Page 95 - Euclid's Elements of Geometry

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ELEMENTS BOOK 3

τῷ Α τῇ ὑπὸ ΑΒΔ γωνίᾳ ἴσην, ἐντὸς τοῦ ΑΒΓ τμήματος [And], similarly, even if angle ABD is equal to BAD,
πεσεῖται τὸ κέντρον ἐπὶ τῆς ΔΒ, καὶ ἔσται δηλαδὴ τὸ ΑΒΓ (since) AD becomes equal to each of BD [Prop. 1.6] and
τμῆμα μεῖζον ἡμικυκλίου. DC, the three (straight-lines) DA, DB, and DC will be
Κύκλου ἄρα τμήματος δοθέντος προσαναγέγραπται ὁ equal to one another. And point D will be the center
κύκλος· ὅπερ ἔδει ποιῆσαι. of the completed circle. And ABC will manifestly be a
semi-circle.
kþ at the point A on it [Prop. 1.23], then the center will fall
And if ABD is less than BAD, and we construct (an-
gle BAE), equal to angle ABD, on the straight-line BA,

on DB, inside the segment ABC. And segment ABC will
manifestly be greater than a semi-circle.
Thus, a circle has been completed from the given seg-
ment of a circle. (Which is) the very thing it was required
to do.

.
Proposition 26
᾿Εν τοῖς ἴσοις κύκλοις αἱ ἴσαι γωνίαι ἐπὶ ἴσων περιφε- In equal circles, equal angles stand upon equal cir-
ρειῶν βεβήκασιν, ἐάν τε πρὸς τοῖς κέντροις ἐάν τε πρὸς cumferences whether they are standing at the center or
ταῖς περιφερείαις ὦσι βεβηκυῖαι. at the circumference.
Α A

∆ D
Η Θ G H

Β Γ Ε Ζ B C E F

Κ Λ K L
῎Εστωσαν ἴσοι κύκλοι οἱ ΑΒΓ, ΔΕΖ καὶ ἐν αὐτοῖς ἴσαι Let ABC and DEF be equal circles, and within them
γωνίαι ἔστωσαν πρὸς μὲν τοῖς κέντροις αἱ ὑπὸ ΒΗΓ, ΕΘΖ, let BGC and EHF be equal angles at the center, and
πρὸς δὲ ταῖς περιφερείαις αἱ ὑπὸ ΒΑΓ, ΕΔΖ· λέγω, ὅτι ἴση BAC and EDF (equal angles) at the circumference. I
ἐστὶν ἡ ΒΚΓ περιφέρεια τῇ ΕΛΖ περιφερείᾳ. say that circumference BKC is equal to circumference
᾿Επεζεύχθωσαν γὰρ αἱ ΒΓ, ΕΖ. ELF.
Καὶ ἐπεὶ ἴσοι εἰσὶν οἱ ΑΒΓ, ΔΕΖ κύκλοι, ἴσαι εἰσὶν αἱ For let BC and EF have been joined.
ἐκ τῶν κέντρων· δύο δὴ αἱ ΒΗ, ΗΓ δύο ταῖς ΕΘ, ΘΖ ἴσαι· And since circles ABC and DEF are equal, their radii
καὶ γωνία ἡ πρὸς τῷ Η γωνίᾳ τῇ πρὸς τῷ Θ ἴση· βάσις ἄρα are equal. So the two (straight-lines) BG, GC (are) equal
ἡ ΒΓ βάσει τῇ ΕΖ ἐστιν ἴση. καὶ ἐπεὶ ἴση ἐστὶν ἡ πρὸς τῷ to the two (straight-lines) EH, HF (respectively). And
Α γωνία τῇ πρὸς τῷ Δ, ὅμοιον ἄρα ἐστὶ τὸ ΒΑΓ τμῆμα τῷ the angle at G (is) equal to the angle at H. Thus, the base
ΕΔΖ τμήματι· καί εἰσιν ἐπὶ ἴσων εὐθειῶν [τῶν ΒΓ, ΕΖ]· τὰ BC is equal to the base EF [Prop. 1.4]. And since the
δὲ ἐπὶ ἴσων εὐθειῶν ὅμοια τμήματα κύκλων ἴσα ἀλλήλοις angle at A is equal to the (angle) at D, the segment BAC
ἐστίν· ἴσον ἄρα τὸ ΒΑΓ τμῆμα τῷ ΕΔΖ. ἔστι δὲ καὶ ὅλος ὁ is thus similar to the segment EDF [Def. 3.11]. And
ΑΒΓ κύκλος ὅλῳ τῷ ΔΕΖ κύκλῳ ἴσος· λοιπὴ ἄρα ἡ ΒΚΓ they are on equal straight-lines [BC and EF]. And simi-
περιφέρεια τῇ ΕΛΖ περιφερείᾳ ἐστὶν ἴση. lar segments of circles on equal straight-lines are equal to
᾿Εν ἄρα τοῖς ἴσοις κύκλοις αἱ ἴσαι γωνίαι ἐπὶ ἴσων περι- one another [Prop. 3.24]. Thus, segment BAC is equal to
φερειῶν βεβήκασιν, ἐάν τε πρὸς τοῖς κέντροις ἐάν τε πρὸς (segment) EDF. And the whole circle ABC is also equal
ταῖς περιφερείας ὦσι βεβηκυῖαι· ὅπερ ἔδει δεῖξαι. to the whole circle DEF. Thus, the remaining circum-
ference BKC is equal to the (remaining) circumference
ELF.
Thus, in equal circles, equal angles stand upon equal
circumferences, whether they are standing at the center

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