Ancient Greek · ~3 min
1. Ad prop. XIX. Ἄλλως τὸ ιθ΄.
Euclid, Data (demonstrationes alterae) 1. Ad prop. XIX. Ἄλλως τὸ ιθ΄.
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Δυνατὸν δέ ἐστί καὶ οὕτως. ἔστω τρία μεγέθη τὰ ΑΒ, Γ, Δ, καὶ τὸ μὲν ΑΒ τοῦ δοθέντι μεῖζον ἔστω- ἢ ἐν λόγῳ, τὸ δὲ Γ τοῦ Δ δοθέντι μεῖζον ἢ ἐν λόγῳ· λέγω, ὅτι καὶ τὸ ΑΒ τοῦ Δ δοθέντι μεῖζόν ἐστιν ἢ ἐν λόγῳ. ἐπεὶ γὰρ τὸ ΑΒ τοῦ Γ δοθέντι μεῖζόν ἐστιν ἢ ἐν λόγῳ, ἀφῃρήσθω τὸ δοθὲν μέγεθος τὸ ΑΕ· λοιποῦ ἄρα τοῦ ΕΒ πρὸς τὸ Γ λόγος ἐστὶ δοθέίς· τὸ δὲ Γ τοῦ Δ δοθέντι μεῖζόν ἐστιν ἢ ἐν λόγῳ· καὶ τὸ ΕΒ ἄρα τοῦ Δ δοθέντι μεῖζόν ἐστιν ἢ ἐν λόγῳ. ἀφῃρήσθω οὖν τὸ δοθὲν μέγεθος τὸ ΕΖ λοιποῦ ἄρα τοῦ ΖΒ πρὸς τὸ Δ λόγος ἐστὶ δοθείς. καί ἐστι δοθὲν τὸ ΑΖ τὸ ΑΒ ἄρα τοῦ Δ δοθέντι μεῖζόν ἐστιν ἢ ἐν λόγῳ.