# Scholia in opticorum recensionem Theonis (scholia vetera) 18

Scholia in Euclidem, Scholia in opticorum recensionem Theonis (scholia vetera), Section 18. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/scholia-in-euclidem/scholia-in-opticorum-recensionem-theonis-scholia-vetera/18

18. Κάθετος ἄρα ἐστίν p. 162, 3— πῶς ἡ ΚΜ κάθετός ἐστιν ἐπὶ τὴν ΜΛ, δείξομεν οὕτως· ἐπεὶ ἀπὸ τοῦ Κ ἐπὶ τὸ ὑποκείμενον ἐπίπεδον κάθετος ἦκται ἡ Κ Α, καὶ πρὸς πάσας ἄρα τὰς ἁπτομένας αὐτῆς εὐθείας καὶ οὔσας ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ ἡ Κ Α ὀρθὰς ποιήσει γωνίας. ἐπεὶ οὖν ἐπὶ τὴν ΖΛ κάθετος ἦκται ἡ ΑΜ, καὶ πρὸς τὴν ΑΜ ἡ ΚΑ ὀρθὴν ποιήσει γωνίαν. ἐπεζεύχθω ἀπὸ τοῦ Α καὶ ἐπὶ τὸ Λ ἡ ΑΛ· καὶ πρὸς ἄρα τὴν ΑΛ ἡ ΑΚ ὀρθὴν ποιήσει γωνίαν. ἐπεὶ οὖν τρίγωνόν ἐστιν ὀρθογώνιον τὸ ΚΑΛ ὀρθὴν ἔχον τὴν ὑπὸ ΚΑΛ γωνίαν, τὸ ἄρα ἀπὸ τῆς ΚΛ ὑποτεινούσης τὴν ὀρθὴν γωνίαν ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΚΑ, ΑΛ. πάλιν ἐπεὶ τρίγωνόν ἐστιν ὀρθογώνιον τὸ ΑΜΛ ὀρθὴν ἔχον τὴν ὑπὸ ΑΜΛ γωνίαν, τὸ ἄρα ἀπὸ τῆς ΑΛ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΑΜ, ΜΑ. τὸ ἄρα ἀπὸ τῆς ΚΛ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΚΑ, ΑΜ, ΜΛ. ἀλλὰ τοῖς ἀπὸ τῶν ΚΑ, ΑΜ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΚΜ· τρίγωνον γάρ ἐστιν ὀρθογώνιον τὸ ΚΑΜ ὀρθὴν ἔχον τὴν ὑπὸ ΚΑΜ γωνίαν. τὸ ἄρα ἀπὸ τῆς ΚΛ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΚΜ, ΜΛ, καὶ διὰ τὸ μηʹ τοῦ πρώτου τῶν Στοιχείων ἡ ὑπὸ ΚΜΛ γωνία ὀρθή ἐστιν· ὅπερ ἔδει δεῖξαι.

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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/.