Page 459 - Euclid's Elements of Geometry

ST EW iaþ.

lbþ and (have) the same height, are equal to one another.

ELEMENTS BOOK 11

(Which is) the very thing it was required to show.

.
Proposition 32
B D
Τὰ ὑπὸ τὸ αὐτὸ ὕψος ὄντα στερεὰ παραλληλεπίπεδα Parallelepiped solids which (have) the same height
πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις. are to one another as their bases. D K
B
E Z

A G H E L F

Let AB and CD be parallelepiped solids (having) the
῎Εστω ὑπὸ τὸ αὐτὸ ὕψος στερεὰ παραλληλεπίπεδα τὰ A C G H
ΑΒ, ΓΔ· λέγω, ὅτι τὰ ΑΒ, ΓΔ στερεὰ παραλληλεπίπεδα same height. I say that the parallelepiped solids AB and
πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις, τουτέστιν ὅτι ἐστὶν ὡς ἡ CD are to one another as their bases. That is to say, as
ΑΕ βάσις πρὸς τὴν ΓΖ βάσιν, οὕτως τὸ ΑΒ στερεὸν πρὸς base AE is to base CF, so solid AB (is) to solid CD.
τὸ ΓΔ στερεόν. For let FH, equal to AE, have been applied to FG (in
Παραβεβλήσθω γὰρ παρὰ τὴν ΖΗ τῷ ΑΕ ἴσον τὸ ΖΘ, the angle FGH equal to angle LCG) [Prop. 1.45]. And
καὶ ἀπὸ βάσεως μὲν τῆς ΖΘ, ὕψους δὲ τοῦ αὐτοῦ τῷ ΓΔ let the parallelepiped solid GK, (having) the same height
στερεὸν παραλληλεπίπεδον συμπεπληρώσθω τὸ ΗΚ. ἴσον as CD, have been completed on the base FH. So solid
δή ἐστι τὸ ΑΒ στερεὸν τῷ ΗΚ στερεῷ· ἐπί τε γὰρ ἴσων AB is equal to solid GK. For they are on the equal bases
βάσεών εἰσι τῶν ΑΕ, ΖΘ καὶ ὑπὸ τὸ αὐτο ὕψος. καὶ ἐπεὶ AE and FH, and (have) the same height [Prop. 11.31].
στερεὸν παραλληλεπίπεδον τὸ ΓΚ ἐπιπέδῳ τῷ ΔΗ τέτμηται And since the parallelepiped solid CK has been cut by
lgþ
παραλλήλῳ ὄντι τοῖς ἀπεναντίον ἐπιπέδοις, ἔστιν ἄρα ὡς ἡ the plane DG, which is parallel to the opposite planes (of
ΓΖ βάσις πρὸς τὴν ΖΘ βάσιν, οὕτως τὸ ΓΔ στερεὸν πρὸς τὸ CK), thus as the base CF is to the base FH, so the solid
ΔΘ στερεόν. ἴση δὲ ἡ μὲν ΖΘ βάσις τῇ ΑΕ βάσει, τὸ δὲ ΗΚ CD (is) to the solid DH [Prop. 11.25]. And base FH (is)
στερεὸν τῷ ΑΒ στερεῷ· ἔστιν ἄρα καὶ ὡς ἡ ΑΕ βάσις πρὸς equal to base AE, and solid GK to solid AB. And thus
τὴν ΓΖ βάσιν, οὕτως τὸ ΑΒ στερεὸν πρὸς τὸ ΓΔ στερεόν. as base AE is to base CF, so solid AB (is) to solid CD.
Τὰ ἄρα ὑπὸ τὸ αὐτὸ ὕψος ὄντα στερεὰ παραλληλεπίπεδα Thus, parallelepiped solids which (have) the same
πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις· ὅπερ ἔδει δεῖξαι. height are to one another as their bases. (Which is) the
very thing it was required to show.

.
Proposition 33
Τὰ ὅμοια στερεὰ παραλληλεπίπεδα πρὸς ἄλληλα ἐν τρι- Similar parallelepiped solids are to one another as the
πλασίονι λόγῳ εἰσὶ τῶν ὁμολόγων πλευρῶν. cubed ratio of their corresponding sides.
῎Εστω ὅμοια στερεὰ παραλληλεπίπεδα τὰ ΑΒ, ΓΔ, Let AB and CD be similar parallelepiped solids, and
ὁμόλογος δὲ ἔστω ἡ ΑΕ τῇ ΓΖ· λέγω, ὅτι τὸ ΑΒ στερεὸν let AE correspond to CF. I say that solid AB has to solid
πρὸς τὸ ΓΔ στερεὸν τριπλασίονα λόγον ἔχει, ἤπερ ἡ ΑΕ CD the cubed ratio that AE (has) to CF.
πρὸς τὴν ΓΖ. For let EK, EL, and EM have been produced in a
᾿Εκβεβλήσθωσαν γὰρ ἐπ᾿ εὐθείας ταῖς ΑΕ, ΗΕ, ΘΕ αἱ straight-line with AE, GE, and HE (respectively). And
ΕΚ, ΕΛ, ΕΜ, καὶ κείσθω τῇ μὲν ΓΖ ἴση ἡ ΕΚ, τῇ δὲ ΖΝ let EK be made equal to CF, and EL equal to FN, and,
ἴση ἡ ΕΛ, καὶ ἔτι τῇ ΖΡ ἴση ἡ ΕΜ, καὶ συμπεπληρώσθω τὸ further, EM equal to FR. And let the parallelogram KL
ΚΛ παραλληλόγραμμον καὶ τὸ ΚΟ στερεόν. have been completed, and the solid KP.

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