Page 456 - Euclid's Elements of Geometry

ST EW iaþ.

ELEMENTS BOOK 11

αὐτῶν εὐθειῶν· λέγω, ὅτι ἴσον ἐστὶ τὸ ΓΜ στερεὸν τῷ ΓΝ LM, LN, CD, CE, BH, and BK, not be on the same
στερεῷ. straight-lines. I say that the solid CM is equal to the
᾿Εκβεβλήσθωσαν γὰρ αἱ ΝΚ, ΔΘ καὶ συμπιπτέτωσαν solid CN.
ἀλλήλαις κατὰ τὸ Ρ, καὶ ἔτι ἐκβεβλήσθωσαν αἱ ΖΜ, ΗΕ For let NK and DH have been produced, and let
ἐπὶ τὰ Ο, Π, καὶ ἐπεζεύχθωσαν αἱ ΑΞ, ΛΟ, ΓΠ, ΒΡ. ἴσον them have joined one another at R. And, further, let FM
δή ἐστι τὸ ΓΜ στερεόν, οὗ βάσις μὲν τὸ ΑΓΒΛ παραλ- and GE have been produced to P and Q (respectively).
ληλόγραμμον, ἀπεναντίον δὲ τὸ ΖΔΘΜ, τῷ ΓΟ στερεῷ, οὗ And let AO, LP, CQ, and BR have been joined. So, solid
βάσις μὲν τὸ ΑΓΒΛ παραλληλόγραμμον, ἀπεναντίον δὲ τὸ CM, whose base (is) parallelogram ACBL, and oppo-
ΞΠΡΟ· ἐπί τε γὰρ τῆς αὐτῆς βάσεώς εἰσι τῆς ΑΓΒΛ καὶ site (face) FDHM, is equal to solid CP, whose base (is)
ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι αἱ ΑΖ, ΑΞ, ΛΜ, ΛΟ, parallelogram ACBL, and opposite (face) OQRP. For
ΓΔ, ΓΠ, ΒΘ, ΒΡ ἐπὶ τῶν αὐτῶν εἰσιν εὐθειῶν τῶν ΖΟ, ΔΡ. they are on the same base, ACBL, and (have) the same
ἀλλὰ τὸ ΓΟ στερεόν, οὗ βάσις μέν ἐστι τὸ ΑΓΒΛ παραλ- height, and the (ends of the straight-lines) standing up in
ληλόγραμμον, ἀπεναντίον δὲ τὸ ΞΠΡΟ, ἴσον ἐστὶ τῷ ΓΝ them, AF, AO, LM, LP, CD, CQ, BH, and BR, are on
στερεῷ, οὗ βάσις μὲν τὸ ΑΓΒΛ παραλληλόγραμμον, ἀπε- the same straight-lines, FP and DR [Prop. 11.29]. But,
ναντίον δὲ τὸ ΗΕΚΝ· ἐπί τε γὰρ πάλιν τῆς αὐτῆς βάσεώς solid CP, whose base is parallelogram ACBL, and oppo-
εἰσι τῆς ΑΓΒΛ καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι αἱ site (face) OQRP, is equal to solid CN, whose base (is)
ΑΗ, ΑΞ, ΓΕ, ΓΠ, ΛΝ, ΛΟ, ΒΚ, ΒΡ ἐπὶ τῶν αὐτῶν εἰσιν parallelogram ACBL, and opposite (face) GEKN. For,
εὐθειῶν τῶν ΗΠ, ΝΡ. ὥστε καὶ τὸ ΓΜ στερεὸν ἴσον ἐστὶ again, they are on the same base, ACBL, and (have)
τῷ ΓΝ στερεῷ. the same height, and the (ends of the straight-lines)
Τὰ ἄρα ἐπὶ τῆς αὐτῆς βάσεως στερεὰ παραλληλεπίπεδα standing up in them, AG, AO, CE, CQ, LN, LP, BK,
laþ base, and (have) the same height, and in which the (ends
καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι οὐκ εἰσὶν ἐπὶ τῶν and BR, are on the same straight-lines, GQ and NR
αὐτῶν εὐθειῶν, ἴσα ἀλλήλοις ἐστίν· ὅπερ ἔδει δεῖξαι. [Prop. 11.29]. Hence, solid CM is also equal to solid
CN.
Thus, parallelepiped solids (which are) on the same

of the straight-lines) standing up are not on the same
straight-lines, are equal to one another. (Which is) the
very thing it was required to show.

Proposition 31
.
Τὰ ἐπὶ ἴσων βάσεων ὄντα στερεὰ παραλληλεπίπεδα καὶ Parallelepiped solids which are on equal bases, and
ὑπὸ τὸ αὐτὸ ὕψος ἴσα ἀλλήλοις ἐστίν. (have) the same height, are equal to one another.
῎Εστω ἐπὶ ἴσων βάσεων τῶν ΑΒ, ΓΔ στερεὰ παραλλη- Let the parallelepiped solids AE and CF be on the
λεπίπεδα τὰ ΑΕ, ΓΖ ὑπὸ τὸ αὐτὸ ὕψος. λέγω, ὅτι ἴσον ἐστὶ equal bases AB and CD (respectively), and (have) the
τὸ ΑΕ στερεὸν τῷ ΓΖ στερεῷ. same height. I say that solid AE is equal to solid CF.
῎Εστωσαν δὴ πρότερον αἱ ἐφεστηκυῖαι αἱ ΘΚ, ΒΕ, ΑΗ, So, let the (straight-lines) standing up, HK, BE, AG,
ΛΜ, ΟΠ, ΔΖ, ΓΞ, ΡΣ πρὸς ὀρθὰς ταῖς ΑΒ, ΓΔ βάσεσιν, LM, PQ, DF, CO, and RS, first of all, be at right-angles
καὶ ἐκβεβλήσθω ἐπ᾿ εὐθείας τῇ ΓΡ εὐθεῖα ἡ ΡΤ, καὶ συ- to the bases AB and CD. And let RT have been produced
νεστάτω πρὸς τῇ ΡΤ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ in a straight-line with CR. And let (angle) T RU, equal
Ρ τῇ ὑπὸ ΑΛΒ γωνίᾳ ἴση ἡ ὑπὸ ΤΡΥ, καὶ κείσθω τῇ μὲν to angle ALB, have been constructed on the straight-line
ΑΛ ἴση ἡ ΡΤ, τῇ δὲ ΛΒ ἴση ἡ ΡΥ, καὶ συμπεπληρώσθω ἥ RT , at the point R on it [Prop. 1.23]. And let RT be
τε ΡΧ βάσις καὶ τὸ ΨΥ στερεόν. made equal to AL, and RU to LB. And let the base RW,
and the solid XU, have been completed.

456