Page 442 - Euclid's Elements of Geometry

ST EW iaþ.

ELEMENTS BOOK 11

τῷ ΔΕ πρὸς ὀρθὰς ἀχθεῖσα ἡ ΖΗ ἐδείχθη τῷ ὑποκειμένῳ at right-angles to the reference plane [Prop. 11.8]. And
ἐπιπέδῳ πρὸς ὀρθάς· τὸ ἄρα ΔΕ ἐπίπεδον ὀρθόν ἐστι πρὸς a plane is at right-angles to a(nother) plane when the
τὸ ὑποκείμενον. ὁμοίως δὴ δειχθήσεται καὶ πάντα τὰ διὰ straight-lines drawn at right-angles to the common sec-
τῆς ΑΒ ἐπίπεδα ὀρθὰ τυγχανοντα πρὸς τὸ ὑποκείμενον tion of the planes, (and lying) in one of the planes, are
ἐπίπεδον. at right-angles to the remaining plane [Def. 11.4]. And
FG, (which was) drawn at right-angles to the common
section of the planes, CE, in one of the planes, DE, was
D H A plane DE is at right-angles to the reference (plane). So,
shown to be at right-angles to the reference plane. Thus,
similarly, it can be shown that all of the planes (passing)
at random through AB (are) at right-angles to the refer-
ence plane.
G
A
D
G Z B E C F B E

ijþ

᾿Εὰν ἄρα εὐθεῖα ἐπιπέδῳ τινὶ πρὸς ὀρθὰς ᾖ, καὶ πάντα τὰ Thus, if a straight-line is at right-angles to some plane
δι᾿ αὐτῆς ἐπίπεδα τῷ αὐτῷ ἐπιπέδῳ πρὸς ὀρθὰς ἔσται· ὅπερ then all of the planes (passing) through it will also be at
ἔδει δεῖξαι. right-angles to the same plane. (Which is) the very thing
it was required to show.

.
Proposition 19
᾿Εὰν δύο ἐπίπεδα τέμνοντα ἄλληλα ἐπιπέδῳ τινὶ πρὸς If two planes cutting one another are at right-angles
B Z E B
ὀρθὰς ᾖ, καὶ ἡ κοινὴ αὐτῶν τομὴ τῷ αὐτῷ ἐπιπέδῳ πρὸς to some plane then their common section will also be at
ὀρθὰς ἔσται. right-angles to the same plane.

D E F

A G D

C
A
Δύο γὰρ ἐπίπεδα τὰ ΑΒ, ΒΓ τῷ ὑποκειμένῳ ἐπιπέδῳ For let the two planes AB and BC be at right-angles
πρὸς ὀρθὰς ἔστω, κοινὴ δὲ αὐτῶν τομὴ ἔστω ἡ ΒΔ· λέγω, to a reference plane, and let their common section be
ὅτι ἡ ΒΔ τῷ ὑποκειμένῳ ἐπιπέδῳ πρὸς ὀρθάς ἐστιν. BD. I say that BD is at right-angles to the reference

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