Page 192 - Euclid's Elements of Geometry

ST EW þ.

ELEMENTS BOOK 6

αὐτὰ δὴ καὶ ὁσαπλασίων ἐστὶν ἡ ΝΕ περιφέρεια τῆς ΕΖ, το- times is angle BGL also (divisible) by BGC. And so, for
σαυταπλασίων ἐστὶ καὶ ἡ ὑπὸ ΝΘΕ γωνία τῆς ὑπὸ ΕΘΖ. εἰ the same (reasons), as many times as circumference NE
ἄρα ἴση ἐστὶν ἡ ΒΛ περιφέρεια τῇ ΕΝ περιφερείᾳ, ἴση ἐστὶ is (divisible) by EF, so many times is angle NHE also
καὶ γωνία ἡ ὑπὸ ΒΗΛ τῇ ὑπὸ ΕΘΝ, καὶ εἰ μείζων ἐστὶν ἡ ΒΛ (divisible) by EHF. Thus, if circumference BL is equal
περιφέρεια τῆς ΕΝ περιφερείας, μείζων ἐστὶ καὶ ἡ ὑπὸ ΒΗΛ to circumference EN then angle BGL is also equal to
γωνία τῆς ὑπὸ ΕΘΝ, καὶ εἰ ἐλάσσων, ἐλάσσων. τεσσάρων EHN [Prop. 3.27], and if circumference BL is greater
δὴ ὄντων μεγεθῶν, δύο μὲν περιφερειῶν τῶν ΒΓ, ΕΖ, δύο than circumference EN then angle BGL is also greater
†
δὲ γωνιῶν τῶν ὑπὸ ΒΗΓ, ΕΘΖ, εἴληπται τῆς μὲν ΒΓ περι- than EHN, and if (BL is) less (than EN then BGL is
φερείας καὶ τῆς ὑπὸ ΒΗΓ γωνίας ἰσάκις πολλαπλασίων ἥ τε also) less (than EHN). So there are four magnitudes,
ΒΛ περιφέρεια καὶ ἡ ὑπὸ ΒΗΛ γωνία, τῆς δὲ ΕΖ περιφερείας two circumferences BC and EF, and two angles BGC
καὶ τῆς ὑπὸ ΕΘΖ γωνίας ἥ τε ΕΝ περιφέρια καὶ ἡ ὑπὸ ΕΘΝ and EHF. And equal multiples have been taken of cir-
γωνία. καὶ δέδεικται, ὅτι εἰ ὑπερέχει ἡ ΒΛ περιφέρεια τῆς cumference BC and angle BGC, (namely) circumference
ΕΝ περιφερείας, ὑπερέχει καὶ ἡ ὑπὸ ΒΗΛ γωνία τῆς ὑπο BL and angle BGL, and of circumference EF and an-
ΕΘΝ γωνίας, καὶ εἰ ἴση, ἴση, καὶ εἰ ἐλάσσων, ἐλάσσων. gle EHF, (namely) circumference EN and angle EHN.
ἔστιν ἄρα, ὡς ἡ ΒΓ περιφέρεια πρὸς τὴν ΕΖ, οὕτως ἡ ὑπὸ And it has been shown that if circumference BL exceeds
ΒΗΓ γωνία πρὸς τὴν ὑπὸ ΕΘΖ. ἀλλ᾿ ὡς ἡ ὑπὸ ΒΗΓ γωνία circumference EN then angle BGL also exceeds angle
πρὸς τὴν ὑπὸ ΕΘΖ, οὕτως ἡ ὑπὸ ΒΑΓ πρὸς τὴν ὑπὸ ΕΔΖ. EHN, and if (BL is) equal (to EN then BGL is also)
διπλασία γὰρ ἑκατέρα ἑκατέρας. καὶ ὡς ἄρα ἡ ΒΓ περιφέρεια equal (to EHN), and if (BL is) less (than EN then BGL
πρὸς τὴν ΕΖ περιφέρειαν, οὕτως ἥ τε ὑπὸ ΒΗΓ γωνία πρὸς is also) less (than EHN). Thus, as circumference BC
τὴν ὑπὸ ΕΘΖ καὶ ἡ ὑπὸ ΒΑΓ πρὸς τὴν ὑπὸ ΕΔΖ. (is) to EF, so angle BGC (is) to EHF [Def. 5.5]. But as
᾿Εν ἄρα τοῖς ἴσοις κύκλοις αἱ γωνίαι τὸν αὐτὸν ἔχουσι angle BGC (is) to EHF, so (angle) BAC (is) to EDF
λόγον ταῖς περιφερείαις, ἐφ᾿ ὧν βεβήκασιν, ἐάν τε πρὸς τοῖς [Prop. 5.15]. For the former (are) double the latter (re-
κέντροις ἐάν τε πρὸς ταῖς περιφερείαις ὦσι βεβηκυῖαι· ὅπερ spectively) [Prop. 3.20]. Thus, also, as circumference BC
ἔδει δεῖξαι. (is) to circumference EF, so angle BGC (is) to EHF,
and BAC to EDF.
Thus, in equal circles, angles have the same ratio as
the (ratio of the) circumferences on which they stand,
whether they are standing at the centers (of the circles)
or at the circumferences. (Which is) the very thing it was
required to show.

† This is a straight-forward generalization of Prop. 3.27

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