Page 486 - Euclid's Elements of Geometry

ST EW ibþ.

ELEMENTS BOOK 12

ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν· ἔστιν ἄρα ὡς ἡ ΒΜ thus equal to solid EHQP. And the bases of equal par-
βάσις πρὸς τὴν ΕΠ βάσιν, οὕτως τὸ τοῦ ΕΘΠΟ στερεοῦ allelepiped solids are reciprocally proportional to their
ὕψος πρὸς τὸ τοῦ ΒΗΜΛ στερεοῦ ὕψος. ἀλλ᾿ ὡς ἡ ΒΜ heights [Prop. 11.34]. Thus, as base BM is to base EQ,
βάσις πρὸς τὴν ΕΠ, οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΔΕΖ so the height of solid EHQP (is) to the height of solid
τρίγωνον. καὶ ὡς ἄρα τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΔΕΖ BGML. But, as base BM (is) to base EQ, so triangle
τρίγωνον, οὕτως τὸ τοῦ ΕΘΠΟ στερεοῦ ὕψος πρὸς τὸ τοῦ ABC (is) to triangle DEF [Prop. 1.34]. And, thus, as
ΒΗΜΛ στερεοῦ ὕψος. ἀλλὰ τὸ μὲν τοῦ ΕΘΠΟ στερεοῦ triangle ABC (is) to triangle DEF, so the height of solid
ὕψος τὸ αὐτὸ ἐστι τῷ τῆς ΔΕΖΘ πυραμίδος ὕψει, τὸ δὲ EHQP (is) to the height of solid BGML [Prop. 5.11].
τοῦ ΒΗΜΛ στερεοῦ ὕψος τὸ αὐτό ἐστι τῷ τῆς ΑΒΓΗ πυ- But, the height of solid EHQP is the same as the height
ραμίδος ὕψει· ἔστιν ἄρα ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ of pyramid DEFH, and the height of solid BGML is
βάσιν, οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς τὸ τῆς the same as the height of pyramid ABCG. Thus, as base
ΑΒΓΗ πυραμίδος ὕψος. τῶν ΑΒΓΗ, ΔΕΖΘ ἄρα πυραμίδων ABC is to base DEF, so the height of pyramid DEFH
ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν. (is) to the height of pyramid ABCG. Thus, the bases
᾿Αλλὰ δὴ τῶν ΑΒΓΗ, ΔΕΖΘ πυραμίδων ἀντιπεπονθέτ- of pyramids ABCG and DEFH are reciprocally propor-
ωσαν αἱ βάσεις τοῖς ὕψεσιν, καὶ ἔστω ὡς ἡ ΑΒΓ βάσις πρὸς tional to their heights.
τὴν ΔΕΖ βάσιν, οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς And so, let the bases of pyramids ABCG and DEFH
τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος· λέγω, ὅτι ἴση ἐστὶν ἡ ΑΒΓΗ be reciprocally proportional to their heights, and (thus)
πυραμὶς τῇ ΔΕΖΘ πυραμίδι. let base ABC be to base DEF, as the height of pyramid
Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεί ἐστιν ὡς ἡ DEFH (is) to the height of pyramid ABCG. I say that
ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὸ τῆς ΔΕΖΘ πυ- pyramid ABCG is equal to pyramid DEFH.
ραμίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος, ἀλλ᾿ ὡς For, with the same construction, since as base ABC
ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὸ ΒΜ παραλ- is to base DEF, so the height of pyramid DEFH (is) to
ληλόγραμμον πρὸς τὸ ΕΠ παραλληλόγραμμον, καὶ ὡς ἄρα the height of pyramid ABCG, but as base ABC (is) to
τὸ ΒΜ παραλληλόγραμμον πρὸς τὸ ΕΠ παραλληλόγραμμον, base DEF, so parallelogram BM (is) to parallelogram
οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ EQ [Prop. 1.34], thus as parallelogram BM (is) to paral-
πυραμίδος ὕψος. ἀλλὰ τὸ [μὲν] τῆς ΔΕΖΘ πυραμίδος ὕψος lelogram EQ, so the height of pyramid DEFH (is) also
τὸ αὐτό ἐστι τῷ τοῦ ΕΘΠΟ παραλληλεπιπέδου ὕψει, τὸ δὲ to the height of pyramid ABCG [Prop. 5.11]. But, the
τῆς ΑΒΓΗ πυραμίδος ὕψος τὸ αὐτό ἐστι τῷ τοῦ ΒΗΜΛ height of pyramid DEFH is the same as the height of
παραλληλεπιπέδου ὕψει· ἔστιν ἄρα ὡς ἡ ΒΜ βάσις πρὸς parallelepiped EHQP, and the height of pyramid ABCG
τὴν ΕΠ βάσιν, οὕτως τὸ τοῦ ΕΘΠΟ παραλληλεπιπέδου is the same as the height of parallelepiped BGML. Thus,
ὕψος πρὸς τὸ τοῦ ΒΗΜΛ παραλληλεπιπέδου ὕψος. ὧν as base BM is to base EQ, so the height of parallelepiped
δὲ στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις EHQP (is) to the height of parallelepiped BGML. And
τοῖς ὕψεσιν, ἴσα ἐστὶν ἐκεῖνα· ἴσον ἄρα ἐστὶ τὸ ΒΗΜΛ those parallelepiped solids whose bases are reciprocally
στερεὸν παραλληλεπίπεδον τῷ ΕΘΠΟ στερεῷ παραλληλε- proportional to their heights are equal [Prop. 11.34].
πιπέδῳ. καί ἐστι τοῦ μὲν ΒΗΜΛ ἕκτον μέρος ἡ ΑΒΓΗ Thus, the parallelepiped solid BGML is equal to the par-
πυραμίς, τοῦ δὲ ΕΘΠΟ παραλληλεπιπέδου ἕκτον μέρος ἡ allelepiped solid EHQP. And pyramid ABCG is a sixth
ΔΕΖΘ πυραμίς· ἴση ἄρα ἡ ΑΒΓΗ πυραμὶς τῇ ΔΕΖΘ πυ- part of BGML, and pyramid DEFH a sixth part of par-
iþ whose bases are reciprocally proportional to their heights
ραμίδι. allelepiped EHQP. Thus, pyramid ABCG is equal to
Τῶν ἄρα ἴσων πυραμίδων καὶ τριγώνους βάσεις ἐχουσῶν pyramid DEFH.
ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν· καὶ ὧν πυραμίδων Thus, the bases of equal pyramids which also have
τριγώνους βάσεις ἐχουσῶν ἀντιπεπόνθασιν αἱ βάσεις τοῖς triangular bases are reciprocally proportional to their
ὕψεσιν, ἴσαι εἰσὶν ἐκεῖναι· ὅπερ ἔδει δεῖξαι. heights. And those pyramids having triangular bases

are equal. (Which is) the very thing it was required to
show.

.
Proposition 10
Πᾶς κῶνος κυλίνδρου τρίτον μέρος ἐστὶ τοῦ τὴν αὐτὴν Every cone is the third part of the cylinder which has
βάσιν ἔχοντος αὐτῷ καὶ ὕψος ἴσον. the same base as it, and an equal height.
᾿Εχέτω γὰρ κῶνος κυλίνδρῷ βάσιν τε τὴν αὐτὴν τὸν For let there be a cone (with) the same base as a cylin-

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