ΠΑΠΠΟΥ ΑΛΕΞΑΝΔΡΕΩΣ ΣΥΝΑΓΩΓΗΣ Γ.

103 Καὶ τούτοις ἀκολούθως δύο τῶν Ε∠ ∠Μ δοθεισῶν ληψόμεθα τὰς δύο μέσας ἀνάλογον ἐν τῇ συνεχεῖ ἀναλογίᾳ. ἐκκείσθωσαν γὰρ ταῖς Ε∠ ∠Ζ ∠Μ ἴσαι αἱ Ε∠ ∠Ζ ∠Μ. καὶ ἐπεὶ δέδεικται ὡς τὸ ἀπὸ τῆς Ε∠ πρὸς τὸ ἀπὸ τῆς ∠Ζ, οὕτως ἡ ∠Ζ πρὸς τὴν ∠Μ, δῆλον ὡς ἡ ∠Μ οὐκ ἔστι τρίτη ἀνάλογον τῶν Ε∠ ∠Ζ. Εἰ γὰρ δυνατόν, ἔστω ἡ ∠Μ τρίτη ἀνάλογον τῶν Ε∠ ∠Ζ. ἐπεὶ οὖν ἡ ∠Μ τρίτη ἀναλογόν ἐστι τῶν Ε∠ ∠Ζ, ἔστιν ὡς ἡ Ε∠ πρὸς τὴν ∠Ζ, οὕτως ἡ ∠Ζ πρὸς τὴν ∠Μ· καὶ ὡς ἄρα τὸ ἀπὸ τῆς Ε∠ πρὸς τὸ ἀπὸ τῆς ∠Ζ, οὕτως ἡ Ε∠ πρὸς ∠Μ, ἀλλʼ ἦν ὡς τὸ ἀπὸ τῆς Ε∠ πρὸς τὸ ἀπὸ τῆς ∠Ζ, οὕτως ἡ ∠Ζ πρὸς ∠Μ· διʼ ἴσου ἄρα ἐστὶν ὡς ἡ Ε∠ πρὸς τὴν ∠Μ, οὕτως ἡ ∠Ζ πρὸς τὴν ∠Μ· ἑκατέρα ἄρα τῶν Ε∠ ∠Ζ πρὸς τὴν ∠Μ τὸν αὐτὸν ἔχει λόγον· ἴση ἄρα ἐστιν ἡ Ε∠ τῇ ∠Ζ, ὅπερ ἀδύνατον (μείζων γὰρ ἡ Ε∠ τῆς ΔΖ)· οὐκ ἄρα ἡ ∠Μ τρίτη ἀνάλογόν ἐστιν τῶν Ε∠ ∠Ζ. Εἰλήφθω τῶν Ε∠ ∠Ζ τρίτη ἀνάλογον ἡ ΟΠ. ἐπεὶ οὖν τῶν Ε∠ ∠Ζ τρίτη ἀνάλογόν ἐστιν ΟΠ, ἔστιν ὡς ἡ Ε∠ πρὸς τὴν ∠Ζ, οὕτως ἡ ∠Ζ πρὸς τὴν ΟΠ. καὶ ὡς ἄρα τὸ ἀπὸ τῆς ∠Ε πρὸς τὸ ἀπὸ τῆς ∠Ζ, οὕτως ἡ Ε∠ πρὸς τὴν ΟΠ. ἀλλʼ ἦν ὡς τὸ ἀπὸ τῆς Ε∠ πρὸς τὸ ἀπὸ τῆς ∠Ζ, οὕτως ἡ ∠Ζ πρὸς τὴν ∠Μ· διʼ ἴσου ἄρα ἐστὶν ὡς ἡ Ε∠ πρὸς τὴν ΟΠ, οὕτως ἡ Ζ∠ πρὸς τὴν ∠Μ· καὶ ἐναλλὰξ ἄρα ἐστὶν ὡς ἡ Ε∠ πρὸς τὴν ∠Ζ, οὕτως ἡ ΟΠ πρὸς τὴν ΔΜ. ἀλλʼ ἦν ὡς ἡ Ε∠ πρὸς τὴν ∠Z, οὕτως ἡ ∠Ζ πρὸς τὴν ΟΠ· καὶ ὡς ἄρα ἔ ∠Ζ πρὸς τὴν ΟΠ, οὕτως ἡ ΟΠ πρὸς τὴν ∠Μ· αἱ Ε∠ ∠Ζ ΟΠ ∠Μ ἄρα τέσσαρες οὖσαι ἐν τῇ συνεχεῖ καὶ ἐφεξῆς ἀναλογίᾳ εἰσί· τῶν Ε∠ ∠Μ ἄρα δύο μέσαι ἀνάλογόν εἰσιν αἱ ∠Ζ ΟΠ.