πα΄.

Ἐὰν τρεῖς εὐθεῖαι ἀνάλογον οὖσαι τρισὶν εὐθείαις ἀνάλογον οὔσαις τὰς ἄκρας ἐν δεδομένῳ λόγῳ ἔχωσιν, καὶ τὰς μέσας ἐν δεδομένῳ λόγῳ ἕξουσιν· καὶ ἐὰν ἡ ἄκρα πρὸς τὴν ἄκραν λόγον ἔχῃ δεδομένον, καὶ ἡ μέση πρὸς τὴν μέσην καὶ ἡ λοιπὴ ἄκρα πρὸς τὴν λοιπὴν ἄκραν λόγον ἕξει δεδομένον. τρεῖς γὰρ εὐθεῖαι ἀνάλογον οὖσαι αἱ Α, Β, Γ τρι- σὶν εὐθείαις ἀνάλογον οὔσαις ταῖς Δ, Ε, Ζ τὰς ἄκρας ἐν δεδομένῳ λόγῳ ἐχέτωσαν, καὶ ἔστω λόγος τῆς μὲν Α πρὸς τὴν Δ δοθείς, τῆς δὲ Γ πρὸς τὴν Ζ λόγος δο- θείς· λέγω, ὅτι καὶ τῆς Β πρὸς τὴν Ε λόγος ἐστὶ δοθείς. ἐπεὶ γὰρ λόγος ἐστὶ τῆς μὲν Α πρὸς τὴν Δ δοθείς, τῆς δὲ Γ πρὸς τὴν Ζ δοθείς, λόγος ἄρα τοῦ ὑπὸ τῶν Α, Γ πρὸς τὸ ὑπὸ τῶν Δ, Ζ δοθείς. ἀλλὰ τῷ μὲν ὑπὸ τῶν Α, Γ ἴσον ἐστὶ τὸ ἀπὸ τῆς Β, τῷ δὲ ὑπὸ τῶν Δ, Ζ ἴσον ἐστὶ τὸ ἀπὸ τῆς Ε. λόγος ἄρα ἐστὶ τοῦ ἀπὸ τῆς Β πρὸς τὸ ἀπὸ τῆς Ε δοθείς· ὥστε καὶ τῆς Β πρὸς τὴν Ε λόγος ἐστὶ δοθείς. ἔστω δὴ πάλιν τῆς μὲν Α πρὸς τὴν Δ λόγος δοθείς, τῆς δὲ Β πρὸς τὴν Ε λόγος δοθείς· λέγω, ὅτι καὶ τῆς Γ πρὸς τὴν Ζ λόγος ἐστὶ δοθείς. ἐπεὶ λόγος ἐστὶ τῆς μὲν Α πρὸς τὴν Δ, τῆς δὲ Β πρὸς τὴν Ε δοθείς, λόγος ἐστὶ καὶ τοῦ ἀπὸ τῆς Β πρὸς τὸ ἀπὸ τῆς Ε δοθείς. ἀλλὰ τῷ μὲν ἀπὸ τῆς Β ἴσον τὸ ὑπὸ τῶν Α, Γ, τῷ δὲ ἀπὸ τῆς Ε ἴσον ἐστὶ τὸ ὑπὸ τῶν Δ, Ζ· λόγος ἄρα ἐστὶ τοῦ ὑπὸ τῶν Α, Γ πρὸς τὸ ὑπὸ τῶν Δ, Ζ δοθείς. καὶ μιᾶς πλευρᾶς τῆς Α πρὸς μίαν πλευρὰν τὴν Δ λόγος ἐστὶ δοθείς· καὶ λοιπῆς ἄρα τῆς Γ πρὸς λοιπὴν τὴν Ζ λό ος ἐστὶ δοθείς.