Page 97 - Euclid's Elements of Geometry

ST EW gþ.

ence AGB to (the lesser) DHE. ELEMENTS BOOK 3
Γ Ζ C F

Κ Λ K L

Α Β ∆ Ε A B D E

Η Θ G H
Εἰλήφθω γὰρ τὰ κέντρα τῶν κύκλων τὰ Κ, Λ, καὶ For let the centers of the circles, K and L, have been
ἐπεζεύχθωσαν αἱ ΑΚ, ΚΒ, ΔΛ, ΛΕ. found [Prop. 3.1], and let AK, KB, DL, and LE have
Καὶ ἐπεὶ ἴσοι κύκλοι εἰσίν, ἴσαι εἰσὶ καὶ αἱ ἐκ τῶν been joined.
κέντρων· δύο δὴ αἱ ΑΚ, ΚΒ δυσὶ ταῖς ΔΛ, ΛΕ ἴσαι εἰσίν· And since (ABC and DEF) are equal circles, their
καὶ βάσις ἡ ΑΒ βάσει τῇ ΔΕ ἴση· γωνία ἄρα ἡ ὑπὸ ΑΚΒ radii are also equal [Def. 3.1]. So the two (straight-
γωνίᾳ τῇ ὑπὸ ΔΛΕ ἴση ἐστίν. αἱ δὲ ἴσαι γωνίαι ἐπὶ ἴσων lines) AK, KB are equal to the two (straight-lines) DL,
περιφερειῶν βεβήκασιν, ὅταν πρὸς τοῖς κέντροις ὦσιν· ἴση LE (respectively). And the base AB (is) equal to the
ἄρα ἡ ΑΗΒ περιφέρεια τῇ ΔΘΕ. ἐστὶ δὲ καὶ ὅλος ὁ ΑΒΓ base DE. Thus, angle AKB is equal to angle DLE
κύκλος ὅλῳ τῷ ΔΕΖ κύκλῳ ἴσος· καὶ λοιπὴ ἄρα ἡ ΑΓΒ [Prop. 1.8]. And equal angles stand upon equal circum-
περιφέρεια λοιπῇ τῇ ΔΖΕ περιφερείᾳ ἴση ἐστίν. ferences, when they are at the centers [Prop. 3.26]. Thus,
kjþ equal circumferences, the greater (circumference being
᾿Εν ἄρα τοῖς ἴσοις κύκλοις αἱ ἴσαι εὐθεῖαι ἴσας πε- circumference AGB (is) equal to DHE. And the whole
ριφερείας ἀφαιροῦσι τὴν μὲν μείζονα τῇ μείζονι τὴν δὲ circle ABC is also equal to the whole circle DEF. Thus,
ἐλάττονα τῇ ἐλάττονι· ὅπερ ἔδει δεῖξαι. the remaining circumference ACB is also equal to the
remaining circumference DFE.
Thus, in equal circles, equal straight-lines cut off

equal) to the greater, and the lesser to the lesser. (Which
is) the very thing it was required to show.

.
Proposition 29
᾿Εν τοῖς ἴσοις κύκλοις τὰς ἴσας περιφερείας ἴσαι εὐθεῖαι In equal circles, equal straight-lines subtend equal cir-
ὑποτείνουσιν. cumferences.
Α ∆ A D

Κ Λ K L

Β Γ Ε Ζ B C E F

Η Θ G H
῎Εστωσαν ἴσοι κύκλοι οἱ ΑΒΓ, ΔΕΖ, καὶ ἐν αὐτοῖς ἴσαι Let ABC and DEF be equal circles, and within them
περιφέρειαι ἀπειλήφθωσαν αἱ ΒΗΓ, ΕΘΖ, καὶ ἐπεζεύχθωσαν let the equal circumferences BGC and EHF have been
αἱ ΒΓ, ΕΖ εὐθεῖαι· λέγω, ὅτι ἴση ἐστὶν ἡ ΒΓ τῇ ΕΖ. cut off. And let the straight-lines BC and EF have been
Εἰλήφθω γὰρ τὰ κέντρα τῶν κύκλων, καὶ ἔστω τὰ Κ, joined. I say that BC is equal to EF.
Λ, καὶ ἐπεζεύχθωσαν αἱ ΒΚ, ΚΓ, ΕΛ, ΛΖ. For let the centers of the circles have been found
Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΒΗΓ περιφέρεια τῇ ΕΘΖ περιφερείᾳ, [Prop. 3.1], and let them be (at) K and L. And let BK,

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