# Synagoge 90

Pappus Alexandrinus, Synagoge, ΠΑΠΠΟΥ ΑΛΕΞΑΝΔΡΕΩΣ ΣΥΝΑΓΩΓΗΣ Γ.. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/pappus-alexandrinus/synagoge/90

90 ν΄. Εἰς τὴν δοθεῖσαν σφαῖραν ὀκτάεδρον ἐγγράψαι. Ἐγγεγράφθω καὶ ἔστω σημεία τῶν γωνιῶν αὐτοῦ ἐν τῇ ἐπιφανείᾳ τῆς σφαίρας τὰ Α Β Γ ∠ Ε Ζ, καὶ ἐκβληθέντα τὰ διʼ αὐτῶν ἐπίπεδα ποιείτω κύκλους τούς ΑΒΓ θένταΕΖ. ἐπεὶ ἀπὸ τοῦ ∠ ἴσαι πρὸς τὴν ἐπιφάνειαν προσπεπτώκασιν αἱ ∠Α ∠Β ∠Ε ∠Ζ, ἔσται τὰ Α Ζ Β ἐν ἑνὶ ἐπιπέδῳ . καὶ εἰσὶν ἴσαι ἀλλήλαις αἱ ΑΒ ΒΖ ΖΕ, καὶ εἰσὶν ἐν κύκλῳ τετράγωνον ἄρα τὸ ΑΕΖΒ, καὶ παράλληλος ἡ ΕΖ τῇ ΑΒ. ὁμοίως καὶ ἡ μὲν ∠Ε τῇ ΒΓ, ἡ δὲ ∠Ζ τῇ ΑΓ· παράλληλοι ἄρα καὶ οἱ κύκλοι· καὶ ἴσοι ἀλλήλοις, ἐπεὶ καὶ τὰ ἐν αὐτοῖς ἰσόπλευρα τρίγωνα ἴσα ἐστίν. καὶ ἐπεὶ ἐν σφαίρᾳ ἴσοι καὶ παράλληλοι κύκλοι εἰσίν, καὶ ἐν αὐτοῖς ἴσαι εὐθεῖαι καὶ παράλληλοι αἱ ΑΒ ΕΖ καὶ εἰσὶν οὐκ ἐπὶ τὰ αὐτὰ μέρη τῶν κέντρων, ἔσται ἐπιζευγνυμένη ἡ ΑΖ διάμετρος τῆς σφαίρας . καὶ εἰσὶν ἴσαι αἱ ΑΕ ΕΖ διπλάσιον ἄρα τὸ ἀπὸ ΑΖ τοῦ ἀπὸ ΖΕ. τὸ δʼ ἀπὸ τῆς διαμέτρου τοῦ ∠ΕΖ κύκλου ἐπίτριτον τοῦ ἀπὸ τῆς ΕΖ· ἡμιόλιον ἄρα τὸ ἀπὸ τῆς ΑΖ τοῦ ἀπὸ τῆς διαμέτρου τοῦ ∠ΕΖ κύκλου δοθεῖσα ἄρα ἡ διάμετρος, καὶ ὁ κύκλος, ὥστε καὶ ὁ ΑΒΓ καὶ τὰ ἐπʼ αὐτῶν σημεῖα.

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Pappus Alexandrinus, Synagoge, ed. Friedrich Hultsch. 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/tlg2032.tlg001.local001/.