Dioptra

[*](p. 282)

κη. Τὰ δὲ ὑπερτεθέντα νῦν δείξομεν. τραπεζίου δοθέντος τοῦ ΑΒΓ∠, παράλληλον ἔχοντος τῇ Α∠ τὴν ΒΓ, καὶ ἔτι ἑκατέραν αὐτῶν καὶ τὴν μὲν ἐπʼ [*](8 εὕρομεν τὸν ΘΕΖ 10 supplevi 12 αἱ ΖΗ ΝΜ) [*](13 supplevi 15 ἐπὶ τὸ Ξ, sed Ξ ex Ζ fec. m. 1 18 καὶ ἔτι: correxi πρὸς τῶ 19 τριγωνω 28 [μὲν] delevi)

αὐτὰς κάθετον δοθεῖσαν, ἀγαγεῖν παράλληλον τῇ Α∠, ὡς τὴν ΕΖ, ἀπολαμβάνουσαν τὸ Α∠ΕΖ τραπέζιον δοθὲν τῷ μεγέθει. γεγονέτω δὴ καὶ ἐκβεβλήσθωσαν αἱ ΒΑ, Γ∠ ἐπὶ τὸ Η· καὶ κάθετος ἡ ΗΘ. ἐπεὶ οὖν ἑκατέρα τῶν Α∠, ΒΓ δοθεῖσά ἐστι τῷ μεγέθει, λόγος ἄρα τῆς ΒΓ πρὸς Α∠ δοθείς, ὥστε καὶ τῆς ΘΗ πρὸς ΗΚ, καὶ τῆς ΘΚ ἄρα πρὸς ΚΗ· καὶ ἔστι δοθεῖσα ἡ ΘΚ, δοθεῖσα ἄρα καὶ ἡ ΚΗ. ἀλλὰ καὶ ἡ Α∠ δοθεῖσα. δέδοται οὖν καὶ τὸ Α∠Η τρίγωνον τῷ μεγέθει· δέδοται ἄρα καὶ ὅλον τὸ ΗΕΖ τρίγωνον· λόγος ἄρα τοῦ ΗΕΖ τριγώνου πρὸς τὸ ΗΑ∠ τρίγωνον δοθείς, ὥστε καὶ τοῦ ἀπὸ ΛΗ πρὸς τὸ ἀπὸ ΚΗ λόγος ἐστὶ δοθείς· καὶ ἔστιν δοθὲν τὸ ἀπὸ ΗΚ, δοθὲν ἄρα καὶ τὸ ἀπὸ ΗΛ· δοθεῖσα ἄρα ἡ ΗΛ. ἀλλὰ καὶ ἡ ΗΘ, καὶ λοιπὴ ἄρα ἡ ΑΘ δοθεῖσά ἐστι. θέσει ἄρα ἡ ΕΖ· ἀλλὰ καὶ ἡ ΗΚ δοθεῖσα, καὶ λοιπὴ ἄρα ἡ ΚΛ δοθεῖσά ἐστι. θέσει ἄρα καὶ ἡ ΕΖ. συντεθήσεται [*](fol. 75v) δὴ | οὕτως. ἔστω ἡ μὲν ΒΓ μοιρῶν ιδ, ἡ δὲ Α∠ μοιρῶν ἑπτὰ, ἡ δὲ ἐπʼ αὐτὴν κάθετος μοιρῶν ϛ. [*](p. 284) ἐπεὶ οὖν διπλασία ἐστὶν ἡ ΒΓ τῆς Α∠, ὅλη ἄρα ἡ ΗΘ τῆς ΗΚ ἐστὶ διπλασίων· καὶ ἔστιν ἡ ΚΘ μοιρῶν ϛ· ἔσται ἄρα καὶ ἡ λοιπὴ μοιρῶν ϛ· ἀλλὰ καὶ ἡ Α∠ μοιρῶν ζ· τὸ ἄρα Α∠Η τρίγωνον ἔσται μοιρῶν κα. δέον οὖν ἔστω τὸ ἀφαιρούμενον τραπέζιον ποιεῖν μοιρῶν ιθ· ὅλον ἄρα τὸ ΗΕΖ τρίγωνον ἔσται μοιρῶν υ· καὶ ἐπεὶ ἡ ΗΚ μοιρῶν ἐστὶν ϛ, τὸ ἄρα ἀπʼ αὐτῆς μοιρῶν ἐστὶ λϛ. πολλαπλασιάζω οὖν τὰ λϛ ἐπὶ τὰ [*](12 πρὸς τῶ 15 ἡ ΛΗ δοθεισα θεσις, tum una littera erasa est 17 καὶ ἡ ΕΒ 19 επαυτ. σ (post τ una litt. eva- nuit) 20—21 ἄρα ἡ ΠΟ 27 μοιρῶν εστι λU+A7FC (in ultima litt. aliquid correctum est)) υ· γίνεται αυμ· καὶ παραβάλλω παρὰ τὸν κα, γίνεται ξη U+2220 ιδʹ· καὶ τούτων πλευρὰ τετραγωνικὴ γίνεται ὡς ἔγγιστα η καὶ β· ἔσται οὖν ἡ ΗΛ μοιρῶν η καὶ β, ὧν ἡ ΗΚ μοιρῶν ϛ· λοιπὴ ἄρα ἡ ΚΛ μοιρῶν β καὶ β· ὥστʼ ἐὰν ἀπὸ τῆς καθέτου ἀφέλω μοίρας δύο καὶ β, καὶ παράλληλον ἀγάγω, ἔσται τὸ ἀφαιρούμενον τραπέζιον μοιρῶν ιθ.