Page 468 - Euclid's Elements of Geometry

ST EW iaþ.

ELEMENTS BOOK 11

στερεὰ παραλληλεπίπεδα ὅμοιά τε καὶ ὁμοίως ἀναγραφόμενα and similarly described, parallelepiped solids on them
ἀνάλογον ἔσται· καὶ ἐὰν τὰ ἀπ᾿ αὐτῶν στερεὰ παραλλη- will also be proportional. And if the similar, and similarly
λεπίπεδα ὅμοιά τε καὶ ὁμοίως ἀναγραφόμενα ἀνάλογον ᾖ, described, parallelepiped solids on them are proportional
K L
καὶ αὐταὶ αἱ εὐθεῖαι ἀνάλογον ἔσονται. then the straight-lines themselves will be proportional.

A B G D A B M C D N

E Z H E F

G
H
῎Εστωσαν τέσσαρες εὐθεῖαι ἀνάλογον αἱ ΑΒ, ΓΔ, ΕΖ, Let AB, CD, EF, and GH, be four proportional
ΗΘ, ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΗΘ, καὶ straight-lines, (such that) as AB (is) to CD, so EF (is)
ἀναγεγράφθωσαν ἀπὸ τῶν ΑΒ, ΓΔ, ΕΖ, ΗΘ ὅμοιά τε καὶ to GH. And let the similar, and similarly laid out, par-
ὁμοίως κείμενα στερεὰ παραλληλεπίπεδα τὰ ΚΑ, ΛΓ, ΜΕ, allelepiped solids KA, LC, ME and NG have been de-
ΝΗ· λέγω, ὅτι ἐστὶν ὡς τὸ ΚΑ πρὸς τὸ ΛΓ, οὕτως τὸ ΜΕ scribed on AB, CD, EF, and GH (respectively). I say
πρὸς τὸ ΝΗ. that as KA is to LC, so ME (is) to NG.
᾿Επεὶ γὰρ ὅμοιόν ἐστι τὸ ΚΑ στερεὸν παραλληλεπίπεδον For since the parallelepiped solid KA is similar to LC,
τῷ ΛΓ, τὸ ΚΑ ἄρα πρὸς τὸ ΛΓ τριπλασίονα λόγον ἔχει ἤπερ KA thus has to LC the cubed ratio that AB (has) to CD
ἡ ΑΒ πρὸς τὴν ΓΔ. διὰ τὰ αὐτὰ δὴ καὶ τὸ ΜΕ πρὸς τὸ ΝΗ [Prop. 11.33]. So, for the same (reasons), ME also has to
τριπλασίονα λόγον ἔχει ἤπερ ἡ ΕΖ πρὸς τὴν ΗΘ. καί ἐστιν NG the cubed ratio that EF (has) to GH [Prop. 11.33].
ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΗΘ. καὶ ὡς And since as AB is to CD, so EF (is) to GH, thus, also,
ἄρα τὸ ΑΚ πρὸς τὸ ΛΓ, οὕτως τὸ ΜΕ πρὸς τὸ ΝΗ. as AK (is) to LC, so ME (is) to NG.
᾿Αλλὰ δὴ ἔστω ὡς τὸ ΑΚ στερεὸν πρὸς τὸ ΛΓ στερεόν, And so let solid AK be to solid LC, as solid ME (is)
οὕτως τὸ ΜΕ στερεὸν πρὸς τὸ ΝΗ· λέγω, ὅτι ἐστὶν ὡς ἡ to NG. I say that as straight-line AB is to CD, so EF (is)
ΑΒ εὐθεῖα πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΗΘ. to GH.
᾿Επεὶ γὰρ πάλιν τὸ ΚΑ πρὸς τὸ ΛΓ τριπλασίονα λόγον For, again, since KA has to LC the cubed ratio that
ἔχει ἤπερ ἡ ΑΒ πρὸς τὴν ΓΔ, ἔχει δὲ καὶ τὸ ΜΕ πρὸς τὸ AB (has) to CD [Prop. 11.33], and ME also has to NG
lhþ required to show.
ΝΗ τριπλασίονα λόγον ἤπερ ἡ ΕΖ πρὸς τὴν ΗΘ, καί ἐστιν the cubed ratio that EF (has) to GH [Prop. 11.33], and
ὡς τὸ ΚΑ πρὸς τὸ ΛΓ, οὕτως τὸ ΜΕ πρὸς τὸ ΝΗ, καὶ ὡς as KA is to LC, so ME (is) to NG, thus, also, as AB (is)
ἄρα ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΕΖ πρὸς τὴν ΗΘ. to CD, so EF (is) to GH.
᾿Εὰν ἄρα τέσσαρες εὐθεῖαι ἀνάλογον ὦσι καὶ τὰ ἑξῆς Thus, if four straight-lines are proportional, and so
τῆς προτάσεως· ὅπερ ἔδει δεῖξαι. on of the proposition. (Which is) the very thing it was

† This proposition assumes that if two ratios are equal then the cube of the former is also equal to the cube of the latter, and vice versa.

Proposition 38
.
᾿Εὰν κύβου τῶν ἀπεναντίον ἐπιπέδων αἱ πλευραὶ δίχα If the sides of the opposite planes of a cube are cut
τμηθῶσιν, διὰ δὲ τῶν τομῶν ἐπίπεδα ἐκβληθῇ, ἡ κοινὴ τομὴ in half, and planes are produced through the pieces, then
τῶν ἐπιπέδων καὶ ἡ τοῦ κύβου διάμετρος δίχα τέμνουσιν the common section of the (latter) planes and the diam-
ἀλλήλας. eter of the cube cut one another in half.

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