Page 426 - Euclid's Elements of Geometry

ST EW iaþ.

ELEMENTS BOOK 11

περιφερείας. radius AB then the diameters (ABD and ABC) will cut
off unequal circumferences of the circle.
Γ C

∆ D

Β B

Α A

bþ

Εὐθείας ἄρα γραμμῆς μέρος μέν τι οὐκ ἔστιν ἐν τῷ ὑπο- Thus, some part of a straight-line cannot be in a refer-
κειμένῳ ἐπιπέδῳ, τὸ δὲ ἐν μετεωροτέρῳ· ὅπερ ἔδει δεῖξαι. ence plane, and (some part) in a more elevated (plane).
(Which is) the very thing it was required to show.
† The proofs of the first three propositions in this book are not at all rigorous. Hence, these three propositions should properly be regarded as
additional axioms.
‡ This assumption essentially presupposes the validity of the proposition under discussion.

.
Proposition 2
᾿Εὰν δύο εὐθεῖαι τέμνωσιν ἀλλήλας, ἐν ἑνί εἰσιν ἐπιπέδῳ, If two straight-lines cut one another then they are in
καὶ πᾶν τρίγωνον ἐν ἑνί ἐστιν ἐπιπέδῳ. one plane, and every triangle (formed using segments of
both lines) is in one plane.
Α ∆ A D

Ε E

Ζ Η F G

Γ Θ Κ Β C H K B
Δύο γὰρ εὐθεῖαι αἱ ΑΒ, ΓΔ τεμνέτωσαν ἀλλήλας κατὰ For let the two straight-lines AB and CD have cut
τὸ Ε σημεῖον. λέγω, ὅτι αἱ ΑΒ, ΓΔ ἐν ἑνί εἰσιν ἐπιπέδῳ, one another at point E. I say that AB and CD are in one
καὶ πᾶν τρίγωνον ἐν ἑνί ἐστιν ἐπιπέδῳ. plane, and that every triangle (formed using segments of
Εἰλήφθω γὰρ ἐπὶ τῶν ΕΓ, ΕΒ τυχόντα σημεῖα τὰ Ζ, Η, both lines) is in one plane.
καὶ ἐπεζεύχθωσαν αἱ ΓΒ, ΖΗ, καὶ διήχθωσαν αἱ ΖΘ, ΗΚ· For let the random points F and G have been taken
λέγω πρῶτον, ὅτι τὸ ΕΓΒ τρίγωνον ἐν ἑνί ἐστιν ἐπιπέδῳ. εἰ on EC and EB (respectively). And let CB and FG
γάρ ἐστι τοῦ ΕΓΒ τριγώνου μέρος ἤτοι τὸ ΖΘΓ ἢ τὸ ΗΒΚ have been joined, and let FH and GK have been drawn
ἐν τῷ ὑποκειμένῳ [ἐπιπέδῳ], τὸ δὲ λοιπὸν ἐν ἄλλῳ, ἔσται καὶ across. I say, first of all, that triangle ECB is in one (ref-
μιᾶς τῶν ΕΓ, ΕΒ εὐθειῶν μέρος μέν τι ἐν τῷ ὑποκειμένῳ erence) plane. For if part of triangle ECB, either FHC

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