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Euclid · Elements §11.prop.34#1

Reciprocal Proportion of Bases and Heights in Parallelepipeds

Passage 262 of 316 · Greek

Summary

The first part of the proof showing that in equal solid parallelepipeds the bases are reciprocally proportional to their heights, and vice versa. It develops the argument by distinguishing between the cases of equal and unequal bases.

§11.prop.34#1τῶν ἴσων στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν· καὶ ὧν στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν, ἴσα ἐστὶν ἐκεῖνα.
In equal solid parallelepipeds the bases are reciprocally proportional to their heights; and those solid parallelepipeds in which the bases are reciprocally proportional to their heights are equal.
ἔστω ἴσα στερεὰ παραλληλεπίπεδα τὰ ΑΒ, ΓΔ· λέγω, ὅτι τῶν ΑΒ, ΓΔ στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν, καί ἐστιν ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως τὸ τοῦ ΓΔ στερεοῦ ὕψος πρὸς τὸ τοῦ ΑΒ στερεοῦ ὕψος.
Let AB, GD be equal solid parallelepipeds; I say that the bases of the solid parallelepipeds AB, GD are reciprocally proportional to their heights, and as the base EQ is to the base NP, so is the height of the solid GD to the height of the solid AB.
ἔστωσαν γὰρ πρότερον αἱ ἐφεστηκυῖαι αἱ ΑΗ, ΕΖ, ΛΒ, ΘΚ, ΓΜ, ΝΞ, ΟΔ, ΠΡ πρὸς ὀρθὰς ταῖς βάσεσιν αὐτῶν· λέγω, ὅτι ἐστὶν ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως ἡ ΓΜ πρὸς τὴν ΑΗ. εἰ μὲν οὖν ἴση ἐστιν ἡ ΕΘ βάσις τῇ ΝΠ βάσει, ἔστι δὲ καὶ τὸ ΑΒ στερεὸν τῷ ΓΔ στερεῷ ἴσον, ἔσται καὶ ἡ ΓΜ τῇ ΑΗ ἴση.
For let the standing straight lines AH, EZ, LB, QK, GM, NX, OD, PR first be at right angles to their bases; I say that as the base EQ is to the base NP, so is GM to AH. Now if the base EQ is equal to the base NP, and the solid AB is also equal to the solid GD, GM will also be equal to AH.
τὰ γὰρ ὑπὸ τὸ αὐτὸ ὕψος στερεὰ παραλληλεπίπεδα πρὸς ἄλληλά ἐστιν ὡς αἱ βάσεις. καὶ ἔσται ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ, οὕτως ἡ ΓΜ πρὸς τὴν ΑΗ, καὶ φανερόν, ὅτι τῶν ΑΒ, ΓΔ στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν.
For solid parallelepipeds of the same height are to one another as their bases; and as the base EQ is to the base NP, so will GM be to AH, and it is manifest that the bases of the solid parallelepipeds AB, GD are reciprocally proportional to their heights.
μὴ ἔστω δὴ ἴση ἡ ΕΘ βάσις τῇ ΝΠ βάσει, ἀλλʼ ἔστω μείζων ἡ ΕΘ. ἔστι δὲ καὶ τὸ ΑΒ στερεὸν τῷ ΓΔ στερεῷ ἴσον· μείζων ἄρα ἐστὶ καὶ ἡ ΓΜ τῆς ΑΗ.
Next, let the base EQ not be equal to the base NP, but let EQ be greater. And the solid AB is also equal to the solid GD; therefore GM is also greater than AH.
κείσθω οὖν τῇ ΑΗ ἴση ἡ ΓΤ, καὶ συμπεπληρώσθω ἀπὸ βάσεως μὲν τῆς ΝΠ, ὕψους δὲ τοῦ ΓΤ, στερεὸν παραλληλεπίπεδον τὸ ΦΓ. καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΑΒ στερεὸν τῷ ΓΔ στερεῷ, ἔξωθεν δὲ τὸ ΓΦ, τὰ δὲ ἴσα πρὸς τὸ αὐτὸ τὸν αὐτὸν ἔχει λόγον, ἔστιν ἄρα ὡς τὸ ΑΒ στερεὸν πρὸς τὸ ΓΦ στερεόν, οὕτως τὸ ΓΔ στερεὸν πρὸς τὸ ΓΦ στερεόν.
Let GT then be made equal to AH, and let the solid parallelepiped FG be completed on the base NP and with the height GT. And since the solid AB is equal to the solid GD, and GF is another solid, and equal things have to the same the same ratio, therefore as the solid AB is to the solid GF, so is the solid GD to the solid GF.
ἀλλʼ ὡς μὲν τὸ ΑΒ στερεὸν πρὸς τὸ ΓΦ στερεόν, οὕτως ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν· ἰσοϋψῆ γὰρ τὰ ΑΒ, ΓΦ στερεά· ὡς δὲ τὸ ΓΔ στερεὸν πρὸς τὸ ΓΦ στερεόν, οὕτως ἡ ΜΠ βάσις πρὸς τὴν ΤΠ βάσιν καὶ ἡ ΓΜ πρὸς τὴν ΓΤ·
But as the solid AB is to the solid GF, so is the base EQ to the base NP, for the solids AB, GF are of the same height; and as the solid GD is to the solid GF, so is the base MP to the base TP, and GM to GT; therefore also as the base EQ is to the base NP, so is MG to GT.
καὶ ὡς ἄρα ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως ἡ ΜΓ πρὸς τὴν ΓΤ. ἴση δὲ ἡ ΓΤ τῇ ΑΗ·
And GT is equal to AH; therefore also as the base EQ is to the base NP, so is MG to AH.
καὶ ὡς ἄρα ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως ἡ ΜΓ πρὸς τὴν ΑΗ. τῶν ΑΒ, ΓΔ ἄρα στερεῶν παραλληλεπιπέδων ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν.
Therefore of the solid parallelepipeds AB, GD the bases are reciprocally proportional to their heights.
πάλιν δὴ τῶν ΑΒ, ΓΔ στερεῶν παραλληλεπιπέδων ἀντιπεπονθέτωσαν αἱ βάσεις τοῖς ὕψεσιν, καὶ ἔστω ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως τὸ τοῦ ΓΔ στερεοῦ ὕψος πρὸς τὸ τοῦ ΑΒ στερεοῦ ὕψος· λέγω, ὅτι ἴσον ἐστὶ τὸ ΑΒ στερεὸν τῷ ΓΔ στερεῷ.
Again, let the bases of the solid parallelepipeds AB, GD be reciprocally proportional to their heights, and let the base EQ be to the base NP as the height of the solid GD is to the height of the solid AB; I say that the solid AB is equal to the solid GD.
ἔστωσαν πάλιν αἱ ἐφεστηκυῖαι πρὸς ὀρθὰς ταῖς βάσεσιν, καὶ εἰ μὲν ἴση ἐστὶν ἡ ΕΘ βάσις τῇ ΝΠ βάσει, καί ἐστιν ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως τὸ τοῦ ΓΔ στερεοῦ ὕψος πρὸς τὸ τοῦ ΑΒ στερεοῦ ὕψος, ἴσον ἄρα ἐστὶ καὶ τὸ τοῦ ΓΔ στερεοῦ ὕψος τῷ τοῦ ΑΒ στερεοῦ ὕψει.
Let the standing straight lines be again at right angles to the bases; and if the base EQ is equal to the base NP, and as the base EQ is to the base NP, so is the height of the solid GD to the height of the solid AB, then the height of the solid GD is also equal to the height of the solid AB.
τὰ δὲ ἐπὶ ἴσων βάσεων στερεὰ παραλληλεπίπεδα καὶ ὑπὸ τὸ αὐτὸ ὕψος ἴσα ἀλλήλοις ἐστίν· ἴσον ἄρα ἐστὶ τὸ ΑΒ στερεὸν τῷ ΓΔ στερεῷ.
But solid parallelepipeds on equal bases and of the same height are equal to one another; therefore the solid AB is equal to the solid GD.
μὴ ἔστω δὴ ἡ ΕΘ βάσις τῇ ΝΠ ἴση, ἀλλʼ ἔστω μείζων ἡ ΕΘ· μεῖζον ἄρα ἐστὶ καὶ τὸ τοῦ ΓΔ στερεοῦ ὕψος τοῦ τοῦ ΑΒ στερεοῦ ὕψους, τουτέστιν ἡ ΓΜ τῆς ΑΗ. κείσθω τῇ ΑΗ ἴση πάλιν ἡ ΓΤ, καὶ συμπεπληρώσθω ὁμοίως τὸ ΓΦ στερεόν.
Now let the base EQ not be equal to the base NP, but let EQ be greater; therefore the height of the solid GD is also greater than the height of the solid AB, that is, GM than AH. Let GT be again made equal to AH, and let the solid GF be completed in like manner.
ἐπεί ἐστιν ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως ἡ ΜΓ πρὸς τὴν ΑΗ, ἴση δὲ ἡ ΑΗ τῇ ΓΤ, ἔστιν ἄρα ὡς ἡ ΕΘ βάσις πρὸς τὴν ΝΠ βάσιν, οὕτως ἡ ΓΜ πρὸς τὴν ΓΤ.
Since as the base EQ is to the base NP, so is MG to AH, and AH is equal to GT, therefore as the base EQ is to the base NP, so is GM to GT.

Notes

  1. 11.prop.34#1ἀντιπεπόνθασιν αἱ βάσεις τοῖς ὕψεσιν — The verb ἀντιπεπονθέναι (to be reciprocally proportional) is used with the nominative "bases" (αἱ βάσεις) and the dative "heights" (τοῖς ὕψεσιν) to indicate that the bases are reciprocally proportional to the heights.
  2. 11.prop.34#1ὧν στερεῶν παραλληλεπιπέδων ... ἴσα ἐστὶν ἐκεῖνα — A syntactic structure where the antecedent (which should originally be the subject of the main clause, ἐκεῖνα τὰ στερεὰ παραλληλεπίπεδα) is incorporated into the relative clause introduced by the relative pronoun ὧν and attracted into the genitive case.
  3. 11.prop.34#1τὰ δὲ ἴσα πρὸς τὸ αὐτὸ τὸν αὐτὸν ἔχει λόγον — Presenting a general rule of proportion (derived from Book 5, Proposition 7) that equal magnitudes have the same ratio to the same magnitude. The subject is 'equal things' (τὰ ἴσα), and the target is 'to the same' (πρὸς τὸ αὐτό).

Cite this passage

Euclid, Elements §11.prop.34#1. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:11.prop.34%231

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