Page 483 - Euclid's Elements of Geometry

ST EW ibþ.

ELEMENTS BOOK 12

καὶ ἡ πυραμὶς ἄρα, ἧς βάσις μὲν τὸ ΑΒΔ τρίγωνον, κο- And, thus, the pyramid whose base (is) triangle ABD,
ρυφὴ δὲ τὸ Γ σημεῖον, ἴση ἐστὶ πυραμίδι, ἧς βάσις μέν and apex the point C, is equal to the pyramid whose base
ἐστι τὸ ΔΕΒ τρίγωνον, κορυφὴ δὲ τὸ Γ σημεῖον. ἀλλὰ is triangle DEB, and apex the point C [Prop. 12.5]. But,
ἡ πυραμίς, ἧς βάσις μέν ἐστι τὸ ΔΕΒ τρίγωνον, κορυφὴ the pyramid whose base is triangle DEB, and apex the
δὲ τὸ Γ σημεῖον, ἡ αὐτή ἐστι πυραμίδι, ἧς βάσις μέν ἐστι point C, is the same as the pyramid whose base is trian-
τὸ ΕΒΓ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον· ὑπὸ γὰρ τῶν gle EBC, and apex the point D. For they are contained
αὐτῶν ἐπιπέδων περιέχεται. καὶ πυραμὶς ἄρα, ἧς βάσις μέν by the same planes. And, thus, the pyramid whose base
ἐστι τὸ ΑΒΔ τρίγωνον, κορυφὴ δὲ τὸ Γ σημεῖον, ἴση ἐστὶ is ABD, and apex the point C, is equal to the pyramid
πυραμίδι, ἧς βάσις μέν ἐστι τὸ ΕΒΓ τρίγωνον, κορυφὴ whose base is EBC and apex the point D. Again, since
δὲ τὸ Δ σημεῖον. πάλιν, ἐπεὶ παραλληλόγραμμόν ἐστι τὸ FCBE is a parallelogram, and CE is its diagonal, trian-
ΖΓΒΕ, διάμετρος δέ ἐστιν αὐτοῦ ἡ ΓΕ, ἴσον ἐστὶ τὸ ΓΕΖ gle CEF is equal to triangle CBE [Prop. 1.34]. And,
τρίγωνον τῷ ΓΒΕ τριγώνῳ. καὶ πυραμὶς ἄρα, ἧς βάσις thus, the pyramid whose base is triangle BCE, and apex
μέν ἐστι τὸ ΒΓΕ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον, ἴση the point D, is equal to the pyramid whose base is trian-
ἐστὶ πυραμίδι, ἧς βάσις μέν ἐστι τὸ ΕΓΖ τρίγωνον, κορυφὴ gle ECF, and apex the point D [Prop. 12.5]. And the
δὲ τὸ Δ σημεῖον. ἡ δὲ πυραμίς, ἧς βάσις μέν ἐστι τὸ ΒΓΕ pyramid whose base is triangle BCE, and apex the point
τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον, ἴση ἐδείχθη πυραμίδι, ἧς D, was shown (to be) equal to the pyramid whose base is
βάσις μέν ἐστι τὸ ΑΒΔ τρίγωνον, κορυφὴ δὲ τὸ Γ σημεῖον· triangle ABD, and apex the point C. Thus, the pyramid
καὶ πυραμὶς ἄρα, ἧς βάσις μέν ἐστι τὸ ΓΕΖ τρίγωνον, κο- whose base is triangle CEF, and apex the point D, is
ρυφὴ δὲ τὸ Δ σημεῖον, ἴση ἐστὶ πυραμίδι, ἧς βάσις μέν also equal to the pyramid whose base [is] triangle ABD,
[ἐστι] τὸ ΑΒΔ τρίγωνον, κορυφὴ δὲ τὸ Γ σημεῖον· διῄρηται and apex the point C. Thus, the prism ABCDEF has
ἄρα τὸ ΑΒΓΔΕΖ πρίσμα εἰς τρεῖς πυραμίδας ἴσας ἀλλήλαις been divided into three pyramids having triangular bases
τριγώνους ἐχούσας βάσεις. (which are) equal to one another.
Καὶ ἐπεὶ πυραμίς, ἧς βάσις μέν ἐστι τὸ ΑΒΔ τρίγωνον, And since the pyramid whose base is triangle ABD,
κορυφὴ δὲ τὸ Γ σημεῖον, ἡ αὐτή ἐστι πυραμίδι, ἧς βάσις and apex the point C, is the same as the pyramid whose
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τὸ ΓΑΒ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον· ὑπὸ γὰρ τῶν base is triangle CAB, and apex the point D. For they are
αὐτῶν ἐπιπέδων περιέχονται· ἡ δὲ πυραμίς, ἧς βάσις τὸ contained by the same planes. And the pyramid whose
ΑΒΔ τρίγωνον, κορυφὴ δὲ τὸ Γ σημεῖον, τρίτον ἐδείχθη base (is) triangle ABD, and apex the point C, was shown
τοῦ πρίσματος, οὗ βάσις τὸ ΑΒΓ τρίγωνον, ἀπεναντίον δὲ (to be) a third of the prism whose base is triangle ABC,
τὸ ΔΕΖ, καὶ ἡ πυραμὶς ἄρα, ἧς βάσις τὸ ΑΒΓ τρίγωνον, and opposite (plane) DEF, thus the pyramid whose base
κορυφὴ δὲ τὸ Δ σημεῖον, τρίτον ἐστὶ τοῦ πρίσματος τοῦ is triangle ABC, and apex the point D, is also a third of
ἔχοντος βάσις τὴν αὐτὴν τὸ ΑΒΓ τρίγωνον, ἀπεναντίον δὲ the pyramid having the same base, triangle ABC, and
hþ
τὸ ΔΕΖ. opposite (plane) DEF.

.
Corollary
᾿Εκ δὴ τούτου φανερόν, ὅτι πᾶσα πυραμὶς τρίτον μέρος And, from this, (it is) clear that any pyramid is the
ἐστὶ τοῦ πρίσματος τοῦ τὴν αὐτὴν βάσιν ἔχοντος αὐτῇ καὶ third part of the prism having the same base as it, and an
ὕψος ἴσον· ὅπερ ἔδει δεῖξαι. equal height. (Which is) the very thing it was required to
show.
Proposition 8
.
Αἱ ὅμοιαι πυραμίδες καὶ τριγώνους ἔχουσαι βάσεις ἐν Similar pyramids which also have triangular bases are
τριπλασίονι λόγῳ εἰσὶ τῶν ὁμολόγων πλευρῶν. in the cubed ratio of their corresponding sides.
῎Εστωσαν ὅμοιαι καὶ ὁμοίως κείμεναι πυραμίδες, ὧν Let there be similar, and similarly laid out, pyramids
βάσεις μέν εἰσι τὰ ΑΒΓ, ΔΕΖ τρίγωνα, κορυφαὶ δὲ τὰ Η, whose bases are triangles ABC and DEF, and apexes
Θ σημεῖα· λέγω, ὅτι ἡ ΑΒΓΗ πυραμὶς πρὸς τὴν ΔΕΖΘ the points G and H (respectively). I say that pyramid
πυραμίδα τριπλασίονα λόγον ἔχει ἤπερ ἡ ΒΓ πρὸς τὴν ΕΖ. ABCG has to pyramid DEFH the cubed ratio of that
BC (has) to EF.

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