Ancient Greek · ~6 min
ιθ΄.
Euclid, Data ιθ΄.
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Ἐὰν ᾖ τρία μεγέθη, καὶ τὸ μὲν πρῶτον τοῦ δευ- τέρου δοθέντι μεῖζον ᾖ ἢ ἐν λόγῳ, ᾖ δὲ καὶ τὸ δεύ- τερον τοῦ τρίτου δοθέντι μεῖζον ἢ ἐν λόγῳ, καὶ τὸ πρῶτον τοῦ τρίτου δοθέντι μεῖζον ἔσται ἢ ἐν λόγῳ. ἔστω τρία μεγέθη τὰ ΑΒ, ΓΔ, Ε, καὶ τὸ μὲν ΑΒ τοῦ ΓΔ δοθέντι μεῖζον ἔστω ἢ ἐν λόγῳ, τὸ δὲ ΓΔ τοῦ Ε δοθέντι μεῖζον ἔστω ἢ ἐν λόγῳ. λέγω, ὅτι καὶ τὸ ΑΒ τοῦ Ε δοθέντι μεῖζόν ἐστιν ἢ ἐν λόγῳ. ἐπεὶ γὰρ τὸ ΓΔ τοῦ Ε δοθέντι μεῖζόν ἐστιν ἢ ἐν λόγῳ, ἀφῃρήσθω τὸ δοθὲν μέγεθος τὸ ΓΖ· λοιποῦ ἄρα τὸῦ ΖΔ πρὸς τὸ Ε λόγος ἐστὶ δοθείς. πάλιν; ἐπεὶ τὸ ΑΒ τοῦ ΓΔ δοθέντι μεῖζόν ἐστιν ἢ ἐν λόγῳ, ἀφῃρήσθω τὸ δοθὲν μέγεθος τὸ ΑΗ· λοιποῦ ἄρα τοῦ ΗΒ πρὸς τὸ ΓΔ λόγος ἐστὶ δοθείς. ὁ αὐτὸς αὐτῷ γεγονέτω τοῦ ΗΘ πρὸς τὸ ΓΖ· λόγος ἄρα καὶ τοῦ ΗΘ πρὸς τὸ ΓΖ δοθείς. δοθὲν δὲ τὸ ΓΖ· δοθὲν ἄρα καὶ τὸ ΗΘ. ἔστι δὲ καὶ τὸ ΗΑ δοθέν· καὶ ὅλον ἄρα τὸ ΘΑ δοθέν ἐστιν. καὶ ἐπεὶ ὡς τὸ ΗΒ πρὸς τὸ ΓΔ, οὕτως τὸ ΗΘ πρὸς τὸ ΓΖ, καὶ λοιποῦ τοῦ ΘΒ πρὸς λοιπὸν τὸ ΖΔ λόγος ἐστὶ δοθείς. τοῦ δὲ ΖΔ πρὸς τὸ Ε λόγος ἐστὶ δοθείς· καὶ τοῦ ΘΒ ἄρα πρὸς τὸ Ε λόγος ἐστὶ δοθείς. καὶ δοθὲν τὸ ΘΑ· τὸ ΒΑ ἄρα τοῦ Ε δοθέντι μεῖζόν ἐστιν ἢ ἐν λόγῳ.