# Conica ιβ′.

Apollonius of Perga, Conica, ιβ′.. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/apollonius-of-perga/conica/%CE%B9%CE%B2%E2%80%B2.

Τῶν αὐτῶν ὄντων ἐὰν περιέχωσιν αἱ τῆς μιᾶς εὐθείας συμπτώσεις τὰς τῆς ἑτέρας, καὶ τὸ ληφθὲν σημεῖον ἐν τῇ ἐφεξῆς γωνίᾳ τῆς ὑπὸ τῶν ἀσυμπτώτων περιεχομένης ᾖ, ἡ διὰ τῶν διαιρέσεων ἀγομένη εὐθεῖα ἐκβαλλομένη τῇ ἀντικειμένῃ τομῇ συμπεσεῖται, καὶ αἱ ἀπὸ τῶν συμπτώσεων ἐπὶ τὸ Δ σημεῖον ἀγόμεναι εὐθεῖαι ἐφάψονται τῶν ἀντικειμένων. ἔστω ὑπερβολὴ ἡ ΕΗ, ἀσύμπτωτοι δὲ αἱ ΝΞ, Ο Π, καὶ κέντρον τὸ Ρ, καὶ τὸ Δ σημεῖον ἔστω ἐν τῇ ὑπὸ ΞΡΠ γωνίᾳ, καὶ ἤχθωσαν αἱ ΔΕ, ΔΖ τέμνουσαι τὴν ὑπερβολὴν ἑκατέρα κατὰ δύο σημεῖα, καὶ περιεχέσθω τὰ E, Θ ὑπὸ τῶν Z, H, καὶ ἔστω, ὡς μὲν ἡ EΔ πρὸς ΔΘ, ἡ ΕΚ πρὸς ΚΘ, ὡς δὲ ἡ ΖΔ πρὸς ΔΗ, ἡ ΖΛ πρὸς ΛH. δεικτέον, ὅτι ἡ διὰ τῶν K, Λ συμπεσεῖταί τε τῇ EZ τομῇ καὶ τῇ ἀντικειμένῃ, καὶ αἱ ἀπὸ τῶν συμπτώσεων ἐπὶ τὸ Δ ἐφάψονται τῶν τομῶν. ἔστω δὴ ἀντικειμένη ἡ Μ, καὶ ἀπὸ τοῦ Δ ἤχθωσαν ἐφαπτόμεναι τῶν τομῶν αἱ Δ Μ, ΔΣ, καὶ ἐπιζευχθεῖσα ἡ ΜΣ, εἰ δυνατόν, μὴ ἐρχέσθω διὰ τῶν Κ, Λ, ἀλλ’ ἤτοι διὰ τοῦ ἑτέρου αὐτῶν ἢ δι’ οὐδετέρου. ἐρχέσθω πρότερον διὰ τοῦ Κ καὶ τεμνέτω τὴν ΖΗ κατὰ τὸ Χ. ἔστιν ἄρα, ὡς ἡ ΖΔ πρὸς ΔΗ, ἡ ΧΖ πρὸς ΧΗ· ὅπερ ἄτοπον· ὑπόκειται γάρ, ὡς ἡ ΖΔ πρὸς ΔΗ, ἡ ΖΛ πρὸς ΛΗ. ἐὰν δὲ μηδὲ δι’ ἑτέρου τῶν Κ, Λ ἔρχηται ἡ ΜΣ, ἐφ’ ἑκατέρας τῶν ΕΔ, ΔΖ τὸ ἀδύνατον συμβαίνει.

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Apollonius of Perga, Conica, ed. J. L. Heiberg. 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/tlg0550.tlg001.local001/.