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Euclid · Elements §11.prop.27-11.prop.28

Construction of Similar Parallelepipeds and Bisection

Passage 256 of 316 · Greek

Summary

In Proposition 27 of Book 11, a method is shown to construct a solid parallelepiped similar and similarly situated to a given one on a given straight line, and in Proposition 28, it is proved that a solid parallelepiped is bisected by a plane passing through the diagonals of its opposite faces.

§11.prop.27ἀπὸ τῆς δοθείσης εὐθείας τῷ δοθέντι στερεῷ παραλληλεπιπέδῳ ὅμοιόν τε καὶ ὁμοίως κείμενον στερεὸν παραλληλεπίπεδον ἀναγράψαι.
On a given straight line, to describe a solid parallelepiped similar and similarly situated to a given solid parallelepiped.
ἔστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ, τὸ δὲ δοθὲν στερεὸν παραλληλεπίπεδον τὸ ΓΔ· δεῖ δὴ ἀπὸ τῆς δοθείσης εὐθείας τῆς ΑΒ τῷ δοθέντι στερεῷ παραλληλεπιπέδῳ τῷ ΓΔ ὅμοιόν τε καὶ ὁμοίως κείμενον στερεὸν παραλληλεπίπεδον ἀναγράψαι.
Let the given straight line be AB, and the given solid parallelepiped CD; it is required indeed to describe on the given straight line AB a solid parallelepiped similar and similarly situated to the given solid parallelepiped CD.
συνεστάτω γὰρ πρὸς τῇ ΑΒ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Α τῇ πρὸς τῷ Γ στερεᾷ γωνίᾳ ἴση ἡ περιεχομένη ὑπὸ τῶν ΒΑΘ, ΘΑΚ, ΚΑΒ, ὥστε ἴσην εἶναι τὴν μὲν ὑπὸ ΒΑΘ γωνίαν τῇ ὑπὸ ΕΓΖ, τὴν δὲ ὑπὸ ΒΑΚ τῇ ὑπὸ ΕΓΗ, τὴν δὲ ὑπὸ ΚΑΘ τῇ ὑπὸ ΗΓΖ· καὶ γεγονέτω ὡς μὲν ἡ ΕΓ πρὸς τὴν ΓΗ, οὕτως ἡ ΒΑ πρὸς τὴν ΑΚ, ὡς δὲ ἡ ΗΓ πρὸς τὴν ΓΖ, οὕτως ἡ ΚΑ πρὸς τὴν ΑΘ. καὶ διʼ ἴσου ἄρα ἐστὶν ὡς ἡ ΕΓ πρὸς τὴν ΓΖ, οὕτως ἡ ΒΑ πρὸς τὴν ΑΘ. καὶ συμπεπληρώσθω τὸ ΘΒ παραλληλόγραμμον καὶ τὸ ΑΛ στερεόν.
For let there be constructed on the straight line AB and at the point A on it, equal to the solid angle at C, the angle contained by BAT, TAK, KAB, so that the angle BAT is equal to ECF, the angle BAK to ECG, and the angle KAT to GCF; and let it be: as EC is to CG, so BA to AK, and as CG is to CF, so KA to AT. Therefore also, ex aequali, as EC is to CF, so BA to AT. And let the parallelogram TB and the solid AL be completed.
καὶ ἐπεί ἐστιν ὡς ἡ ΕΓ πρὸς τὴν ΓΗ, οὕτως ἡ ΒΑ πρὸς τὴν ΑΚ, καὶ περὶ ἴσας γωνίας τὰς ὑπὸ ΕΓΗ, ΒΑΚ αἱ πλευραὶ ἀνάλογόν εἰσιν, ὅμοιον ἄρα ἐστὶ τὸ ΗΕ παραλληλόγραμμον τῷ ΚΒ παραλληλογράμμῳ.
And since as EC is to CG, so BA to AK, and the sides about the equal angles ECG, BAK are proportional, therefore the parallelogram EG is similar to the parallelogram KB.
διὰ τὰ αὐτὰ δὴ καὶ τὸ μὲν ΚΘ παραλληλόγραμμον τῷ ΗΖ παραλληλογράμμῳ ὅμοιόν ἐστι καὶ ἔτι τὸ ΖΕ τῷ ΘΒ· τρία ἄρα παραλληλόγραμμα τοῦ ΓΔ στερεοῦ τρισὶ παραλληλογράμμοις τοῦ ΑΛ στερεοῦ ὅμοιά ἐστιν.
For the same reasons indeed, the parallelogram KT is also similar to the parallelogram GF, and further FE to TB; therefore three parallelograms of the solid CD are similar to three parallelograms of the solid AL.
ἀλλὰ τὰ μὲν τρία τρισὶ τοῖς ἀπεναντίον ἴσα τέ ἐστι καὶ ὅμοια, τὰ δὲ τρία τρισὶ τοῖς ἀπεναντίον ἴσα τέ ἐστι καὶ ὅμοια· ὅλον ἄρα τὸ ΓΔ στερεὸν ὅλῳ τῷ ΑΛ στερεῷ ὅμοιόν ἐστιν.
But the three are equal and similar to the three opposite, and the three are equal and similar to the three opposite; therefore the whole solid CD is similar to the whole solid AL.
ἀπὸ τῆς δοθείσης ἄρα εὐθείας τῆς ΑΒ τῷ δοθέντι στερεῷ παραλληλεπιπέδῳ τῷ ΓΔ ὅμοιόν τε καὶ ὁμοίως κείμενον ἀναγέγραπται τὸ ΑΛ· ὅπερ ἔδει ποιῆσαι.
Therefore, on the given straight line AB, there has been described the solid AL similar and similarly situated to the given solid parallelepiped CD; which was required to do.
§11.prop.28ἐὰν στερεὸν παραλληλεπίπεδον ἐπιπέδῳ τμηθῇ κατὰ τὰς διαγωνίους τῶν ἀπεναντίον ἐπιπέδων, δίχα τμηθήσεται τὸ στερεὸν ὑπὸ τοῦ ἐπιπέδου.
If a solid parallelepiped be cut by a plane through the diagonals of the opposite planes, the solid will be bisected by the plane.
στερεὸν γὰρ παραλληλεπίπεδον τὸ ΑΒ ἐπιπέδῳ τῷ ΓΔΕΖ τετμήσθω κατὰ τὰς διαγωνίους τῶν ἀπεναντίον ἐπιπέδων τὰς ΓΖ, ΔΕ· λέγω, ὅτι δίχα τμηθήσεται τὸ ΑΒ στερεὸν ὑπὸ τοῦ ΓΔΕΖ ἐπιπέδου.
For let the solid parallelepiped AB be cut by the plane CDEF through the diagonals CF, DE of the opposite planes; I say that the solid AB will be bisected by the plane CDEF.
ἐπεὶ γὰρ ἴσον ἐστὶ τὸ μὲν ΓΗΖ τρίγωνον τῷ ΓΖΒ τριγώνῳ, τὸ δὲ ΑΔΕ τῷ ΔΕΘ, ἔστι δὲ καὶ τὸ μὲν ΓΑ παραλληλόγραμμον τῷ ΕΒ ἴσον· ἀπεναντίον γάρ· τὸ δὲ ΗΕ τῷ ΓΘ, καὶ τὸ πρίσμα ἄρα τὸ περιεχόμενον ὑπὸ δύο μὲν τριγώνων τῶν ΓΗΖ, ΑΔΕ, τριῶν δὲ παραλληλογράμμων τῶν ΗΕ, ΑΓ, ΓΕ ἴσον ἐστὶ τῷ πρίσματι τῷ περιεχομένῳ ὑπὸ δύο μὲν τριγώνων τῶν ΓΖΒ, ΔΕΘ, τριῶν δὲ παραλληλογράμμων τῶν ΓΘ, ΒΕ, ΓΕ· ὑπὸ γὰρ ἴσων ἐπιπέδων περιέχονται τῷ τε πλήθει καὶ τῷ μεγέθει.
For since the triangle CGF is equal to the triangle CFB, and ADE to DEH, and the parallelogram CA is also equal to EB—for they are opposite—and EG to CH, therefore the prism contained by two triangles CGF, ADE and three parallelograms EG, AC, CE is equal to the prism contained by two triangles CFB, DEH and three parallelograms CH, BE, CE; for they are contained by planes equal both in multitude and in magnitude.
ὥστε ὅλον τὸ ΑΒ στερεὸν δίχα τέτμηται ὑπὸ τοῦ ΓΔΕΖ ἐπιπέδου· ὅπερ ἔδει δεῖξαι.
Therefore the whole solid AB is bisected by the plane CDEF; which was required to prove.

Notes

  1. 10συνεστάτω ... ἴση ἡ περιεχομένη — The subject of the third-person singular imperative `συνεστάτω` (let there be constructed) is the subsequent feminine singular nominative `ἡ περιεχομένη` (the solid angle contained by...). The adjective `ἴση` (equal, nominative) modifies this subject and takes the dative `τῇ ... στερεᾷ γωνίᾳ` (to the solid angle at C) to indicate equality.
  2. 13ὥστε ἴσην εἶναι τὴν μὲν ὑπὸ ΒΑΘ γωνίαν — Inside the result/purpose clause introduced by the conjunction `ὥστε`, an accusative-and-infinitive construction is used, where `τὴν ... γωνίαν` is the subject accusative of `εἶναι`, and `ἴσην` is the predicate accusative.
  3. 14τὸ πρίσμα ἄρα τὸ περιεχόμενον ... ἴσον ἐστὶ τῷ πρίσματι — Clarifies the structure of this long sentence. The nominative `τὸ πρίσμα ...` (the prism) is the subject, qualified by the passive participle `τὸ περιεχόμενον` (contained) which governs the agent phrase `ὑπὸ δύο μὲν...` (by two... and three...). The predicate is `ἴσον ἐστὶ` (is equal), taking the dative `τῷ πρίσματι` (to the prism) as the object of comparison.
  4. 19τῷ τε πλήθει καὶ τῷ μεγέθει — These are the dative forms of the nouns `πλῆθος` (multitude) and `μέγεθος` (magnitude), functioning as datives of respect (in respect to, in terms of) to specify the scope of the adjective `ἴσων` (equal).

Cite this passage

Euclid, Elements §11.prop.27-11.prop.28. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:11.prop.27-11.prop.28

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