Page 481 - Euclid's Elements of Geometry

ST EW ibþ.

ELEMENTS BOOK 12

Εἰ γὰρ δυνατόν, ἔστω πρὸς μεῖζον τὸ Χ· ἀνάπαλιν ἄρα not to some solid less than pyramid DEFH. So, simi-
ἐστὶν ὡς ἡ ΔΕΖ βάσις πρὸς τὴν ΑΒΓ βάσιν, οὕτως τὸ Χ larly, we can show that base DEF is not to base ABC,
στερεὸν πρὸς τὴν ΑΒΓΗ πυραμίδα. ὡς δὲ τὸ Χ στερεὸν as pyramid DEFH (is) to some solid less than pyramid
πρὸς τὴν ΑΒΓΗ πυραμίδα, οὕτως ἡ ΔΕΖΘ πυραμὶς πρὸς ABCG either.
ἔλασσόν τι τῆς ΑΒΓΗ πυραμίδος, ὡς ἔμπροσθεν ἐδείχθη· So, I say that neither is base ABC to base DEF, as
καὶ ὡς ἄρα ἡ ΔΕΖ βάσις πρὸς τὴν ΑΒΓ βάσιν, οὕτως ἡ pyramid ABCG (is) to some solid greater than pyramid
ΔΕΖΘ πυραμὶς πρὸς ἔλασσόν τι τῆς ΑΒΓΗ πυραμίδος· DEFH.
ὅπερ ἄτοπον ἐδείχθη. οὐκ ἄρα ἐστὶν ὡς ἡ ΑΒΓ βάσις πρὸς For, if possible, let it be (in this ratio) to some
τὴν ΔΕΖ βάσιν, οὕτως ἡ ΑΒΓΗ πυραμὶς πρὸς μεῖζόν τι greater (solid), W. Thus, inversely, as base DEF
τῆς ΔΕΖΘ πυραμίδος στερεόν. ἐδείχθη δέ, ὅτι οὐδὲ πρὸς (is) to base ABC, so solid W (is) to pyramid ABCG
ἔλασσον. ἔστιν ἄρα ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, [Prop. 5.7. corr.]. And as solid W (is) to pyramid ABCG,
οὕτως ἡ ΑΒΓΗ πυραμὶς πρὸς τὴν ΔΕΖΘ πυραμίδα· ὅπερ so pyramid DEFH (is) to some (solid) less than pyramid
ἔδει δεῖξαι. ABCG, as shown before [Prop. 12.2 lem.]. And, thus,
as base DEF (is) to base ABC, so pyramid DEFH (is)
þ ABC is not to base DEF, as pyramid ABCG (is) to
to some (solid) less than pyramid ABCG [Prop. 5.11].
The very thing was shown (to be) absurd. Thus, base

some solid greater than pyramid DEFH. And, it was
shown that neither (is it in this ratio) to a lesser (solid).
Thus, as base ABC is to base DEF, so pyramid ABCG
(is) to pyramid DEFH. (Which is) the very thing it was
required to show.

Proposition 6
.
M N
Αἱ ὐπὸ τὸ αὐτὸ ὕψος οὖσαι πυραμίδες καὶ πολυγώνους Pyramids which are of the same height, and have
ἔχουσαι βάσεις πρὸς ἀλλήλας εἰσὶν ὡς αἱ βάσεις. polygonal bases, are to one another as their bases.

D G B H C B K H

E A Z D A L G

F
E
῎Εστωσαν ὑπὸ τὸ αὐτὸ ὕψος πυραμίδες, ὧν [αἱ] βάσεις Let there be pyramids of the same height whose bases
μὲν τὰ ΑΒΓΔΕ, ΖΗΘΚΛ πολύγωνα, κορυφαὶ δὲ τὰ Μ, (are) the polygons ABCDE and FGHKL, and apexes
Ν σημεῖα· λέγω, ὅτι ἐστὶν ὡς ἡ ΑΒΓΔΕ βάσις πρὸς τὴν the points M and N (respectively). I say that as base
ΖΗΘΚΛ βάσιν, οὕτως ἡ ΑΒΓΔΕΜ πυραμὶς πρὸς τὴν ΖΗΘ- ABCDE is to base FGHKL, so pyramid ABCDEM (is)
ΚΛΝ πυραμίδα. to pyramid FGHKLN.
᾿Επεζεύχθωσαν γὰρ αἱ ΑΓ, ΑΔ, ΖΘ, ΖΚ. ἐπεὶ οὖν For let AC, AD, FH, and FK have been joined.
δύο πυραμίδες εἰσὶν αἱ ΑΒΓΜ, ΑΓΔΜ τριγώνους ἔχου- Therefore, since ABCM and ACDM are two pyramids
σαι βάσεις καὶ ὕψος ἴσον, πρὸς ἀλλήλας εἰσὶν ὡς αἱ βάσεις· having triangular bases and equal height, they are to
ἔστιν ἄρα ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΑΓΔ βάσιν, οὕτως ἡ one another as their bases [Prop. 12.5]. Thus, as base
ΑΒΓΜ πυραμὶς πρὸς τὴν ΑΓΔΜ πυραμίδα. καὶ συνθέντι ABC is to base ACD, so pyramid ABCM (is) to pyra-
ὡς ἡ ΑΒΓΔ βάσις πρὸς τὴν ΑΓΔ βάσιν, οὕτως ἡ ΑΒΓΔΜ mid ACDM. And, via composition, as base ABCD

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