§6.prop.33ἐν τοῖς ἴσοις κύκλοις αἱ γωνίαι τὸν αὐτὸν ἔχουσι λόγον ταῖς περιφερείαις, ἐφʼ ὧν βεβήκασιν, ἐάν τε πρὸς τοῖς κέντροις ἐάν τε πρὸς ταῖς περιφερείαις ὦσι βεβηκυῖαι.
In equal circles angles have the same ratio as the circumferences on which they stand, whether they stand at the centres or at the circumferences.
ἔστωσαν ἴσοι κύκλοι οἱ ΑΒΓ, ΔΕΖ, καὶ πρὸς μὲν τοῖς κέντροις αὐτῶν τοῖς Η, Θ γωνίαι ἔστωσαν αἱ ὑπὸ ΒΗΓ, ΕΘΖ, πρὸς δὲ ταῖς περιφερείαις αἱ ὑπὸ ΒΑΓ, ΕΔΖ· λέγω, ὅτι ἐστὶν ὡς ἡ ΒΓ περιφέρεια πρὸς τὴν ΕΖ περιφέρειαν, οὕτως ἥ τε ὑπὸ ΒΗΓ γωνία πρὸς τὴν ὑπὸ ΕΘΖ καὶ ἡ ὑπὸ ΒΑΓ πρὸς τὴν ὑπὸ ΕΔΖ.
κείσθωσαν γὰρ τῇ μὲν ΒΓ περιφερείᾳ ἴσαι κατὰ τὸ ἑξῆς ὁσαιδηποτοῦν αἱ ΓΚ, ΚΛ, τῇ δὲ ΕΖ περιφερείᾳ ἴσαι ὁσαιδηποτοῦν αἱ ΖΜ, ΜΝ, καὶ ἐπεζεύχθωσαν αἱ ΗΚ, ΗΛ, ΘΜ, ΘΝ.
ἐπεὶ οὖν ἴσαι εἰσὶν αἱ ΒΓ, ΓΚ, ΚΛ περιφέρειαι ἀλλήλαις, ἴσαι εἰσὶ καὶ αἱ ὑπὸ ΒΗΓ, ΓΗΚ, ΚΗΛ γωνίαι ἀλλήλαις· ὁσαπλασίων ἄρα ἐστὶν ἡ ΒΛ περιφέρεια τῆς ΒΓ, τοσαυταπλασίων ἐστὶ καὶ ἡ ὑπὸ ΒΗΛ γωνία τῆς ὑπὸ ΒΗΓ. διὰ τὰ αὐτὰ δὴ καὶ ὁσαπλασίων ἐστὶν ἡ ΝΕ περιφέρεια τῆς ΕΖ, τοσαυταπλασίων ἐστὶ καὶ ἡ ὑπὸ ΝΘΕ γωνία τῆς ὑπὸ ΕΘΖ. εἰ ἄρα ἴση ἐστὶν ἡ ΒΛ περιφέρεια τῇ ΕΝ περιφερείᾳ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΒΗΛ τῇ ὑπὸ ΕΘΝ, καὶ εἰ μείζων ἐστὶν ἡ ΒΛ περιφέρεια τῆς ΕΝ περιφερείας, μείζων ἐστὶ καὶ ἡ ὑπὸ ΒΗΛ γωνία τῆς ὑπὸ ΕΘΝ, καὶ εἰ ἐλάσσων, ἐλάσσων.
Let ΑΒΓ, ΔΕΖ be equal circles, and let the angles ΒΗΓ, ΕΘΖ be at their centres Η, Θ, and the angles ΒΑΓ, ΕΔΖ at the circumferences; I say that, as the circumference ΒΓ is to the circumference ΕΖ, so is the angle ΒΗΓ to the angle ΕΘΖ, and the angle ΒΑΓ to the angle ΕΔΖ. For let any number of circumferences ΓΚ, ΚΛ consecutive to it be made equal to the circumference ΒΓ, and any number of circumferences ΖΜ, ΜΝ equal to the circumference ΕΖ, and let ΗΚ, ΗΛ, ΘΜ, ΘΝ be joined. Since then the circumferences ΒΓ, ΓΚ, ΚΛ are equal to one another, the angles ΒΗΓ, ΓΗΚ, ΚΗΛ are also equal to one another; therefore, whatever multiple the circumference ΒΛ is of the circumference ΒΓ, the same multiple also is the angle ΒΗΛ of the angle ΒΗΓ. For the same reasons also, whatever multiple the circumference ΝΕ is of the circumference ΕΖ, the same multiple also is the angle ΝΘΕ of the angle ΕΘΖ. If therefore the circumference ΒΛ is equal to the circumference ΕΝ, the angle ΒΗΛ is also equal to the angle ΕΘΝ; and if the circumference ΒΛ is greater than the circumference ΕΝ, the angle ΒΗΛ is also greater than the angle ΕΘΝ; and if less, less.
τεσσάρων δὴ ὄντων μεγεθῶν, δύο μὲν περιφερειῶν τῶν ΒΓ, ΕΖ, δύο δὲ γωνιῶν τῶν ὑπὸ ΒΗΓ, ΕΘΖ, εἴληπται τῆς μὲν ΒΓ περιφερείας καὶ τῆς ὑπὸ ΒΗΓ γωνίας ἰσάκις πολλαπλασίων ἥ τε ΒΛ περιφέρεια καὶ ἡ ὑπὸ ΒΗΛ γωνία, τῆς δὲ ΕΖ περιφερείας καὶ τῆς ὑπὸ ΕΘΖ γωνίας ἥ τε ΕΝ περιφέρεια καὶ ἡ ὑπὸ ΕΘΝ γωνία.
Since then there are four magnitudes, two circumferences ΒΓ, ΕΖ, and two angles ΒΗΓ, ΕΘΖ, there have been taken, as equimultiples of the circumference ΒΓ and the angle ΒΗΓ, the circumference ΒΛ and the angle ΒΗΛ, and as equimultiples of the circumference ΕΖ and the angle ΕΘΖ, the circumference ΕΝ and the angle ΕΘΝ.
καὶ δέδεικται, ὅτι εἰ ὑπερέχει ἡ ΒΛ περιφέρεια τῆς ΕΝ περιφερείας, ὑπερέχει καὶ ἡ ὑπὸ ΒΗΛ γωνία τῆς ὑπὸ ΕΘΝ γωνίας, καὶ εἰ ἴση, ἴση, καὶ εἰ ἐλάσσων, ἐλάσσων.
And it has been proved that, if the circumference ΒΛ exceeds the circumference ΕΝ, the angle ΒΗΛ also exceeds the angle ΕΘΝ; and if equal, equal; and if less, less.
ἔστιν ἄρα, ὡς ἡ ΒΓ περιφέρεια πρὸς τὴν ΕΖ, οὕτως ἡ ὑπὸ ΒΗΓ γωνία πρὸς τὴν ὑπὸ ΕΘΖ. ἀλλʼ ὡς ἡ ὑπὸ ΒΗΓ γωνία πρὸς τὴν ὑπὸ ΕΘΖ, οὕτως ἡ ὑπὸ ΒΑΓ πρὸς τὴν ὑπὸ ΕΔΖ· διπλασία γὰρ ἑκατέρα ἑκατέρας.
Therefore, as the circumference ΒΓ is to the circumference ΕΖ, so is the angle ΒΗΓ to the angle ΕΘΖ. But as the angle ΒΗΓ is to the angle ΕΘΖ, so is the angle ΒΑΓ to the angle ΕΔΖ; for each is double of each.
καὶ ὡς ἄρα ἡ ΒΓ περιφέρεια πρὸς τὴν ΕΖ περιφέρειαν, οὕτως ἥ τε ὑπὸ ΒΗΓ γωνία πρὸς τὴν ὑπὸ ΕΘΖ καὶ ἡ ὑπὸ ΒΑΓ πρὸς τὴν ὑπὸ ΕΔΖ.
ἐν ἄρα τοῖς ἴσοις κύκλοις αἱ γωνίαι τὸν αὐτὸν ἔχουσι λόγον ταῖς περιφερείαις, ἐφʼ ὧν βεβήκασιν, ἐάν τε πρὸς τοῖς κέντροις ἐάν τε πρὸς ταῖς περιφερείαις ὦσι βεβηκυῖαι· ὅπερ ἔδει δεῖξαι.
Therefore also, as the circumference ΒΓ is to the circumference ΕΖ, so is the angle ΒΗΓ to the angle ΕΘΖ, and the angle ΒΑΓ to the angle ΕΔΖ. Therefore, in equal circles angles have the same ratio as the circumferences on which they stand, whether they stand at the centres or at the circumferences; which was to be proved.