κδ΄.

Ἐὰν τρεῖς εὐθεῖαι ἀνάλογον ὦσιν, ἡ δὲ πρώτη πρὸς τὴν τρίτην λόγον ἔχῃ δεδομένον, καὶ πρὸς τὴν δευ- τέραν λόγον ἕξει δεδομένον. ἔστωσαν τρεῖς εὐθεῖαι ἀνάλογον αἱ Α, Β, Γ, ὡς ἡ Α πρὸς τὴν Β, οὕτως ἡ Β πρὸς τὴν Γ, ἡ δὲ Α πρὸς τὴν Γ λόγον ἐχέτω δεδομένον· λέγω, ὅτι καὶ πρὸς τὴν Β λόγον ἕξει δεδομένον. ἐκκείσθω γὰρ δοθεῖσα ἡ Δ. καὶ ἐπεὶ λόγος ἐστὶ τῆς Α πρὸς τὴν Γ δοθείς, ὁ αὐτὸς αὐτῷ γεγονέτω ὁ τῆς Δ πρὸς τὴν Ζ· λόγος ἄρα καὶ τῆς Δ πρὸς τὴν Ζ δοθείς· δοθεῖσα δὲ ἡ Δ· δοθεῖσα ἄρα καὶ ἡ Ζ. εἰλήφθω τῶν Δ, Ζ μέση ἀνάλογον ἡ Ε· τὸ ἄρα ὑπὸ τῶν Δ, Ζ ἴσον ἐστὶ τῷ ἀπὸ τῆς Ε. δοθὲν δὲ τὸ ῦπὸ τῶν Δ, Ζ· δοθεῖσα γὰρ ἐκατέρα αὐτῶν· δοθὲν ἄρα καὶ τὸ ἀπὸ Ε· δοθεῖσα ἄρα ἐστὶν ἡ Ε. ἔστι δὲ καὶ ἡ Δ δοθεῖσα· λόγος ἄρα ἐστὶ τῆς Δ πρὸς τὴν E δοθείς. καὶ ἐπεί ἐστιν ὡς ἡ Α πρὸς τὴν Γ, οὕτως ἡ Δ πρὸς τὴν Ζ, ἀλλʼ ὡς μὲν ἡ Α πρὸς τὴν Γ, οὕτως τὸ ἀπὸ τῆς Α πρὸς τὸ ὑπὸ τῶν Α, Γ, ὡς δὲ ἡ Δ πρὸς τὴν Ζ, οὕτως τὸ ἀπὸ τῆς Δ πρὸς τὸ ὑπὸ τῶν Δ, Ζ, ὡς ἄρα τὸ ἀπὸ τῆς Α πρὸς τὸ ὑπὸ τῶν Α, Γ, οὕτως τὸ ἀπὸ τῆς Δ πρὸς τὸ ὑπὸ τῶν Δ, Ζ. ἀλλὰ τῷ μὲν ὑπὸ τῶν Α, Γ ἴσον ἐστὶ τὸ ἀπὸ τῆς Β· αἱ γὰρ Α, Β, Γ ἀνάλογσν εἰσιν· τῷ δὲ ὑπὸ τῶν Δ, Ζ ἴσον ἐστὶ τὸ ἀπὸ τῆς Ε· ὡς ἄρα τὸ ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς Β, οὕτως τὸ ἀπὸ τῆς Δ πρὸς τὸ ἀπὸ τῆς Ε· καὶ ὡς ἄρα ἡ Α πρὸς τὴν Β, οὕτως ἡ Δ πρὸς τὴν Ε. λόγος δὲ τῆς Δ πρὸς τὴν Ε δοθείς· λόγος ἄρα καὶ τῆς Α πρὸς τὴν Β δοθείς.