# Sphaerica 5.ΠΡΟΤΑΣΙΣ ε′. ΘΕΩΡ.

Theodosius of Bithynia, Sphaerica, ΒΙΒΛΙΟΝ ΠΡΩΤΟΝ., Chapter 5. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/theodosius-of-bithynia/sphaerica/5.%CE%A0%CE%A1%CE%9F%CE%A4%CE%91%CE%A3%CE%99%CE%A3%20%CE%B5%E2%80%B2.%20%CE%98%CE%95%CE%A9%CE%A1.

Ἐὰν σφαῖρα ἐπιπέδου ἅπτηται μὴ τέμνοντος, ἀπὸ δὲ τῆς ἁφῆς τῷ ἐπιπέδῳ πρὸς ὀρθὰς ἀνασταθῇ εὐθεῖα, ἐπὶ τῆς ἀνασταθείσης ἔσται τὸ κέντρον τῆς σφαίρας. Σφαῖρα γὰρ ἐπιπέδου ἁπτέσθω μὴ τέμνοντος κατὰ τὸ Α σημεῖον, καὶ ἀνεστάτω ἀπὸ τοῦ Α σημείου τῷ ἐπιπέδῳ πρὸς ὀρθὰς ἡ ΑΒ· λέγω, ὅτι ἐπὶ τῆς ΑΒ ἐστι τὸ κέντρον τῆς σφαίρας. Μὴ γὰρ, ἀλλ’, εἰ δυνατὸν, ἔστω τὸ Γ, καὶ ἐπεζεύχθω ἡ ΓΑ. Ἐπεὶ οὖν σφαῖρα ἐπιπέδου ἅπτεται μὴ τέμνοντος κατὰ τὸ Α σημεῖον, ἀπὸ δὲ τοῦ κέντρου τῆς σφαίρας ἐπὶ τὴν ἁφὴ ἐπέζευκται ἡ ΓΑ, ἡ ΓΑ ἄρα ὀρθή ἐστι πρὸς τὸ ἐπίπεδον· ἔστι δὲ καὶ ἡ ΒΑ ὀρθὴ πρὸς τὸ αὐτὸ ἐπίπεδον· ἀπὸ τοῦ αὐτοῦ ἄρα σημείου τοῦ Α τῷ αὐτῷ ἐπιπέδῳ ὑποκειμένῳ δύο πρὸς ὀρθὰς ἀνιστάμεναί εἰσιν αἱ ΑΒ, ΑΓ εὐθεῖαι, ὅπερ ἐστὶν ἀδύνατον· οὐκ ἄρα τὸ Γ σημεῖον κέντρον ἐστὶ τῆς σφαίρας. Ὁμοίως δὴ δείξομεν, ὅτι οὐδὲ ἄλλο, πλὴν τὸ ἐπὶ τῆς ΒΑ. Ἐπὶ τῆς ΒΑ ἄρα ἐστὶ τὸ κέντρον τῆς σφαίρας.

---

Theodosius of Bithynia, Sphaerica, ed. Ernst Nizze. 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/tlg1719.tlg001.local001/.