β, καὶ ἐπεζεύχθω ἡ ΒΓ· ἡ ΑΓ ἄρα τῆς ΒΓ δυνάμει τριπλασίων ἐστίν (ὡς γὰρ ἡ ΑΓ πρὸς Γ∠, οὕτως τὸ ἀπὸ τῆς ΑΓ πρὸς τὸ ἀπὸ τῆς ΓΒ διὰ τὴν ὁμοιότητα τῶν ΑΒΓ Β∠Γ τριγώνων). τετμήσθω δʼ ἡ ΒΓ ἄκρον καὶ μέσον λόγον τῷ Θ, καὶ τὸ μεῖζον τμῆμα ἔστω ἡ ΒΘ, καὶ γεγενήσθω ἡ ΓΕ τῆς ΕΖ δυνάμει πενταπλῆ (δυνατὸν δὲ τοῦτο ἡ γὰρ ΕΓ τῆς Ε∠ τριπλῆ )· λέγω ὅτι λόγος ἐστὶν τῆς ΒΘ πρὸς τὴν ΓΖ δυνάμει ὃν ε΄ πρὸς γ΄. Κείσθω τῇ ΕΖ ἴση ἡ ΕΗ, καὶ ἡ ΖΗ ἄκρον καὶ μέσον λόγον τετμήσθω τῷ Κ, καὶ ἔστω μείζων ἡ ΖΚ. καὶ ἐπεὶ εὐθεῖα ἡ ΓΕ ἑαυτῆς τμήματος τῆς ΕΖ πενταπλάσιον δύναται, καὶ ἡ διπλῆ τῆς ΕΖ ἄκρον καὶ μέσον λόγον τέτμηται τῷ Κ, ἡ ΚΖ ἄρα ἴση ἐστὶν τῇ ΖΓ διὰ τὸ θεώρημα τοῦ ιγ΄ στοιχείων ὥστε καὶ ἡ ΓΗ. ἄκρον καὶ μέσον λόγον τέτμηται τῷ Ζ, καὶ τὸ μεῖζον αὐτῆς τμῆμά ἐστιν ἡ ΖΗ. ἀλλὰ διὰ τὸ προδεχθέν ἐστιν ὡς ἡ ΓΒ πρὸς τὴν ΒΘ, ἡ ΓΗ πρὸς τὴν ΗΖ, τουτέστιν ἡ ΖΗ πρὸς ΖΓ καὶ ἐναλλὰξ ἄρα ἐστὶν ὡς ἡ ΒΓ πρὸς τὴν ΖΗ, ἡ ΒΘ πρὸς τὴν ΓΖ. ἐπεὶ οὖν ἡ ΑΓ τῆς μὲν BΓ· δυνάμει τριπλῆ ἐστιν, τῆς δὲ ΗΖ πενταπλῆ, οἵων ἄρα δυνάμει ἡ ΑΓ ιε΄, τοιούτων ἡ μὲν ΒΓ ε΄, ἡ δὲ ΖΗ γ΄· ἡ ΒΓ ἄρα πρὸς ΖΗ λόγον ἔχει δυνάμει ὃν ε΄ πρὸς γʼ ὥστε καὶ ἡ ΒΘ πρὸς ΖΓ λόγον ἔχει δυνάμει ὃν ε΄ πρὸς 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/.