# Commentarii in libros de sphaera et cylindro Ὡς Μέναιχμος.

Eutocius, Commentarii in libros de sphaera et cylindro, Ὡς Μέναιχμος.. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/eutocius/commentarii-in-libros-de-sphaera-et-cylindro/%E1%BD%A9%CF%82%20%CE%9C%CE%AD%CE%BD%CE%B1%CE%B9%CF%87%CE%BC%CE%BF%CF%82.

Ἔστωσαν αἱ δοθεῖσαι δύο εὐθεῖαι αἱ Α, Ε · δεῖ δὴ τῶν Α, Ε δύο μέσας ἀνάλογον εὑρεῖν. Γεγονέτω, καὶ ἔστωσαν αἱ Β, Γ, καὶ ἐκκείσθω θέσει εὐθεῖα ἡ △Η πεπερασμένη κατὰ τὸ △, καὶ πρὸς τῷ △ τῇ Γ ἴση κείσθω ἡ △Ζ καὶ ἤχθω πρὸς ὀρθὰς ἡ ΖΘ, καὶ τῇ Β ἴση κείσθω ἡ ΖΘ. Ἐπεὶ οὖν τρεῖς εὐθεῖαι ἀνάλογον αἱ Α, Β, Γ, τὸ ὑπὸ τῶν Α, Γ ἴσον ἐστὶ τῷ ἀπὸ τῆς Β · τὸ ἄρα ὑπὸ δοθείσης τῆς Α καὶ τῆς Γ, τουτέστι τῆς △Ζ, ἴσον ἐστὶ τῷ ἀπὸ τῆς Β, τουτέστι τῷ ἀπὸ τῆς ΖΘ. Ἐπὶ παραβολῆς ἄρα τὸ Θ διὰ τοῦ γεγραμμένης. Ἤχθωσαν παράλληλοι αἱ ΘΚ, △Κ. Καὶ ἐπεὶ δοθὲν τὸ ὑπὸ Β, Γ, ἴσον γάρ ἐστι τῷ ὑπὸ Α, Ε, δοθὲν ἄρα καὶ τὸ ὑπὸ ΚΘΖ. Ἐπὶ ὑπερβολῆς ἄρα τὸ Θ ἐν ἀσυμπτώτοις ταῖς Κ△, △Ζ. Δοθὲν ἄρα τὸ Θ · ὥστε καὶ τὸ Ζ. Συντεθήσεται δὴ οὕτως. Ἔστωσαν αἱ μὲν δοθεῖσαι εὐθεῖαι αἱ Α, Ε, ἡ δὲ τῇ θέσει ἡ △Η πεπαρασμένη κατὰ τὸ △, καὶ γεγράφθω διὰ τοῦ παραβολή, ἧς ἄξων μὲν ἡ △Η, ὀρθία δὲ τοῦ εἴδους πλευρὰ ἡ Α, αἱ δὲ καταγόμεναι ἐπὶ τὴν △Η ἐν ὀρθῇ γωνίᾳ δυνάσθωσαν τὰ παρὰ τὴν Α παρακείμενα χωρία πλάτη ἔχοντα τὰς ἀπολαμβανομένας ὑπʼ αὐτῶν πρὸς τῷ σημείῳ. Γεγράφθω καὶ ἔστω ἡ △Θ, καὶ ὀρθὴ ἡ △Κ, καὶ ἐν ἀσυμπτώτοις ταῖς Κ△, △Ζ γεγράφθω ὑπερβολή, ἀφ᾿ ἧς αἱ παρὰ τὰς Κ△, △Ζ ἀχθεῖσαι ποιήσουσιν τὸ χωρίον ἴσον τῷ ὑπὸ Α, Ε · τεμεῖ δὴ τὴν παραβολήν. Γεμνέτω κατὰ τὸ Θ, καὶ κάθετοι ἤχθωσαν αἱ ΘΚ, ΘΖ. Ἐπεὶ οὖν τὸ ἀπὸ ΖΘ ἴσον ἐστὶ τῷ ὑπὸ Α, △Ζ, ἔστιν ὡς ἡ Α πρὸς τὴν ΖΘ, ἡ ΘΖ πρὸς Ζ△. Πάλιν, ἐπεὶ τὸ ὑπὸ Α, Ε ἴσον ἐστὶ τῷ ὑπὸ ΘΖ△, ἔστιν ὡς ἡ Α πρὸς τὴν ΖΘ, ἡ Ζ△ πρὸς τὴν E. Ἀλλ᾿ ὡς ἡ Α πρὸς τὴν ΖΘ, ἡ ΖΘ πρὸς Ζ△· καὶ ὡς ἄρα ἡ πρὸς τὴν ΖΘ, ἡ ΖΘ πρὸς Ζ△ καὶ ἡ Ζ△ πρὸς Ε. Κείσθω τῇ μὲν ΘΖ ἴση ἡ Β, τῇ δὲ △Ζ ἴση ἡ Γ · ἔστιν ἄρα ὡς ἡ Α πρὸς τὴν Β, ἡ Β πρὸς τὴν Γ καὶ ἡ πρὸς Ε. Αἱ Α, Β, Γ, Ε ἄρα ἑξῆς ἀνάλογόν εἰσιν · ὅπερ ἔδει εὑρεῖν.

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Eutocius, Commentarii in libros de sphaera et cylindro, ed. Charles Mugler. 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/tlg4072.tlg001.local001/.