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Euclid · Elements §11.prop.31#1

Equality of Right Parallelepipeds of Equal Base and Height

Passage 258 of 316 · Greek

Summary

The first part of Proposition 31, proving that solid parallelepipeds on equal bases and with the same height are equal. It addresses the first case where the uprights are at right angles to the bases, constructing equivalent solids to advance the proof.

§11.prop.31#1τὰ ἐπὶ ἴσων βάσεων ὄντα στερεὰ παραλληλεπίπεδα καὶ ὑπὸ τὸ αὐτὸ ὕψος ἴσα ἀλλήλοις ἐστίν.
Solid parallelepipeds which are on equal bases and have the same height are equal to one another.
ἔστω ἐπὶ ἴσων βάσεων τῶν ΑΒ, ΓΔ στερεὰ παραλληλεπίπεδα τὰ ΑΕ, ΓΖ ὑπὸ τὸ αὐτὸ ὕψος· λέγω, ὅτι ἴσον ἐστὶ τὸ ΑΕ στερεὸν τῷ ΓΖ στερεῷ.
Let the solid parallelepipeds AE, GZ be on equal bases AB, GD and have the same height; I say, that the solid AE is equal to the solid GZ.
ἔστωσαν δὴ πρότερον αἱ ἐφεστηκυῖαι αἱ ΘΚ, ΒΕ, ΑΗ, ΛΜ, ΟΠ, ΔΖ, ΓΞ, ΡΣ πρὸς ὀρθὰς ταῖς ΑΒ, ΓΔ βάσεσιν, καὶ ἐκβεβλήσθω ἐπʼ εὐθείας τῇ ΓΡ εὐθεῖα ἡ ΡΤ, καὶ συνεστάτω πρὸς τῇ ΡΤ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Ρ τῇ ὑπὸ ΑΛΒ γωνίᾳ ἴση ἡ ὑπὸ ΤΡΥ, καὶ κείσθω τῇ μὲν ΑΛ ἴση ἡ ΡΤ, τῇ δὲ ΛΒ ἴση ἡ ΡΥ, καὶ συμπεπληρώσθω ἥ τε ΡΧ βάσις καὶ τὸ ΨΥ στερεόν.
Indeed, let first the uprights TK, BE, AH, LM, OP, DZ, GX, RS be at right angles to the bases AB, GD, and let the straight line RT be produced in a straight line with GD, and let there be constructed on the straight line RT and at the point R on it the angle TRY equal to the angle ALB, and let RT be made equal to AL, and RY to LB, and let the base RX and the solid YU be completed.
καὶ ἐπεὶ δύο αἱ ΤΡ, ΡΥ δυσὶ ταῖς ΑΛ, ΛΒ ἴσαι εἰσίν, καὶ γωνίας ἴσας περιέχουσιν, ἴσον ἄρα καὶ ὅμοιον τὸ ΡΧ παραλληλόγραμμον τῷ ΘΛ παραλληλογράμμῳ.
And since the two TR, RY are equal to the two AL, LB, and contain equal angles, the parallelogram RX is also equal and similar to the parallelogram TL.
καὶ ἐπεὶ πάλιν ἴση μὲν ἡ ΑΛ τῇ ΡΤ, ἡ δὲ ΛΜ τῇ ΡΣ, καὶ γωνίας ὀρθὰς περιέχουσιν, ἴσον ἄρα καὶ ὅμοιόν ἐστι τὸ ΡΨ παραλληλόγραμμον τῷ ΑΜ παραλληλογράμμῳ.
And since again AL is equal to RT, and LM to RS, and they contain right angles, the parallelogram RY is also equal and similar to the parallelogram AM.
διὰ τὰ αὐτὰ δὴ καὶ τὸ ΛΕ τῷ ΣΥ ἴσον τέ ἐστι καὶ ὅμοιον· τρία ἄρα παραλληλόγραμμα τοῦ ΑΕ στερεοῦ τρισὶ παραλληλογράμμοις τοῦ ΨΥ στερεοῦ ἴσα τέ ἐστι καὶ ὅμοια.
For the same reasons indeed, LE is also equal and similar to SU; therefore three parallelograms of the solid AE are equal and similar to three parallelograms of the solid YU.
ἀλλὰ τὰ μὲν τρία τρισὶ τοῖς ἀπεναντίον ἴσα τέ ἐστι καὶ ὅμοια, τὰ δὲ τρία τρισὶ τοῖς ἀπεναντίον· ὅλον ἄρα τὸ ΑΕ στερεὸν παραλληλεπίπεδον ὅλῳ τῷ ΨΥ στερεῷ παραλληλεπιπέδῳ ἴσον ἐστίν.
But the three are equal and similar to the three opposite, and the three to the three opposite; therefore the whole solid parallelepiped AE is equal to the whole solid parallelepiped YU.
διήχθωσαν αἱ ΔΡ, ΧΥ καὶ συμπιπτέτωσαν ἀλλήλαις κατὰ τὸ Ω, καὶ διὰ τοῦ Τ τῇ ΔΩ παράλληλος ἤχθω ἡ #22αΤ#5, καὶ ἐκβεβλήσθω ἡ ΟΔ κατὰ τὸ #22α, καὶ συμπεπληρώσθω τὰ ΩΨ, ΡΙ στερεά.
Let DR, XY be drawn and meet one another at W, and through T let #22aT#5 be drawn parallel to DW, and let OD be produced to #22a, and let the solids WY, RI be completed.
ἴσον δή ἐστι τὸ ΨΩ στερεόν, οὗ βάσις μέν ἐστι τὸ ΡΨ παραλληλόγραμμον, ἀπεναντίον δὲ τὸ Ω#4, τῷ ΨΥ στερεῷ, οὗ βάσις μὲν τὸ ΡΨ παραλληλόγραμμον, ἀπεναντίον δὲ τὸ ΥΦ· ἐπί τε γὰρ τῆς αὐτῆς βάσεώς εἰσι τῆς ΡΨ καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι αἱ ΡΩ, ΡΥ, Τ#5, ΤΧ, Σ#2, Σο͂, Ψ#4, ΨΦ ἐπὶ τῶν αὐτῶν εἰσιν εὐθειῶν τῶν ΩΧ, #2Φ. ἀλλὰ τὸ ΨΥ στερεὸν τῷ ΑΕ ἐστιν ἴσον·
Indeed, the solid YW, whose base is the parallelogram RY, and opposite to it W#4, is equal to the solid YU, whose base is the parallelogram RY, and opposite to it UF; for they are on the same base RY and have the same height, and their uprights RW, RY, T#5, TX, S#2, So, Y#4, YF are on the same straight lines WX, #2F.
καὶ τὸ ΨΩ ἄρα στερεὸν τῷ ΑΕ στερεῷ ἐστιν ἴσον.
But the solid YU is equal to AE; therefore the solid YW is also equal to the solid AE.

Notes

  1. §11.prop.31#1τὰ ἐπὶ ἴσων βάσεων ὄντα στερεὰ παραλληλεπίπεδα καὶ ὑπὸ τὸ αὐτὸ ὕψος ἴσα ἀλλήλοις ἐστίν — An instance of the classic Greek grammatical rule ('Schema Atticum') where a neuter plural subject (τὰ στερεὰ παραλληλεπίπεδα) takes a singular verb (ἐστίν).
  2. §11.prop.31#1ἔστωσαν δὴ πρότερον αἱ ἐφεστηκυῖαι ... πρὸς ὀρθὰς ταῖς ΑΒ, ΓΔ βάσεσιν — The verb ἔστωσαν is a third-person plural imperative, standard in geometrical texts for setting up assumptions. The phrase πρὸς ὀρθάς takes the dative ταῖς βάσεσιν to mean 'at right angles to the bases'.
  3. §11.prop.31#1συνεστάτω πρὸς τῇ ΡΤ εὐθείᾳ ... τῇ ὑπὸ ΑΛΒ γωνίᾳ ἴση ἡ ὑπὸ ΤΡΥ — The subject of συνεστάτω (third-person singular passive imperative) is ἡ ὑπὸ ΤΡΥ (with γωνία understood), while τῇ ὑπὸ ΑΛΒ γωνίᾳ ἴση acts as a predicate adjective phrase modifying it. The expression ἡ ὑπὸ ΑΛΒ is an elliptical form of ἡ ὑπὸ ΑΛΒ περιεχομένη γωνία ('the angle contained by A, L, B').
  4. §11.prop.31#1τρία ἄρα παραλληλόγραμμα τοῦ ΑΕ στερεοῦ τρισὶ παραλληλογράμμοις τοῦ ΨΥ στερεοῦ ἴσα τέ ἐστι καὶ ὅμοια — The genitives τοῦ ΑΕ στερεοῦ and τοῦ ΨΥ στερεοῦ are genitives of possession. The dative τρισὶ παραλληλογράμμοις is governed by the adjectives ἴσα καὶ ὅμοια, expressing the objects to which the first three are 'equal and similar'.

Cite this passage

Euclid, Elements §11.prop.31#1. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:11.prop.31%231

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