Page 470 - Euclid's Elements of Geometry

ST EW iaþ. ljþ ELEMENTS BOOK 11

Proposition 39
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D
B
If there are two equal height prisms, and one has a
᾿Εὰν ᾖ δύο πρίσματα ἰσοϋψῆ, καὶ τὸ μὲν ἔχῇ βάσιν πα-
ραλληλόγραμμον, τὸ δὲ τρίγωνον, διπλάσιον δὲ ᾖ τὸ παραλ- parallelogram, and the other a triangle, (as a) base, and
X

ληλόγραμμον τοῦ τριγώνου, ἴσα ἔσται τὰ πρίσματα. the parallelogram is double the triangle, then the prisms
will be equal.
A E G Z H A B C D O H M L P N

F
E
K
G
῎Εστω δύο πρίσματα ἰσοϋψῆ τὰ ΑΒΓΔΕΖ, ΗΘΚΛΜΝ, Let ABCDEF and GHKLMN be two equal height
καὶ τὸ μὲν ἐχέτω βάσιν τὸ ΑΖ παραλληλόγραμμον, τὸ prisms, and let the former have the parallelogram AF,
δὲ τὸ ΗΘΚ τρίγωνον, διπλάσιον δὲ ἔστω τὸ ΑΖ παραλ- and the latter the triangle GHK, as a base. And let par-
ληλόγραμμον τοῦ ΗΘΚ τριγώνου· λέγω, ὅτι ἴσον ἐστὶ τὸ allelogram AF be twice triangle GHK. I say that prism
ΑΒΓΔΕΖ πρίσμα τῷ ΗΘΚΛΜΝ πρίσματι. ABCDEF is equal to prism GHKLMN.
Συμπεπληρώσθω γὰρ τὰ ΑΞ, ΗΟ στερεά. ἐπεὶ διπλάσιόν For let the solids AO and GP have been com-
ἐστι τὸ ΑΖ παραλληλόγραμμον τοῦ ΗΘΚ τριγώνου, ἔστι pleted. Since parallelogram AF is double triangle GHK,
δὲ καὶ τὸ ΘΚ παραλληλόγραμμον διπλάσιον τοῦ ΗΘΚ and parallelogram HK is also double triangle GHK
τριγώνου, ἴσον ἄρα ἐστὶ τὸ ΑΖ παραλληλόγραμμον τῷ ΘΚ [Prop. 1.34], parallelogram AF is thus equal to paral-
παραλληλογράμμῳ. τὰ δὲ ἐπὶ ἴσων βάσεων ὄντα στερεὰ lelogram HK. And parallelepiped solids which are on
παραλληλεπίπεδα καὶ ὑπὸ τὸ αὐτὸ ὕψος ἴσα ἀλλήλοις ἐστίν· equal bases, and (have) the same height, are equal to
ἴσον ἄρα ἐστὶ τὸ ΑΞ στερεὸν τῷ ΗΟ στερεῷ. καί ἐστι one another [Prop. 11.31]. Thus, solid AO is equal to
τοῦ μὲν ΑΞ στερεοῦ ἥμισυ τὸ ΑΒΓΔΕΖ πρίσμα, τοῦ δὲ solid GP. And prism ABCDEF is half of solid AO, and
ΗΟ στερεοῦ ἥμισυ τὸ ΗΘΚΛΜΝ πρίσμα· ἴσον ἄρα ἐστὶ τὸ prism GHKLMN half of solid GP [Prop. 11.28]. Prism
ΑΒΓΔΕΖ πρίσμα τῷ ΗΘΚΛΜΝ πρίσματι. ABCDEF is thus equal to prism GHKLMN.
᾿Εὰν ἄρα ᾖ δύο πρίσματα ἰσοϋψῆ, καὶ τὸ μὲν ἔχῇ βάσιν Thus, if there are two equal height prisms, and one
παραλληλόγραμμον, τὸ δὲ τρίγωνον, διπλάσιον δὲ ᾖ τὸ πα- has a parallelogram, and the other a triangle, (as a) base,
ραλληλόγραμμον τοῦ τριγώνου, ἴσα ἔστὶ τὰ πρίσματα· ὅπερ and the parallelogram is double the triangle, then the
ἔδει δεῖξαι. prisms are equal. (Which is) the very thing it was re-
quired to show.

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