Page 96 - Euclid's Elements of Geometry

ST EW gþ.

kzþ or at the circumference. (Which is) the very thing which

ELEMENTS BOOK 3

it was required to show.

Proposition 27
.
᾿Εν τοῖς ἴσοις κύκλοις αἱ ἐπὶ ἴσων περιφερειῶν βεβηκυῖαι In equal circles, angles standing upon equal circum-
γωνίαι ἴσαι ἀλλήλαις εἰσίν, ἐάν τε πρὸς τοῖς κέντροις ἐάν τε ferences are equal to one another, whether they are
πρὸς ταῖς περιφερείαις ὦσι βεβηκυῖαι. standing at the center or at the circumference.
Α ∆ A D

Η Θ G H

Β Γ Ε Ζ B C E F
Κ K

᾿Εν γὰρ ἴσοις κύκλοις τοῖς ΑΒΓ, ΔΕΖ ἐπὶ ἴσων περι- For let the angles BGC and EHF at the centers G
φερειῶν τῶν ΒΓ, ΕΖ πρὸς μὲν τοῖς Η, Θ κέντροις γωνίαι and H, and the (angles) BAC and EDF at the circum-
βεβηκέτωσαν αἱ ὑπὸ ΒΗΓ, ΕΘΖ, πρὸς δὲ ταῖς περιφερείαις ferences, stand upon the equal circumferences BC and
αἱ ὑπὸ ΒΑΓ, ΕΔΖ· λέγω, ὅτι ἡ μὲν ὑπὸ ΒΗΓ γωνία τῇ ὑπὸ EF, in the equal circles ABC and DEF (respectively). I
ΕΘΖ ἐστιν ἴση, ἡ δὲ ὑπὸ ΒΑΓ τῇ ὑπὸ ΕΔΖ ἐστιν ἴση. say that angle BGC is equal to (angle) EHF, and BAC
Εἰ γὰρ ἄνισός ἐστιν ἡ ὑπὸ ΒΗΓ τῇ ὑπὸ ΕΘΖ, μία αὐτῶν is equal to EDF.
μείζων ἐστίν. ἔστω μείζων ἡ ὑπὸ ΒΗΓ, καὶ συνεστάτω For if BGC is unequal to EHF, one of them is greater.
πρὸς τῇ ΒΗ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Η τῇ ὑπὸ Let BGC be greater, and let the (angle) BGK, equal to
ΕΘΖ γωνίᾳ ἴση ἡ ὑπὸ ΒΗΚ· αἱ δὲ ἴσαι γωνίαι ἐπὶ ἴσων angle EHF, have been constructed on the straight-line
περιφερειῶν βεβήκασιν, ὅταν πρὸς τοῖς κέντροις ὦσιν· ἴση BG, at the point G on it [Prop. 1.23]. But equal angles
ἄρα ἡ ΒΚ περιφέρεια τῇ ΕΖ περιφερείᾳ. ἀλλὰ ἡ ΕΖ τῇ ΒΓ (in equal circles) stand upon equal circumferences, when
ἐστιν ἴση· καὶ ἡ ΒΚ ἄρα τῇ ΒΓ ἐστιν ἴση ἡ ἐλάττων τῇ they are at the centers [Prop. 3.26]. Thus, circumference
μείζονι· ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα ἄνισός ἐστιν ἡ ὑπὸ BK (is) equal to circumference EF. But, EF is equal
ΒΗΓ γωνία τῇ ὑπὸ ΕΘΖ· ἴση ἄρα. καί ἐστι τῆς μὲν ὑπὸ to BC. Thus, BK is also equal to BC, the lesser to the
ΒΗΓ ἡμίσεια ἡ πρὸς τῷ Α, τῆς δὲ ὑπὸ ΕΘΖ ἡμίσεια ἡ πρὸς greater. The very thing is impossible. Thus, angle BGC
khþ cumferences are equal to one another, whether they are
τῷ Δ· ἴση ἄρα καὶ ἡ πρὸς τῷ Α γωνία τῇ πρὸς τῷ Δ. is not unequal to EHF. Thus, (it is) equal. And the
᾿Εν ἄρα τοῖς ἴσοις κύκλοις αἱ ἐπὶ ἴσων περιφερειῶν βε- (angle) at A is half BGC, and the (angle) at D half EHF
βηκυῖαι γωνίαι ἴσαι ἀλλήλαις εἰσίν, ἐάν τε πρὸς τοῖς κέντροις [Prop. 3.20]. Thus, the angle at A (is) also equal to the
ἐάν τε πρὸς ταῖς περιφερείαις ὦσι βεβηκυῖαι· ὅπερ ἔδει δεῖξαι. (angle) at D.
Thus, in equal circles, angles standing upon equal cir-

standing at the center or at the circumference. (Which is)
the very thing it was required to show.

.
Proposition 28
᾿Εν τοῖς ἴσοις κύκλοις αἱ ἴσαι εὐθεῖαι ἴσας περιφερείας In equal circles, equal straight-lines cut off equal cir-
ἀφαιροῦσι τὴν μὲν μείζονα τῇ μείζονι τὴν δὲ ἐλάττονα τῇ cumferences, the greater (circumference being equal) to
ἐλάττονι. the greater, and the lesser to the lesser.
῎Εστωσαν ἴσοι κύκλοι οἱ ΑΒΓ, ΔΕΖ, καὶ ἐν τοῖς κύκλοις Let ABC and DEF be equal circles, and let AB
ἴσαι εὐθεῖαι ἔστωσαν αἱ ΑΒ, ΔΕ τὰς μὲν ΑΓΒ, ΑΖΕ περι- and DE be equal straight-lines in these circles, cutting
φερείας μείζονας ἀφαιροῦσαι τὰς δὲ ΑΗΒ, ΔΘΕ ἐλάττονας· off the greater circumferences ACB and DFE, and the
λέγω, ὅτι ἡ μὲν ΑΓΒ μείζων περιφέρεια ἴση ἐστὶ τῇ ΔΖΕ lesser (circumferences) AGB and DHE (respectively). I
μείζονι περιφερείᾳ ἡ δὲ ΑΗΒ ἐλάττων περιφέρεια τῇ ΔΘΕ. say that the greater circumference ACB is equal to the
greater circumference DFE, and the lesser circumfer-

96