# De utilitate mathematicae 146

Theon Smyrnaeus, De utilitate mathematicae, Paragraph 146. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/theon-smyrnaeus/de-utilitate-mathematicae/146

λαμβάνεται δὲ κοινότερον ἔν τε ἀριθμοῖς [καὶ] ῥητοῖς καὶ ἐν λόγοις καὶ μεγέθεσι [καὶ] συμμέτροις γεω- μετρικῶς οὕτως. ἔστωσαν δύο ὅροι ὧν δεῖ μέσον ἀνά- λογον λαβεῖν γεωμετρικῶς· οἷον αβ βγ καὶ ἐκκείσθωσαν ἐπʼ εὐθείας· καὶ περὶ ὅλην τὴν αγ γεγράφθω ἡμικύκλιον· καὶ ἀπὸ τοῦ β ἀνήχθω τῇ αγ πρὸς ὀρθὰς μέχρι τῆς περιφερείας ἡ βδ· αὕτη δὴ γίνεται μέση τῶν αβ βγ κατὰ τὴν γεωμετρικὴν ἀναλογίαν. ἐπιζευχθεισῶν γὰρ τῶν αδ δγ ὀρθὴ γίνεται ἡ δ γωνία, ἐπεί ἐστιν ἐν ἡμι- κυκλίῳ· καὶ 〈ἐν〉 ὀρθογωνίῳ τῷ αδγ κάθετος ἡ δβ· καὶ τὰ περὶ ταύτην τρίγωνα τῷ τε ὅλῳ καὶ ἀλλήλοις ὅμοιά ἐστιν· ὥστε αἱ περὶ τὰς ἴσας αὐτῶν γωνίας πλευραὶ ἀνάλογόν εἰσιν· ὡς ἄρα ἡ αβ πρὸς τὴν βδ, ἡ δβ πρὸς βγ· τῶν ἄρα αβ βγ μέση ἀνάλογόν ἐστιν ἡ βδ· ὅπερ ἔδει δεῖξαι.

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Theon Smyrnaeus, De utilitate mathematicae, ed. Eduardus Hiller. 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/tlg1724.tlg001.local001/.