Page 482 - Euclid's Elements of Geometry

ST EW ibþ.

ELEMENTS BOOK 12

πυραμὶς πρὸς τὴν ΑΓΔΜ πυραμίδα. ἀλλὰ καὶ ὡς ἡ ΑΓΔ (is) to base ACD, so pyramid ABCDM (is) to pyra-
βάσις πρὸς τὴν ΑΔΕ βάσιν, οὕτως ἡ ΑΓΔΜ πυραμὶς πρὸς mid ACDM [Prop. 5.18]. But, as base ACD (is) to
τὴν ΑΔΕΜ πυραμίδα. δι᾿ ἴσου ἄρα ὡς ἡ ΑΒΓΔ βάσις πρὸς base ADE, so pyramid ACDM (is) also to pyramid
τὴν ΑΔΕ βάσιν, οὕτως ἡ ΑΒΓΔΜ πυραμὶς πρὸς τὴν ΑΔΕΜ ADEM [Prop. 12.5]. Thus, via equality, as base ABCD
πυραμίδα. καὶ συνθέντι πάλιν, ὡς ἡ ΑΒΓΔΕ βάσις πρὸς τὴν (is) to base ADE, so pyramid ABCDM (is) to pyramid
ΑΔΕ βάσιν, οὕτως ἡ ΑΒΓΔΕΜ πυραμὶς πρὸς τὴν ΑΔΕΜ ADEM [Prop. 5.22]. And, again, via composition, as
πυραμίδα. ὁμοίως δὴ δειχθήσεται, ὅτι καὶ ὡς ἡ ΖΗΘΚΛ base ABCDE (is) to base ADE, so pyramid ABCDEM
βάσις πρὸς τὴν ΖΗΘ βάσιν, οὕτως καὶ ἡ ΖΗΘΚΛΝ πυραμὶς (is) to pyramid ADEM [Prop. 5.18]. So, similarly, it can
πρὸς τὴν ΖΗΘΝ πυραμίδα. καὶ ἐπεὶ δύο πυραμίδες εἱσὶν αἱ also be shown that as base FGHKL (is) to base FGH,
ΑΔΕΜ, ΖΗΘΝ τριγώνους ἔχουσαι βάσεις καὶ ὕψος ἴσον, so pyramid FGHKLN (is) also to pyramid FGHN. And
ἔστιν ἄρα ὡς ἡ ΑΔΕ βάσις πρὸς τὴν ΖΗΘ βάσιν, οὕτως since ADEM and FGHN are two pyramids having tri-
ἡ ΑΔΕΜ πυραμὶς πρὸς τὴν ΖΗΘΝ πυραμίδα. ἀλλ᾿ ὡς ἡ angular bases and equal height, thus as base ADE (is) to
ΑΔΕ βάσις πρὸς τὴν ΑΒΓΔΕ βάσιν, οὕτως ἦν ἡ ΑΔΕΜ base FGH, so pyramid ADEM (is) to pyramid FGHN
πυραμὶς πρὸς τὴν ΑΒΓΔΕΜ πυραμίδα. καὶ δι᾿ ἴσου ἄρα ὡς [Prop. 12.5]. But, as base ADE (is) to base ABCDE, so
ἡ ΑΒΓΔΕ βάσις πρὸς τὴν ΖΗΘ βάσιν, οὕτως ἡ ΑΒΓΔΕΜ pyramid ADEM (was) to pyramid ABCDEM. Thus, via
zþ so pyramid ABCDEM (is) also to pyramid FGHKLN
πυραμὶς πρὸς τὴν ΖΗΘΝ πυραμίδα. ἀλλὰ μὴν καὶ ὡς ἡ ΖΗΘ equality, as base ABCDE (is) to base FGH, so pyramid
βάσις πρὸς τὴν ΖΗΘΚΛ βάσιν, οὕτως ἦν καὶ ἡ ΖΗΘΝ πυ- ABCDEM (is) also to pyramid FGHN [Prop. 5.22].
ραμὶς πρὸς τὴν ΖΗΘΚΛΝ πυραμίδα, καὶ δι᾿ ἴσου ἄρα ὡς ἡ But, furthermore, as base FGH (is) to base FGHKL,
ΑΒΓΔΕ βάσις πρὸς τὴν ΖΗΘΚΛ βάσιν, οὕτως ἡ ΑΒΓΔΕΜ so pyramid FGHN was also to pyramid FGHKLN.
πυραμὶς πρὸς τὴν ΖΗΘΚΛΝ πυραμίδα· ὅπερ ἔδει δεῖξαι. Thus, via equality, as base ABCDE (is) to base FGHKL,

[Prop. 5.22]. (Which is) the very thing it was required to
show.
Z
.
Proposition 7
Πᾶν πρίσμα τρίγωνον ἔχον βάσιν διαιρεῖται εἰς τρεῖς πυ- Any prism having a triangular base is divided into
ραμίδας ἴσας ἀλλήλαις τριγώνους βάσεις ἐχούσας. three pyramids having triangular bases (which are) equal
E D E F D
to one another.

G

B A C

A
B
῎Εστω πρίσμα, οὗ βάσις μὲν τὸ ΑΒΓ τρίγωνον, ἀπε- Let there be a prism whose base (is) triangle ABC,
ναντίον δὲ τὸ ΔΕΖ· λέγω, ὅτι τὸ ΑΒΓΔΕΖ πρίσμα διαιρεῖται and opposite (plane) DEF. I say that prism ABCDEF
εἰς τρεῖς πυραμίδας ἴσας ἀλλήλαις τριγώνους ἐχούσας is divided into three pyramids having triangular bases
βάσεις. (which are) equal to one another.
᾿Επεζεύχθωσαν γὰρ αἱ ΒΔ, ΕΓ, ΓΔ. ἐπεὶ παραλ- For let BD, EC, and CD have been joined. Since
ληλόγραμμόν ἐστι τὸ ΑΒΕΔ, διάμετρος δὲ αὐτὸῦ ἐστιν ABED is a parallelogram, and BD is its diagonal, tri-
ἡ ΒΔ, ἴσον ἄρα ἐστι τὸ ΑΒΔ τρίγωνον τῷ ΕΒΔ τρίγωνῳ· angle ABD is thus equal to triangle EBD [Prop. 1.34].

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