Page 455 - Euclid's Elements of Geometry

ST EW iaþ.

ELEMENTS BOOK 11

λεπίπεδα τὰ ΓΜ, ΓΝ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι the same base AB, and (have) the same height, and let
αἱ ΑΗ, ΑΖ, ΛΜ, ΛΝ, ΓΔ, ΓΕ, ΒΘ, ΒΚ ἐπὶ τῶν αὐτῶν the (ends of the straight-lines) standing up in them, AG,
εὐθειῶν ἔστωσαν τῶν ΖΝ, ΔΚ· λέγω, ὅτι ἴσον ἐστὶ τὸ ΓΜ AF, LM, LN, CD, CE, BH, and BK, be on the same
στερεὸν τῷ ΓΝ στερεῷ. straight-lines, FN and DK. I say that solid CM is equal
᾿Επεὶ γὰρ παραλληλόγραμμόν ἐστιν ἑκάτερον τῶν ΓΘ, to solid CN.
ΓΚ, ἴση ἐστὶν ἡ ΓΒ ἑκατέρᾳ τῶν ΔΘ, ΕΚ· ὥστε καὶ ἡ For since CH and CK are each parallelograms, CB
ΔΘ τῇ ΕΚ ἐστιν ἴση. κοινὴ ἀφῃρήσθω ἡ ΕΘ· λοιπὴ ἄρα is equal to each of DH and EK [Prop. 1.34]. Hence,
ἡ ΔΕ λοιπῇ τῇ ΘΚ ἐστιν ἴση. ὥστε καὶ τὸ μὲν ΔΓΕ DH is also equal to EK. Let EH have been subtracted
τρίγωνον τῷ ΘΒΚ τριγώνῳ ἴσον ἐστίν, τὸ δὲ ΔΗ παραλ- from both. Thus, the remainder DE is equal to the re-
ληλόγραμμον τῷ ΘΝ παραλληλογράμμῳ. διὰ τὰ αὐτὰ δὴ καὶ mainder HK. Hence, triangle DCE is also equal to tri-
τὸ ΑΖΗ τρίγωνον τῷ ΜΛΝ τριγώνῳ ἴσον ἐστίν. ἔστι δὲ καὶ angle HBK [Props. 1.4, 1.8], and parallelogram DG to
τὸ μὲν ΓΖ παραλληλόγραμμον τῷ ΒΜ παραλληλογράμμῳ parallelogram HN [Prop. 1.36]. So, for the same (rea-
ἴσον, τὸ δὲ ΓΗ τῷ ΒΝ· ἀπεναντίον γάρ· καὶ τὸ πρίσμα ἄρα sons), traingle AFG is also equal to triangle MLN. And
τὸ περιεχόμενον ὑπὸ δύο μὲν τριγώνων τῶν ΑΖΗ, ΔΓΕ, parallelogram CF is also equal to parallelogram BM,
τριῶν δὲ παραλληλογράμμων τῶν ΑΔ, ΔΗ, ΓΗ ἴσον ἐστὶ and CG to BN [Prop. 11.24]. For they are opposite.
τῷ πρίσματι τῷ περιεχομένῳ ὑπὸ δύο μὲν τριγώνων τῶν Thus, the prism contained by the two triangles AFG and
ΜΛΝ, ΘΒΚ, τριῶν δὲ παραλληλογράμμων τῶν ΒΜ, ΘΝ, DCE, and the three parallelograms AD, DG, and CG, is
ΒΝ. κοινὸν προσκείσθω τὸ στερεὸν, οὗ βάσις μὲν τὸ ΑΒ equal to the prism contained by the two triangles MLN
παραλληλόγραμμον, ἀπεναντίον δὲ τὸ ΗΕΘΜ· ὅλον ἄρα τὸ and HBK, and the three parallelograms BM, HN, and
ΓΜ στερεὸν παραλληλεπίπεδον ὅλῳ τῷ ΓΝ στερεῷ παραλ- BN. Let the solid whose base (is) parallelogram AB, and
lþ base, and (have) the same height, and in which the
ληλεπιπέδῳ ἴσον ἐστίν. (whose) opposite (face is) GEHM, have been added to
Τὰ ἄρα ἐπὶ τῆς αὐτῆς βάσεως ὄντα στερεὰ παραλλη- both (prisms). Thus, the whole parallelepiped solid CM
λεπίπεδα καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι ἐπὶ τῶν is equal to the whole parallelepiped solid CN.
αὐτῶν εἰσιν εὐθειῶν, ἴσα ἀλλήλοις ἐστίν· ὅπερ ἔδει δεῖξαι. Thus, parallelepiped solids which are on the same

(ends of the straight-lines) standing up (are) on the same
straight-lines, are equal to one another. (Which is) the
very thing it was required to show.

.
Proposition 30
R
Τὰ ἐπὶ τῆς αὐτῆς βάσεως ὄντα στερεὰ παραλληλεπίπεδα Parallelepiped solids which are on the same base, and

καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι οὐκ εἰσὶν ἐπὶ τῶν (have) the same height, and in which the (ends of the
H
X E
αὐτῶν εὐθειῶν, ἴσα ἀλλήλοις ἐστίν. straight-lines) standing up are not on the same straight-
lines, are equal to one another.
N
K
P
R
M
H
A Z B G D L G O F B E Q D

C
A
῎Εστω ἐπὶ τῆς αὐτῆς βάσεως τῆς ΑΒ στερεὰ παραλλη- Let the parallelepiped solids CM and CN be on the
λεπίπεδα τὰ ΓΜ, ΓΝ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι αἱ same base, AB, and (have) the same height, and let the
ΑΖ, ΑΗ, ΛΜ, ΛΝ, ΓΔ, ΓΕ, ΒΘ, ΒΚ μὴ ἔστωσαν ἐπὶ τῶν (ends of the straight-lines) standing up in them, AF, AG,

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