Page 10 - Euclid's Elements of Geometry

ST EW aþ.

ELEMENTS BOOK 1

ἴση ἐστὶν ἡ ΑΕ τῇ ΑΔ· ἀλλὰ καὶ ἡ Γ τῇ ΑΔ ἐστιν ἴση. And since point A is the center of circle DEF, AE
ἑκατέρα ἄρα τῶν ΑΕ, Γ τῇ ΑΔ ἐστιν ἴση· ὥστε καὶ ἡ ΑΕ is equal to AD [Def. 1.15]. But, C is also equal to AD.
τῇ Γ ἐστιν ἴση. Thus, AE and C are each equal to AD. So AE is also
equal to C [C.N. 1].
Γ C

∆ D

Ε E
Α Β A B
dþ

Ζ F
Δύο ἄρα δοθεισῶν εὐθειῶν ἀνίσων τῶν ΑΒ, Γ ἀπὸ τῆς Thus, for two given unequal straight-lines, AB and C,
μείζονος τῆς ΑΒ τῇ ἐλάσσονι τῇ Γ ἴση ἀφῄρηται ἡ ΑΕ· ὅπερ the (straight-line) AE, equal to the lesser C, has been cut
ἔδει ποιῆσαι. off from the greater AB. (Which is) the very thing it was
required to do.

Proposition 4
.
᾿Εὰν δύο τρίγωνα τὰς δύο πλευρὰς [ταῖς] δυσὶ πλευραῖς If two triangles have two sides equal to two sides, re-
ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ καὶ τὴν γωνίαν τῇ γωνίᾳ ἴσην spectively, and have the angle(s) enclosed by the equal
ἔχῃ τὴν ὑπὸ τῶν ἴσων εὐθειῶν περιεχομένην, καὶ τὴν straight-lines equal, then they will also have the base
βάσιν τῂ βάσει ἴσην ἕξει, καὶ τὸ τρίγωνον τῷ τριγώνῳ ἴσον equal to the base, and the triangle will be equal to the tri-
ἔσται, καὶ αἱ λοιπαὶ γωνίαι ταῖς λοιπαῖς γωνίαις ἴσαι ἔσονται angle, and the remaining angles subtended by the equal
ἑκατέρα ἑκατέρᾳ, ὑφ᾿ ἃς αἱ ἴσαι πλευραὶ ὑποτείνουσιν. sides will be equal to the corresponding remaining an-
gles.
Α ∆ A D

Β Γ Ε Ζ B C E F

῎Εστω δύο τρίγωνα τὰ ΑΒΓ, ΔΕΖ τὰς δύο πλευρὰς Let ABC and DEF be two triangles having the two
τὰς ΑΒ, ΑΓ ταῖς δυσὶ πλευραῖς ταῖς ΔΕ, ΔΖ ἴσας ἔχοντα sides AB and AC equal to the two sides DE and DF, re-
ἑκατέραν ἑκατέρᾳ τὴν μὲν ΑΒ τῇ ΔΕ τὴν δὲ ΑΓ τῇ ΔΖ spectively. (That is) AB to DE, and AC to DF. And (let)
καὶ γωνίαν τὴν ὑπὸ ΒΑΓ γωνίᾳ τῇ ὑπὸ ΕΔΖ ἴσην. λέγω, the angle BAC (be) equal to the angle EDF. I say that
ὅτι καὶ βάσις ἡ ΒΓ βάσει τῇ ΕΖ ἴση ἐστίν, καὶ τὸ ΑΒΓ the base BC is also equal to the base EF, and triangle
τρίγωνον τῷ ΔΕΖ τριγώνῳ ἴσον ἔσται, καὶ αἱ λοιπαὶ γωνίαι ABC will be equal to triangle DEF, and the remaining
ταῖς λοιπαῖς γωνίαις ἴσαι ἔσονται ἑκατέρα ἑκατέρᾳ, ὑφ᾿ ἃς angles subtended by the equal sides will be equal to the
αἱ ἴσαι πλευραὶ ὑποτείνουσιν, ἡ μὲν ὑπὸ ΑΒΓ τῇ ὑπὸ ΔΕΖ, corresponding remaining angles. (That is) ABC to DEF,
ἡ δὲ ὑπὸ ΑΓΒ τῇ ὑπὸ ΔΖΕ. and ACB to DFE.
†
᾿Εφαρμοζομένου γὰρ τοῦ ΑΒΓ τριγώνου ἐπὶ τὸ ΔΕΖ For if triangle ABC is applied to triangle DEF, the
τρίγωνον καὶ τιθεμένου τοῦ μὲν Α σημείου ἐπὶ τὸ Δ σημεῖον point A being placed on the point D, and the straight-line

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