§12.prop.9τῶν ἴσων πυραμίδων καὶ τριγώνους βάσεις ἐχουσῶν ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν· καὶ ὧν πυραμίδων τριγώνους βάσεις ἐχουσῶν ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν, ἴσαι εἰσὶν ἐκεῖναι.
In equal pyramids which have triangular bases the bases are reciprocally proportional to the heights; and those pyramids having triangular bases in which the bases are reciprocally proportional to the heights are equal.
ἔστωσαν γὰρ ἴσαι πυραμίδες τριγώνους βάσεις ἔχουσαι τὰς ΑΒΓ, ΔΕΖ, κορυφὰς δὲ τὰ Η, Θ σημεῖα·
¦5 For let there be equal pyramids having triangular bases, the triangles ABG and DEZ, and vertices the points H and Th.
λέγω, ὅτι τῶν ΑΒΓΗ, ΔΕΖΘ πυραμίδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν, καί ἐστιν ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος.
I say that in the pyramids ABGH and DEZTh the bases are reciprocally proportional to the heights, and as the base ABG is to the base DEZ, so is the height of the pyramid DEZTh to the height of the pyramid ABGH.
συμπεπληρώσθω γὰρ τὰ ΒΗΜΛ, ΕΘΠΟ στερεὰ παραλληλεπίπεδα.
For let the parallelepipedal solids BHML and EThPO be completed.
καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΒΓΗ πυραμὶς τῇ ΔΕΖΘ πυραμίδι, καί ἐστι τῆς μὲν ΑΒΓΗ πυραμίδος ἑξαπλάσιον τὸ ΒΗ ΜΛ στερεόν, τῆς δὲ ΔΕΖΘ πυραμίδος ἑξαπλάσιον τὸ ΕΘΠΟ στερεόν, ἴσον ἄρα ἐστὶ τὸ ΒΗΜΛ στερεὸν τῷ ΕΘΠΟ στερεῷ.
And since the pyramid ABGH is equal to the pyramid DEZTh, and the solid BHML is sixfold of the pyramid ABGH, and the solid EThPO is sixfold of the pyramid DEZTh, therefore the solid BHML is equal to the solid EThPO.
τῶν δὲ ἴσων στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν· ἔστιν ἄρα ὡς ἡ ΒΜ βάσις πρὸς τὴν ΕΠ βάσιν, οὕτως τὸ τοῦ ΕΘΠΟ στερεοῦ ὕψος πρὸς τὸ τοῦ ΒΗΜΛ στερεοῦ ὕψος.
But in equal parallelepipedal solids the bases are reciprocally proportional to the heights; therefore, as the base BM is to the base EP, so is the height of the solid EThPO to the height of the solid BHML.
ἀλλʼ ὡς ἡ ΒΜ βάσις πρὸς τὴν ΕΠ, οὕτως τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΔΕΖ τρίγωνον.
But as the base BM is to the base EP, so is the triangle ABG to the triangle DEZ.
καὶ ὡς ἄρα τὸ ΑΒΓ τρίγωνον πρὸς τὸ ΔΕΖ τρίγωνον, οὕτως τὸ τοῦ ΕΘΠΟ στερεοῦ ὕψος πρὸς τὸ τοῦ ΒΗΜΛ στερεοῦ ὕψος.
Therefore also, as the triangle ABG is to the triangle DEZ, so is the height of the solid EThPO to the height of the solid BHML.
ἀλλὰ τὸ μὲν τοῦ ΕΘΠΟ στερεοῦ ὕψος τὸ αὐτό ἐστι τῷ τῆς ΔΕΖΘ πυραμίδος ὕψει, τὸ δὲ τοῦ ΒΗΜΛ στερεοῦ ὕψος τὸ αὐτό ἐστι τῷ τῆς ΑΒΓΗ πυραμίδος ὕψει· ἔστιν ἄρα ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος.
But the height of the solid EThPO is the same as the height of the pyramid DEZTh, and the height of the solid BHML is the same as the height of the pyramid ABGH; therefore, as the base ABG is to the base DEZ, so is the height of the pyramid DEZTh to the height of the pyramid ABGH.
τῶν ΑΒΓΗ, ΔΕΖΘ ἄρα πυραμίδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν.
Therefore in the pyramids ABGH and DEZTh the bases are reciprocally proportional to the heights.
ἀλλὰ δὴ τῶν ΑΒΓΗ, ΔΕΖΘ πυραμίδων ἀντιπεπονθέτωσαν αἱ βάσεις τοῖς ὕψεσιν, καὶ ἔστω ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος·
But indeed, let the bases of the pyramids ABGH and DEZTh be reciprocally proportional to the heights, and let it be that, as the base ABG is to the base DEZ, so is the height of the pyramid DEZTh to the height of the pyramid ABGH.
λέγω, ὅτι ἴση ἐστὶν ἡ ΑΒΓΗ πυραμὶς τῇ ΔΕΖΘ πυραμίδι.
I say that the pyramid ABGH is equal to the pyramid DEZTh.
τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεί ἐστιν ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος, ἀλλʼ ὡς ἡ ΑΒΓ βάσις πρὸς τὴν ΔΕΖ βάσιν, οὕτως τὸ ΒΜ παραλληλόγραμμον πρὸς τὸ ΕΠ παραλληλόγραμμον, καὶ ὡς ἄρα τὸ ΒΜ παραλληλόγραμμον πρὸς τὸ ΕΠ παραλληλόγραμμον, οὕτως τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος πρὸς τὸ τῆς ΑΒΓΗ πυραμίδος ὕψος.
For, with the same construction, since as the base ABG is to the base DEZ, so is the height of the pyramid DEZTh to the height of the pyramid ABGH, but as the base ABG is to the base DEZ, so is the parallelogram BM to the parallelogram EP, therefore also, as the parallelogram BM is to the parallelogram EP, so is the height of the pyramid DEZTh to the height of the pyramid ABGH.
ἀλλὰ τὸ τῆς ΔΕΖΘ πυραμίδος ὕψος τὸ αὐτό ἐστι τῷ τοῦ ΕΘΠΟ παραλληλεπιπέδου ὕψει, τὸ δὲ τῆς ΑΒΓΗ πυραμίδος ὕψος τὸ αὐτό ἐστι τῷ τοῦ ΒΗΜΛ παραλληλεπιπέδου ὕψει· ἔστιν ἄρα ὡς ἡ ΒΜ βάσις πρὸς τὴν ΕΠ βάσιν, οὕτως τὸ τοῦ ΕΘΠΟ παραλληλεπιπέδου ὕψος πρὸς τὸ τοῦ ΒΗΜΛ παραλληλεπιπέδου ὕψος.
But the height of the pyramid DEZTh is the same as the height of the parallelepiped EThPO, and the height of the pyramid ABGH is the same as the height of the parallelepiped BHML; therefore, as the base BM is to the base EP, so is the height of the parallelepiped EThPO to the height of the parallelepiped BHML.
ὧν δὲ στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν, ἴσα ἐστὶν ἐκεῖνα· ἴσον ἄρα ἐστὶ τὸ ΒΗΜΛ στερεὸν παραλληλεπίπεδον τῷ ΕΘ ΠΟ στερεῷ παραλληλεπιπέδῳ.
But those parallelepipedal solids in which the bases are reciprocally proportional to the heights are equal; therefore the parallelepipedal solid BHML is equal to the parallelepipedal solid EThPO.
καί ἐστι τοῦ μὲν ΒΗΜΛ ἕκτον μέρος ἡ ΑΒΓΗ πυραμίς, τοῦ δὲ ΕΘΠΟ παραλληλεπιπέδου ἕκτον μέρος ἡ ΔΕΖΘ πυραμίς· ἴση ἄρα ἡ ΑΒΓΗ πυραμὶς τῇ ΔΕΖΘ πυραμίδι.
And the pyramid ABGH is a sixth part of the parallelepiped BHML, and the pyramid DEZTh is a sixth part of the parallelepiped EThPO; therefore the pyramid ABGH is equal to the pyramid DEZTh.
τῶν ἄρα ἴσων πυραμίδων καὶ τριγώνους βάσεις ἐχουσῶν ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν· καὶ ὧν πυραμίδων τριγώνους βάσεις ἐχουσῶν ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν, ἴσαι εἰσὶν ἐκεῖναι· ὅπερ ἔδει δεῖξαι.
Therefore, in equal pyramids which have triangular bases the bases are reciprocally proportional to the heights; and those pyramids having triangular bases in which the bases are reciprocally proportional to the heights are equal; which was to be proved.