# Metrica 17

Hero of Alexandria, Metrica, ΗΡΩΝΟΣ ΜΕΤΡΙΚΩΝ Α. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/hero-of-alexandria/metrica/17

ιζ. Ἔστω δὲ πρότερον τρίγωνον ἰσόπλευρον, οὗ ἑκάστη ἐστὶ πλευρὰ μονάδων ι. καὶ ἔστω τὸ ΑΒΓ. ἤχθω κάθετος ἐπὶ τὴν ΓΒ ἡ Α∠. ἐπεὶ διπλῆ ἐστιν ἡ ΒΓ, τουτέστιν ἡ ΑΒ, τῆς Β∠, τετραπλάσιον ἄρα τὸ ἀπὸ ΑΒ τοῦ ἀπὸ Β∠. ὥστε τριπλάσιον τὸ ἀπὸ Α∠ τοῦ ἀπὸ ∠Β. τοῦ δὲ ἀπὸ ∠Β τετραπλάσιόν ἐστι τὸ ἀπὸ ΒΓ. ἐπίτριτον ἄρα ἔσται τὸ ἀπὸ ΒΓ τοῦ ἀπὸ Α∠. τὸ ἄρα ἀπὸ ΒΓ πρὸς τὸ ἀπὸ Α∠ λόγον ἔχει, ὃν δ πρὸς γ, καὶ πάντα ἐπὶ τὸν ἀπὸ ΒΙ, τουτέστιν τό τε ἀπὸ ΒΓ ἐφʼ ἑαυτὸ καὶ τὸ ἀπὸ Α∠ ἐπὶ τὸ ἀπὸ ΒΓ. ἡ ἄρα ἀπὸ τῆς ΒΓ | δυνάμεως πρὸς τὸ ἀπὸ τῆς ΒΓ ἐπὶ τὸ ἀπὸ τῆς Α∠ λόγον ἔχει, ὃν δ πρὸς γ, τουτέστιν ὃν ιϛ πρὸς ιβ. τὸ δὲ ἀπὸ τῆς ΒΓ ἐπὶ τὸ ἀπὸ τῆς Α∠ τὸ ὑπὸ Α∠ ΒΓ ἐστὶν ἐφʼ ἑαυτό, τουτέστι δύο τρίγωνα ἐφʼ ἑαυτά. ἡ ἄρα ἀπὸ ΒΓ δυναμοδύναμις πρὸς δύο τρίγωνα ἐφʼ ἑαυτὰ λόγον ἔχει, ὃν ιϛ πρὸς ιβ δύο δὲ τρίγωνα ἐφ᾿ ἑαυτὰ ἑνὸς τριγώνου ἐφʼ ἑαυτό ἐστιν τετραπλάσια. ἡ ἄρα ἀπὸ τῆς ΒΓ δυναμοδύναμις πρὸς ἓν τρίγωνον ἐφʼ ἑαυτὸ λόγον ἔχει, ὃν ιϛ πρὸς γ. καὶ ἔστι δοθεῖσα ἡ ἀπὸ ΒΓ δυναμοδύναμις, ἐπεὶ καὶ ἡ ΒΓ. δοθὲν ἄρα ἐστὶ τὸ ἐμβαδὸν τοῦ τριγώνου ἐφʼ ἑαυτό· ὥστε καὶ αὐτὸ τὸ τρίγωνον δοθέν ἐστιν. συντεθήσεται δὲ ἀκολούθως τῇ ἀναλύσει οὕτως. τὰ ι ἐφʼ ἑαυτά· γίγνεται ρ. ταῦτα ἐφʼ ἑαυτά· γίγνεται μ· τούτων λαβὲ γ. γίγνεται αωοε. τούτων πλευρὰν λαβέ· καὶ ἐπεὶ οὐκ ἔχει ῥητὴν πλευρὰν, εἰλήφθω ὡς ἐμάθομεν ἔγγιστα μετὰ διαφόρου. καὶ ἔσται τὸ ἐμβαδὸν μγ γ΄. Λῆμμα. Ἔστω τρίγωνον ὀρθογώνιον τὸ ΑΒΓ ὀρθὴν ἔχον τὴν πρὸς τῷ Γ, δύο δὲ πέμπτων ὀρθῆς τὴν πρὸς τῷ Α. δεῖξαι ὅτι τὸ ἀπὸ συναμφοτέρου τῆς ΒΑ ΑΓ πενταπλάσιόν ἐστι τοῦ ἀπὸ ΑΓ. ἐκβεβλήσθω ἡ ΑΓ ἐπὶ τὸ ∠, καὶ τῇ ΑΓ ἴση κείσθω ἡ Γ∠, καὶ ἐπεζεύχθω ἡ Β∠. ἴση ἄρα ἡ μὲν ΑΒ τῇ Β∠, ἡ δὲ ὑπὸ ΑΒΓ γωνία τῇ ὑπὸ ΓΒ∠. ἀλλὰ ἡ ὑπὸ ΓΒΑ γωνία τριῶν πέμπτων ἐστὶν ὀρθῆς διὰ τὸ τῆν ὑπὸ ΒΑΓ γωνίαν δύο πέμπτων εἶναι· ἡ ἄρα ὑπὸ ΑΒ∠ γωνία ἓξ πέμπτων ἐστὶν ὀρθῆς· πενταγώνου ἄρα ἐστὶ γωνία ἡ ὑπὸ ΑΒ∠. καὶ ἔστιν ἴση ἡ ΑΒ τῇ Β∠· | τῆς ἄρα Α∠ ἄκρον καὶ μέσον λόγον τεμνομένης τὸ μεῖζον τμῆμά ἐστιν ἡ ΑΒ. καὶ ἔστι τῆς Α∠ ἡμίσεια ἡ ΑΓ· τὸ ἄρα ἀπὸ συναμφοτέρου τῆς ΒΑ ΑΓ πενταπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΑΓ.

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Hero of Alexandria, Metrica, ed. Hermann Schöne. 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/tlg0559.tlg006.local001/.