ΠΑΠΠΟΥ ΑΛΕΞΑΝΔΡΕΩΣ ΣΥΝΑΓΩΓΗΣ Γ.

9 Ἐπεὶ τοίνυν δοθείς ἐστιν ὁ τῆς ΚΘ πρὸς ΘΡ λόγος καὶ δοθεῖσά ἐστιν ἡ ΘΚ (τοῦτο γὰρ ὑποκεῖσθαι δεῖ), δοθεῖσα ἄρα καὶ ἡ ΘΡ καὶ λοιπὴ ἡ ΡΚ. ἀλλὰ καὶ ἡ ΣΡ ἡμίσεια οὖσα τῆς ΡΚ· ἦν δὲ καὶ ἡ ΡΘ δοθεῖσα· καὶ ὅλη ἄρα ἡ ΘΣ δοθεῖσά ἐστιν, ὥστε καὶ ὁ λόγος τῆς ΚΘ πρὸς ΘΣ δοθείς ἐστιν. καὶ ἔστιν ὡς ἡ ΚΘ πρὸς τὴν ΘΣ, ἡ ΘΣ πρὸς τὴν ΘΤ, καὶ δοθεῖσα δέδεικται ἡ ΘΣ, δοθεῖσα ἄγα ἔσται καὶ ἡ ΤΘ. διὰ τὰ αὐτὰ δὴ καὶ ἡ ΘΦ δοθεῖσα ἔσται, ὥστε καὶ ἡ διαφορά τῶν ΘΡ ΘΦ εὐθειῶν δοθεῖσά ἐστιν. εὑρήσθω οὖν τὸ Φ μεταξὺ τῶν Θ Ρ, ὡς καὶ διὰ τῶν ἀριθμῶν ἐδείχθη. καὶ ἐπεὶ δέδοται ἡ ΦΡ διαφορὰ καὶ ἡ τὰ Ρ Χ ἐπιζευγνύουσα εὐθεῖα ἴση οὖσα τῇ ΞΚ, δοθέν ἄρα τὸ ΦΧΡ τρίγωνον ὀρθογώνιον τῷ εἴδει καὶ τῷ μεγέθει. δοθεῖσα ἄρα ἡ ὑπὸ ΡΦΧ γωνία, καὶ ἴση ἐστὶν τῇ ὑπὸ ΚΣΨ ἐκτὸς γωνίᾳ· ἐκβληθείσης ἄρα καὶ τῆς ΩΨ ἐπὶ τὸ Ζ, δοθὲν ἔσται τὸ ΣΖΨ τρίγωνον ὀρθογώνιον τῷ εἴδει. ἀλλὰ καὶ τῷ μεγέθει . δοθεῖσα ἄρα καὶ ἡ ΨΖ, παράλληλος οὖσα τῇ ΞΚ καὶ ἐπʼ εὐθείας τῇ ΨΩ· δοθεῖσα ἄρα καὶ ἡ ΩΛ ἴση οὖσα τῇ ΖΚ. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΘΚ τῇ ΜΛ, ἐλάσσων δέ ἡ ΩΛ τῆς ΣΚ ἴση γὰρ ἢ ΩΛ τῇ ΚΖ), καὶ ἔστιν ὡς μὲν ἡ ΚΘ πρὸς ΘΣ, οὕτως ἡ ΣΘ πρὸς τὴν ΘΤ καὶ ἡ ΤΘ πρὸς τὴν ΘΦ, ὡς δὲ ἡ ΛΜ πρὸς ΜΩ, ἡ ΜΩ πρὸς τὴν ΜΑ καὶ ἡ ΑΜ πρὸς τὴν ΜΒ, ἔσται ἄρα μείζων ἡ MΒ τῆς ΘΦ (καὶ τοῦτο γὰρ ἑξῆς δειχθήσεται)· καὶ λοιπὴ ἄρα ἡ ΒΛ τῆς ΦΚ ἐλάσσων. ἐπεὶ οὐν πάλιν δοθεῖσά ἐστιν ἡ ΩΛ ,δοθεῖσα δέ καὶ ἡ ΛΜ (ὅτι καὶ ἡ ΚΘ), καὶ λύγος ἄρα τῆς ΛΜ πρὸς ΜΩ δοθείς. καὶ ἔστιν ὡς ἡ ΛΜ πρὸς τὴν ΜΩ, καὶ ἡ ΩΜ πρὸς τὴν ΜΑ, καὶ δοθεῖσα ἡ ΩΜ· δοθεῖσα ἄρα καὶ ἡ ΜΑ.