# Scholia in opticorum recensionem Theonis (scholia vetera) 64

Scholia in Euclidem, Scholia in opticorum recensionem Theonis (scholia vetera), Section 64. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/scholia-in-euclidem/scholia-in-opticorum-recensionem-theonis-scholia-vetera/64

64. Καὶ αὕτη μὲν ἡ ἀπόδειξις, εἰ μήτε πρὸς ὁρθὰζ ἡ Ε∠ τῇ ΓΑ διαχθῇ· τότε γὰρ ἀπὸ τοῦ Γ ἐπὶ τὴν ∠Ε εὐθεῖαν δυνάμεθα κάθετον ἀγαγεῖν τὴν Γ Ζ, καὶ οὕτως ἡ ἀπόδειξις προχωρεῖ. εἰ δὲ ἡ Ε∠ κάθετος ἐπὶ τὴν ΓΑ διαχθῇ, δειχθήσεται πάλιν ἡ ὑπὸ ΒΑΓ γωνία τῆς ὑπὸ ΒΑΕ ἐλάττων τοῦτον τὸν τρόπον· ἐπεὶ ἡ Βγ ὀρθή ἐστι πρὸς τὸ ὑποκείμενον ἐπίπεδον, καὶ πάντα ἄρα τὰ διʼ αὐτῆς ἐπίπεδα τῷ αὐτῷ ἐπιπέδῳ πρὸς ὀρθὰς ἔσται. ὥστε καὶ τὸ ΒΓΑ τρίγωνον τῷ Ε∠ κύκλῳ πρὸς ὀρθὰς ἔσται. ἐπεὶ οὖν τὸ ΓΑΒ τρίγωνον τῷ κύκλῳ πρὸς ὀρθάς ἐστι καὶ τῇ κοινῇ αὐτῶν τομῇ ἡ Ε Α ἐν ἐνὶ τῶν ἐπιπέδων, ἡ ΕΑ ἄρα καὶ τῷ ΑΒΓ τριγώνῳ πρὸς ὀρθὰς ἔσται· καὶ πρὸς πάσας ἄρα τὰς ἁπτομένας αὐτῆς εὐθείας καὶ οὔσας ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ τῷ ΑΒΓ ὀρθὰς ποιήσει γωνίας. ἅπτεται δὲ αὐτῆς καὶ ἡ ΒΑ· καὶ πρὸς ἄρα τὴν ΒΑ ὀρθὴν ποιήσει γωνίαν. ὀρθὴ ἄρα ἡ ὑπὸ ΒΑΕ ὀξεῖα δὲ ἡ ὑπὸ ΒΑΓ. ἐλάττων ἄρα ἡ ὑπὸ ΒΑΓ τῆς ὑπὸ ΒΑΕ.

---

Scholia in Euclidem, Scholia in opticorum recensionem Theonis (scholia vetera), ed. Johan Ludvig Heiberg, H. Menge. Text: The Greek Library, from the Perseus Digital Library and other open sources, CC BY-SA 4.0. Changes: apostrophes normalised and the text divided into reading sections by Interlinea.
License: CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Source: https://thegreeklibrary.pages.lisn.upsaclay.fr/works/tlg5022.tlg004.local001/.