Fragmenta

κγ΄. Ἔστω τὸ ὑπὸ ΑΒΓ ἴσον τῷ ἀπὸ Β∠ τετραγώνῳ· ὅτι γίνεται γ, τὸ μὲν ὑπὸ συναμφοτέρου τῆς Α∠Γ καὶ τῆς Β∠ ἴσον τῷ ὑπὸ Α∠, ∠Γ, τὸ δὲ ὑπὸ συναμφοτέρου τῆς Α∠Γ καὶ τῆς ΒΓ ἴσον τῷ ἀπὸ ∠Γ τετραγώνῳ, τὸ δὲ ὑπὸ συναμφοτέρου τῆς Α∠Γ καὶ τῆς ΒΑ ἴσον τῷ ἀπὸ Α∠ τετραγώνῳ.

ἐπεὶ γὰρ τὸ ὑπὸ ΑΒΓ ἴσον ἐστὶν τῷ ἀπὸ Β∠, ἀνάλογον καὶ ὅλη πρὸς ὅλην καὶ ἀνάπαλιν καὶ συνθέντι· ἔστιν ἄρα, ὡς συναμφότερος ἡ Γ∠, ∠Α πρὸς τὴν ∠Α, οὕτως ἡ Γ∠ πρὸς τὴν ∠Β· τὸ ἄρα ὑπὸ συναμφοτέρου τῆς Α∠, ∠Γ καὶ τῆς Β∠ ἴσον ἐστὶ τῷ ὑπὸ τῶν Α∠Γ. πάλιν, ἐπεὶ ὅλη ἡ Α∠ πρὸς ὅλην τὴν ∠Γ ἐστιν, ὡς ἡ ∠Β πρὸς τὴν ΒΓ, συνθέντι ἐστίν, ὡς συναμφότερος ἡ Α∠Γ πρὸς τὴν ∠Γ, οὕτως ἡ ∠Γ πρὸς τὴν ΓΒ· τὸ ἄρα ὑπὸ συναμφοτέρου τῆς Α∠Γ καὶ τῆς ΓΒ ἴσον ἐστὶν τῷ ἀπὸ ∠Γ. πάλιν, ἐπεὶ ὅλη ἡ Α∠ πρὸς ὅλην τὴν ∠Γ ἐστιν, ὡς ἡ ΑΒ πρὸς τὴν Β∠, ἀνάπαλιν καὶ συνθέντι ἐστίν, ὡς συναμφότερος ἡ Γ∠Α πρὸς τὴν ∠Α, οὕτως ἡ ∠Α πρὸς τὴν ΑΒ· τὸ ἄρα ὑπὸ συναμφοτέρου τῆς Α∠Γ καὶ τῆς ΑΒ ἴσον ἐστὶν τῷ ἀπὸ Α∠ τετραγώνῳ.

κδ΄. Εὐθεῖα ἡ ΑΒ καὶ δύο σημεῖα τὰ Γ, Δ, καὶ ἔστω τὸ ἀπὸ Γ∠ τετράγωνον ἴσον τῷ δὲς ὑπὸ ΑΓ Β∠· ὅτι καὶ τὸ ἀπὸ ΑΒ τετράγωνον ἴσον ἐστὶν τοῖς ἀπὸ τῶν Α∠, ΓΒ τετραγώνοις.

ἐπεὶ γὰρ τὸ ἀπὸ Γ∠ ἴσον ἐστὶν τῷ θὶς ὑπὸ ΑΓ, ∠Β, [*](6. γ] τρία Hultsch. 26. ΑΓ, Β∠ ὅτι] ΑΓΒ διότι cod., ΑΓ ∠Β· ὅτι Hultsch cum Commandino.)

κε΄. Ἔστω τὸ ὑπὸ τῶν ΑΒΓ ἴσον τῷ ἀπὸ τῆς Β∠· ὅτι γίνεται γ, τὸ μὲν ὑπὸ τῆς τῶν Α∠, ∠Γ ὑπεροχῆς καὶ τῆς Β∠ ἴσον τῷ ὑπὸ Α∠Γ τὸ δὲ ὑπὸ τῆς τῶν Α∠Γ ὑπεροχῆς καὶ τῆς ΒΓ ἴσον τῷ ἀπὸ τῆς ∠Γ τετραγώνῳ, τὸ δὲ ὑπὸ τῆς τῶν Α∠, ∠Γ ὑπεροχῆς καὶ τῆς ΒΑ ἴσον τῷ ἀπὸ τῆς Α∠ τετραγώνῳ.

ἐπεὶ γάρ ἐστιν, ὡς ἡ ΑΒ πρὸς τὴν Β∠, οὕτως ἡ Β∠ πρὸς τὴν ΒΓ, λοιπὴ πρὸς λοιπὴν καὶ διελόντι· ἔστιν οὖν, ὡς ἡ τῶν Α∠, ∠Γ ὑπεροχὴ πρὸς τὴν ∠Γ, οὕτως ἡ Α∠ πρὸς τὴν ∠Β τὸ ἄρα ὑπὸ τῆς τῶν Α∠, ∠ ὑπεροχῆς καὶ τῆς ∠Β ἴσον ἐστὶν τῷ ὑπὸ τῶν Α∠, ∠Γ. πάλιν, ἐπεὶ λοιπὴ ἡ Α∠ πρὸς λοιπὴν τὴν ∠Γ ἐστιν, ὡς ἡ ∠Β πρὸς τὴν ΒΓ, διελόντι ἐστίν, ὡς ἡ τῶν Α∠Γ ὑπεροχὴ πρὸς τὴν ∠Γ, οὕτως ἡ ∠Γ πρὸς τὴν ΓΒ· τὸ ἄρα ὑπὸ τῆς τῶν Α∠, ∠Γ ὑπεροχῆς καὶ τῆς ΒΓ ἴσον ἐστὶν τῷ ἀπὸ τῆς ΔΓ τετραγώνῳ. πάλιν, ἐπεί ἐστιν, ὡς ἡ Α∠ πρὸς τὴν ∠Γ οὕτως ἡ ΑΒ πρὸς τὴν Β∠, ἀνάπαλιν καὶ διελόντι ἐστίν, ὡς ἡ τῶν Απαλιν, ∠Γ ὑπεροχὴ πρὸς τὴν ∠Α, οὕτως ἡ ∠Α πρὸς τὴν ΑΒ τὸ ἄρα ὑπὸ τῆς τῶν Α∠, ∠Γ ὑπεροχῆς καὶ τῆς ΑΒ ἴσον ἐστὶν τῷ ἀπὸ τῆς Α∠ τετραγώνῳ.

κϚ΄. Ἔστω, ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ ἀπὸ Α∠ πρὸς τὸ ἀπὸ ∠ ὅτι τὸ ὑπὸ τῶν ΑΒΓ ἴσον ἐστὶν τῷ ἀπὸ τῆς Β∠ τετραγώνῳ.

κείσθω τῇ Γ∠ ἴση ἡ ∠Ε· τὸ ἄρα ὑπὸ ΕΑΓ μετὰ τοῦ ἀπὸ Γ∠, τουτέστιν τοῦ ὑπὸ Γ∠Ε, ἴσον τῷ ἀπὸ Α∠. ἐπεὶ οὗν ἐστιν, ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ ἀπὸ Α∠ πρὸς τὸ ἀπὸ ∠Γ, διελόντι ἐστίν, ὡς ἡ ΑΓ πρὸς τὴν ΓΒ, τουτέστιν ὡς τὸ ὑπὸ ΕΑΓ πρὸς τὸ ὑπὸ ΕΑ, ΒΓ οὕτως τὸ ὑπὸ ΕΑΓ πρὸς τὸ ὑπὸ Γ∠Ε ἴσον ἄρα ἐστὶν τὸ ὑπὸ ΑΕ, ΒΓ τῷ ὑπὸ Γ∠Ε. ἀνάλογον καὶ διελόντι ἐστίν, ὡς ἡ Α∠ πρὸς τὴν ∠Ε, τουτέστιν πρὸς τὴν ∠Γ, οὕτως ἡ ∠Β πρὸς τὴν ΒΓ καὶ λοιπὴ ἄρα ἡ ΑΒ πρὸς λοιπὴν τὴν Β∠ ἐστιν, ὡς ἡ Β∠ πρὸς τὴν ΒΓ. τὸ ἄρα ὑπὸ ΑΒΓ ἴσον ἐστὶν τῷ ἀπὸ τῆς Β∠ τετραγώνῳ.

κζ΄. Ἔστω δὲ πάλιν, ὡς ἡ ΑΒ πρὸς τὴν ΒΓ. οὕτως τὸ ἀπὸ Α∠ τετράγωνον πρὸς τὸ ἀπὸ ∠Γ τετράγωνον· ὅτι τὸ ὑπὸ ΑΒΓ ἴσον ἐστὶν τῷ ἀπὸ τῆς Β∠ τετραγώνῳ.

κείσθω γὰρ ὁμοίως τῇ Γ∠ ἴση ἡ ∠Ε· τὸ ἄρα ὑπὸ ΓΑΕ μετὰ τοῦ ἀπὸ Γ∠, τουτέστιν τοῦ ὑπὸ Ε∠Γ, ἴσον τῷ ἀπὸ Α∠. καὶ γίνεται κατὰ διαίρεσιν, ὡς ἡ ΑΓ πρὸς τὴν ΓΒ, τουτέστιν ὡς τὸ ὑπὸ ΕΑΓ πρὸς τὸ ὑπὸ ΕΑ, ΓΒ, οὕτως τὸ ὑπὸ ΓΑΕ πρὸς τὸ ὑπὸ Ε∠ ἴσον ἄρα ἐστὶν τὸ ὑπὸ ΑΕ. ΓΒ τῷ ὑπὸ Ε∠Γ. ἀνάλογον καὶ συνθέντι ἐστίν, ὡς ἡ Α∠ πρὸς τὴν ∠Ε, τουτέστιν πρὸς τὴν ∠Γ. οὕτως ἡ ∠Β πρὸς τὴν ΒΓ καὶ ὅλη ἄρα ἡ ΑΒ πρὸς ὅλην τὴν Β∠ ἐστιν, ὡς ἡ Β∠ πρὸς τὴν ΒΓ. τὸ ἄρα ὑπὸ τῶν ΑΒΓ ἴσον ἐστὶν τῷ ἀπὸ τῆς Β∠ τετραγώνῳ.

κη΄. Κύκλου τοῦ ΑΒΓ ἐφαπτέσθωσαν αἱ Α∠, ∠Γ, καὶ ἐπεζεύχθω ἡ ΑΓ. καὶ διήχθω τυχοῦσα ἡ ∠Β· ὅτι γίνεται, ὡς ἡ Β∠ πρὸς τὴν ∠Ε, οὕτως ἡ Β πρὸς τὴν ΖΕ.

ἐπεὶ γὰρ ἴση ἐστὶν ἡ Α∠ τῇ ∠Γ, τὸ ἄρα ὑπὸ ΑΖΓ μετὰ τοῦ ἀπὸ Ζ∠ ἴσον ἐστὶν τῷ ἀπὸ ∠Α. ἀλλὰ τὸ μὲν ὑπὸ ΑΖΓ ἴσον ἐστὶν τῷ ὑπὸ ΒΖΕ, τὸ δὲ ἀπὸ ∠Α. ἐστιν τὸ ὑπὸ Β∠Ε· τὸ ἄρα ὑπὸ ΒΖΕ μετὰ τοῦ ἀπὸ ∠Ζ ἴσον ἐστὶν τῷ ὑπὸ Β∠Ε. ἐὰν δὲ τοῦτο, γίνεται, ὡς ἡ Β∠ πρὸς τὴν ∠Ε, οὕτως ἡ ΒΖ πρὸς τὴν ΖΕ.

κθ΄. Τμήματος δοθέντος τοῦ ἐπὶ τῆς ΑΒ κλάσαι εὐθεῖαν θεῖαν τὴν ΑΓΒ ἐν λόγῳ τῷ δοθέντι.

γεγονέτω, καὶ διήχθω ἀπὸ τοῦ Γ ἐφαπτομένη ἡ Γ∠ ὡς ἄρα τὸ ἀπὸ ΑΓ πρὸς τὸ ἀπὸ ΒΓ, οὕτως ἡ Α∠ πρὸς ∠ Β. λόγος δὲ τοῦ ἀπὸ ΑΓ πρὸς τὸ ἀπὸ ΓΒ, δοθείς· ὥστε καὶ ὁ τῆς Α∠ πρὸς τὴν Β∠ δοθείς. καί ἐστιν δύο [*](7. ἐστιν τὸ] ἴσον τῷ Hultsch. 16. δύο δοθέντα τὰ Α, Β] δύο cod., δοθέντα τὰ Α B Hultsch cum Simsono.)

Συντεθήσεται δὴ τὸ πρόβλημα οὕτως. ἔστω τὸ μὲν τμῆμα τὸ ΑΒΓ, ὁ δὲ λόγος ὁ τῆς πρὸς τὴν Ζ, καὶ πεποιήσθω, ὡς τὸ ἀπὸ Ε πρὸς τὸ ἀπὸ Ζ, οὕτως ἡ Α∠ πρὸς τὴν ∠Β, καὶ ἤχθω ἐφαπτομένη ἡ ∠Γ, καὶ ἐπεζεύχθωσαν αἱ ΑΓ, ΓΒ. λέγω, ὅτι αἱ ΑΓ, ΓΒ ποιοῦσι τὸ πρόβλημα.

ἐπεὶ γάρ ἐστιν, ὡς τὸ ἀπὸ Ε πρὸς τὸ ἀπὸ Ζ, οὕτως ἡ Α∠ πρὸς τὴν ∠Β, ὡς δὲ ἡ Α∠ πρὸς τὴν ∠Β, οὕτως τὸ ἀπὸ ΑΓ πρὸς τὸ ἀπὸ ΓΒ διὰ τὸ ἔφάπτεσθαι τὴν ΓΔ, καὶ ὡς ἄρα τὸ ἀπὸ Ε πρὸς τὸ ἀπὸ Ζ, οὕτως τὸ ἀπὸ ΑΓ πρὸς τὸ ἀπὸ ΓΒ ὥστε καί, ὡς ἡ πρὸς τὴν Ζ, οὕτως ἡ ΑΓ πρὸς τὴν ΓΒ. ἡ ΑΓΒ ἄρα ποιεῖ τὸ πρόβλημα.

λ΄. Κύκλος, οὗ διάμετρος ἡ ΑΒ, καὶ ἀπὸ τυχόντος ἐπ᾿ αὐτὴν κάθετος ἡ ∠Ε, διήχθω ἡ ∠Ζ, ἐπεζεύχθω ἡ ΕΖ καὶ ἐκβεβλήσθω, καί, καθ᾿ ὃ συμπίπτει τῇ διαμέτρῳ, ἔστω τὸ Η· ὅτι ἐστίν, ὡς ἡ ΑΗ πρὸς τὴν ΗΒ, οὕτως ἡ ΑΘ πρὸς τὴν ΘΒ.

ἐπεζεύχθωσαν αἱ ∠Α, ΑΕ, ΑΖ.

ἐπεὶ οὖν ἐπὶ διάμετρον κάθετος ἡ ∠Ε, ἴση ἐστὶν ἡ ὑπὸ ∠ΑΒ τῇ ὑπὸ ΒΑΕ. ἀλλ᾿ ἡ ὑπὸ ∠ΑΒ τῇ ἐν τῷ αὐτῷ τμήματι ἴση ἐστὶν τῇ ὑπὸ ΘΖΒ, ἡ δὲ ὑπὸ ΒΑΕ ἴση ἐστὶν τῇ ἐκτὸς τετραπλεύρου τῇ ὑπὸ ΒΖΗ καὶ ἡ ὑπὸ ΘΖΒ ἄρα γωνία ἴση ἐστὶν τῇ ὑπὸ ΒΖΗ. καί ἐστιν ὀρθὴ ἡ ὑπὸ ΑΖΒ γωνία· διὰ δὴ τὸ λῆμμα γίνεται, ὡς ἡ ΑΗ πρὸς τὴν ΗΒ, οὕτως ἡ Α πρὸς τὴν ΒΘ.