Synagoge, Book 1 (2.9)

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
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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 οὕτως: ἐπεὶ γὰρ δοθεῖσά ἐστιν ἑκατέρα τῶν ΡΚ ΡΧ, δοθεῖσα ἔσται καὶ ἡ ΚΚ καὶ λόγος ἐστὶν δοθεὶς τῆς ΧΚ πρὸς τὴν ΚΨ (ἱ αὐτὸς γάρ ἐστιν τῷ τῆς ΦΚ πρὸς τὴν ΚΣ
λόγῳ δοθέντι)· δοθεῖσα ἄρα καὶ ἡ ΨΚ, ἀλλὰ καὶ ἡ ΨΣ δοθεῖσά ἐστιν, ἐπεὶ καὶ ὡς ἡ ΦΚ πρὸς τὴν ΚΣ, οὕτως ἡ ΦΧ πρὸς τὴν ΨΣ· καὶ δοθεῖσα δέδεικται ἡ ΦΧ· δοθεῖσα οὖν ἐστιν καὶ ἡ ΨΣ. ἦν δὲ καὶ ἡ ὑπὸ ΨΣΚ γωνία δοθεῖσα, ὥστε καὶ τὸ ΨΣΖ τρίγωνον ὀρθογώνιον τῷ εἴδει καὶ τῷ
μεγέθει δεδομένον ἔσται. 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 ἐδείχθη ἴση γὰρ τῇ ΖΚ δοθείσῃ, 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.