Page 432 - Euclid's Elements of Geometry

ST EW iaþ.

ELEMENTS BOOK 11

ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ εὐθεῖαν. ποιείτω ὡς τὴν ΕΖ· δύο (plane), such as EGF. And let a plane have been drawn
ἄρα εὐθεῖαι αἱ ΕΗΖ, ΕΖ χωρίον περιέξουσιν· ὅπερ ἐστὶν through EGF. So it will make a straight cutting in the
ἀδύνατον. οὐκ ἄρα ἡ ἀπὸ τοῦ Ε ἐπὶ τὸ Ζ ἐπιζευγνυμένη reference plane [Prop. 11.3]. Let it make EF. Thus, two
εὐθεῖαι ἐν μετεωροτέρῳ ἐστὶν ἐπιπέδῳ· ἐν τῷ διὰ τῶν ΑΒ, straight-lines (with the same end-points), EGF and EF,
ΓΒ ἄρα παραλλήλων ἐστὶν ἐπιπέδῳ ἡ ἀπὸ τοῦ Ε ἐπὶ τὸ Ζ will enclose an area. The very thing is impossible. Thus,
hþ line joining the two points is in the same plane as the
ἐπιζευγνυμένη εὐθεῖα. the straight-line joining E to F is not in a more elevated
᾿Εὰν ἄρα ὦσι δύο εὐθεῖαι παράλληλοι, ληφθῇ δὲ ἐφ᾿ plane. The straight-line joining E to F is thus in the plane
ἑκατέρας αὐτῶν τυχόντα σημεῖα, ἡ ἐπὶ τὰ σημεῖα ἐπιζευ- through the parallel (straight-lines) AB and CD.
γνυμένη εὐθεῖα ἐν τῷ αὐτῷ ἐπιπέδῳ ἐστὶ ταῖς παραλλήλοις· Thus, if there are two parallel straight-lines, and ran-
ὅπερ ἔδει δεῖξαι. dom points are taken on each of them, then the straight-

parallel (straight-lines). (Which is) the very thing it was
required to show.
Proposition 8
.
A G
᾿Εὰν ὦσι δύο εὐθεῖαι παράλληλοι, ἡ δὲ ἑτέρα αὐτῶν If two straight-lines are parallel, and one of them is at
ἐπιπέδῳ τινὶ πρὸς ὀρθὰς ᾖ, καὶ ἡ λοιπὴ τῷ αὐτῷ ἐπιπέδῳ right-angles to some plane, then the remaining (one) will
πρὸς ὀρθὰς ἔσται. also be at right-angles to the same plane.
C
A

B D

E B D

E
῎Εστωσαν δύο εὐθεῖαι παράλληλοι αἱ ΑΒ, ΓΔ, ἡ δὲ ἑτέρα Let AB and CD be two parallel straight-lines, and let
αὐτῶν ἡ ΑΒ τῷ ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθὰς ἔστω· one of them, AB, be at right-angles to a reference plane.
λέγω, ὅτι καὶ ἡ λοιπὴ ἡ ΓΔ τῷ αὐτῷ ἐπιπέδῳ πρὸς ὀρθὰς I say that the remaining (one), CD, will also be at right-
ἔσται. angles to the same plane.
Συμβαλλέτωσαν γὰρ αἱ ΑΒ, ΓΔ τῷ ὑποκειμένῳ ἐπιπέδῳ For let AB and CD meet the reference plane at points
κατὰ τὰ Β, Δ σημεῖα, καὶ ἐπεζέυχθω ἡ ΒΔ· αἱ ΑΒ, ΓΔ, ΒΔ B and D (respectively). And let BD have been joined.
ἄρα ἐν ἑνί εἰσιν ἐπιπέδῳ. ἤχθω τῇ ΒΑ πρὸς ὀρθὰς ἐν τῷ AB, CD, and BD are thus in one plane [Prop. 11.7].
ὑποκειμένῳ ἐπιπέδῳ ἡ ΔΕ, καὶ κείσθω τῇ ΑΒ ἴση ἡ ΔΕ, Let DE have been drawn at right-angles to BD in the
καὶ ἐπεζεύχθωσαν αἱ ΒΕ, ΑΕ, ΑΔ. reference plane, and let DE be made equal to AB, and
Καὶ ἐπεὶ ἡ ΑΒ ὁρθή ἐστι πρὸς τὸ ὑποκείμενον ἐπίπεδον, let BE, AE, and AD have been joined.
καὶ πρὸς πάσας ἄρα τὰς ἁπτομένας αὐτῆς εὐθείας καὶ οὔσας And since AB is at right-angles to the reference
ἐν τῷ ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν ἡ ΑΒ· ὀρθὴ ἄρα plane, AB is thus also at right-angles to all of the
[ἐστὶν] ἑκατέρα τῶν ὑπὸ ΑΒΔ, ΑΒΕ γωνιῶν. καὶ ἐπεὶ εἰς straight-lines joined to it which are in the reference plane
παραλλήλους τὰς ΑΒ, ΓΔ εὐθεῖα ἐμπέπτωκεν ἡ ΒΔ, αἱ ἄρα [Def. 11.3]. Thus, the angles ABD and ABE [are] each
ὑπὸ ΑΒΔ, ΓΔΒ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν. ὀρθὴ δὲ ἡ right-angles. And since the straight-line BD has met the
ὑπὸ ΑΒΔ· ὀρθὴ ἄρα καὶ ἡ ὑπὸ ΓΔΒ· ἡ ΓΔ ἄρα πρὸς τὴν ΒΔ parallel (straight-lines) AB and CD, the (sum of the)
ὀρθή ἐστιν. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΒ τῇ ΔΕ, κοινὴ δὲ ἡ ΒΔ, angles ABD and CDB is thus equal to two right-angles

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