Page 461 - Euclid's Elements of Geometry

ST EW iaþ.

ELEMENTS BOOK 11

B X corresponding side AE (has) to the corresponding side
πλασίονα λόγον ἔχει ἤπερ ἡ ὁμόλογος αὐτοῦ πλευρὰ ἡ ΑΕ equal to solid CD, and straight-line EK to CF. Thus,
πρὸς τὴν ὁμόλογον πλευρὰν τὴν ΓΖ. solid AB also has to solid CD the cubed ratio which its

CF.
O
B
H H Q
E
A A G E L K

ìri sma M P

Τὰ ἄρα ὅμοια στερεὰ παραλληλεπίπεδα ἐν τριπλασίονι Thus, similar parallelepiped solids are to one another
λόγῳ ἐστὶ τῶν ὁμολόγων πλευρῶν· ὅπερ ἔδει δεῖξαι. as the cubed ratio of their corresponding sides. (Which
is) the very thing it was required to show.
ldþ
.
Corollary
᾿Εκ δὴ τούτου φανερόν, ὅτι ἐὰν τέσσαρες εὐθεῖαι So, (it is) clear, from this, that if four straight-lines are
ἀνάλογον ὦσιν, ἔσται ὡς ἡ πρώτη πρὸς τὴν τετάρτην, οὕτω (continuously) proportional then as the first is to the
τὸ ἀπὸ τῆς πρώτης στερεὸν παραλληλεπίπεδον πρὸς τὸ ἀπὸ fourth, so the parallelepiped solid on the first will be to
τῆς δευτέρας τὸ ὅμοιον καὶ ὁμοίως ἀναγραφόμενον, ἐπείπερ the similar, and similarly described, parallelepiped solid
καὶ ἡ πρώτη πρὸς τὴν τετάρτην τριπλασίονα λόγον ἔχει ἤπερ on the second, since the first also has to the fourth the
πρὸς τὴν δευτέραν. cubed ratio that (it has) to the second.

.
Proposition 34
†
Τῶν ἴσων στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν The bases of equal parallelepiped solids are recip-
αἱ βάσεις τοῖς ὕψεσιν· καὶ ὧν στερεῶν παραλληλεπιπέδων rocally proportional to their heights. And those paral-
ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν, ἵσα ἐστὶν ἐκεῖνα. lelepiped solids whose bases are reciprocally proportional
῎Εστω ἴσα στερεὰ παραλληλεπίπεδα τὰ ΑΒ, ΓΔ· λέγω, to their heights are equal.
ὅτι τῶν ΑΒ, ΓΔ στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν Let AB and CD be equal parallelepiped solids. I say
αἱ βάσεις τοῖς ὕψεσιν, καί ἐστιν ὡς ἡ ΕΘ βάσις πρὸς τὴν that the bases of the parallelepiped solids AB and CD
ΝΠ βάσιν, οὕτως τὸ τοῦ ΓΔ στερεοῦ ὕψος πρὸς τὸ τοῦ ΑΒ are reciprocally proportional to their heights, and (so) as
στερεοῦ ὕψος. base EH is to base NQ, so the height of solid CD (is) to
῎Εστωσαν γὰρ πρότερον αἱ ἐφεστηκυῖαι αἱ ΑΗ, ΕΖ, ΛΒ, the height of solid AB.
ΘΚ, ΓΜ, ΝΞ, ΟΔ, ΠΡ πρὸς ὀρθὰς ταῖς βάσεσιν αὐτῶν· For, first of all, let the (straight-lines) standing up,
λέγω, ὅτι ἐστὶν ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως AG, EF, LB, HK, CM, NO, PD, and QR, be at right-
ἡ ΓΜ πρὸς τὴν ΑΗ. angles to their bases. I say that as base EH is to base
Εἰ μὲν οὖν ἴση ἐστὶν ἡ ΕΘ βάσιν τῇ ΝΠ βάσει, ἔστι δὲ NQ, so CM (is) to AG.
καὶ τὸ ΑΒ στερεὸν τῷ ΓΔ στερεῷ ἴσον, ἔσται καὶ ἡ ΓΜ τῇ Therefore, if base EH is equal to base NQ, and solid
ΑΗ ἴση. τὰ γὰρ ὑπὸ τὸ αὐτὸ ὕψος στερεὰ παραλληλεπίπεδα AB is also equal to solid CD, CM will also be equal to
πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις. καὶ ἔσται ὡς ἡ ΕΘ βάσις AG. For parallelepiped solids of the same height are to
πρὸς τὴν ΝΠ, οὕτως ἡ ΓΜ πρὸς τὴν ΑΗ, καὶ φανερόν, ὅτι one another as their bases [Prop. 11.32]. And as base

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