# Scholia in Euclidis Data Ad prop. LXVII demonstr. quart.

Scholia in Euclidem, Scholia in Euclidis Data, APPENDIX SCHΟLIΟRUM.. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/scholia-in-euclidem/scholia-in-euclidis-data/Ad%20prop.%20LXVII%20demonstr.%20quart.

32. Πῶς μὲν τὴν ὑπὸ ∠ΕΓ δύναμαι συστήσασθαι ἴσην τῇ ὑπὸ Α∠Γ χωρὶς τῶν Ἀπολλωνίου; οὕτως. ἐπεὶ γὰρ ἴση ἐστὶν ἡ ὑπὸ ΑΓ∠ τῇ ὑπὸ Α∠Γ, μείζων ἐστὶν ἡ ὑπὸ ΒΓ∠ τῆς ὑπὸ Α∠Γ. κείσθω οὖν ἴση τῇ ὑπὸ ΒΓ∠ ἡ ὑπὸ Β∠Ε, καὶ ἐκβεβλήσθω ἡ ΒΓ. ἔστι δὲ κοινὴ ἡ πρὸς τῷ Β γωνία τοῦ τε ∠ΒΓ τρι- γώνου καὶ τοῦ ∠ΒΕ. λοιπὴ ἄρα ἡ ὑπὸ Β∠Γ λοιπῇ τῇ ὑπὸ ∠ΕΓ ἐστιν ἴση. 33. Πῶς δὲ δυνατὸν καθόλου ἀπὸ τοῦ δοθέντος σημείου ὡς τοῦ α ἐπὶ τὴν δοθεῖσαν εὐθεῖαν ὡς τὴν βγ καταγαγεῖν εὐθεῖαν ἴσην ποιοῦσαν γωνίαν τῇ δο- θείσῃ τῇ ὑπὸ δεζ; δείξομεν οὕτως. ἡ γὰρ ὑπὸ δεζ ἢ ὀρθή ἐστιν ἢ ὀξεῖα ἢ ἀμβλεῖα. εἰ μὲν οὖν ὀρθή, φανερόν· ἄγω γὰρ ἀπὸ τοῦ α κάθετον τὴν αη· καὶ ἔσται ἴση ἡ ε τῇ η. ἀλλὰ δὴ ἔστω ὀξεῖα ἡ ὑπὸ δεζ. καὶ ἤχθω κάθετος ἀπὸ μὲν τοῦ δ ἐπὶ τὴν εζ ἡ δθ, ἀπὸ δὲ τοῦ α ἐπὶ τὴν βγ ἡ αη, καὶ συνεστάτω πρὸς τῇ αη εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ α τῇ ὑπὸ εδθ ἴση ἡ ὑπὸ ηακ· λοιπὴ ἄρα ἡ ὑπὸ δεζ ἴση ἐστὶ τῇ ὑπὸ ακη. ἀλλὰ δὴ ἔστω ἀμβλεῖα ἡ ὑπὸ δεζ. ἐκ- βληθείσης ἄρα τῆς ζε ὀξεῖα ἔσται ἡ ὑπὸ δελ. κάθετος οὖν ἤχθω ἡ δλ, καὶ τῇ ὑπὸ λδε ἴση κείσθω ἡ ὑπὸ ηακ. λοιπὴ ἄρα ἡ ὑπὸ δελ ἴση ἐστὶ τῇ ὑπὸ ακη, ὥστε καὶ ἡ ἐφεξῆς ἡ ὑπὸ δεζ τῇ ἐφεξῆς τῇ ὑπὸ ακγ ἴση ἐστίν. Τουτέστι τὸ ὑπὸ τῶν ΕΓΒ p. 214, 6–7] ἐὰν γὰρ εὐθεῖα γραμμὴ τμηθῇ, ὡς ἔτυχεν, τὸ ὑπὸ τῆς ὅλης καὶ ἑνὸς τῶν τμημάτων περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ τε ὑπὸ τῶν τμημάτων καὶ τῷ ἀπὸ τοῦ προ- ειρημένου τετραγώνῳ. 35. Ἴσον ἐστὶ τῷ ἀπὸ Β∠ p. 214, 7–8] ἐὰν γὰρ τρεῖς εὐθεῖαι ἀνάλογον, τὸ ὑπὸ πρώτης καὶ τρίτης ἴσον ἐστὶ τῷ ἀπὸ τῆς δευτέρας.

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Scholia in Euclidem, Scholia in Euclidis Data, ed. Heinrich 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.tlg002.local001/.