Euclid · Elements §12.prop.3#2

Equality of the Prisms and Comparison with the Whole

Passage 272 of 316 · Greek

Summary

This chunk shows that each of the divided pyramids is similar to the whole, and proves that the two prisms are equal since they are of equal height with a base ratio of two to one. Finally, it demonstrates that these two prisms are greater than the remaining two pyramids, thus concluding that they are greater than half of the whole.

§12.prop.3#2καί ἐστιν ὡς ἡ ΒΑ πρὸς τὴν ΑΓ, οὕτως ἡ ΚΘ πρὸς τὴν ΘΛ· ὅμοιον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΘΚΛ τριγώνῳ.

And as BA is to AG, so is KQ to QL; therefore the triangle ABG is similar to the triangle QKL.

καὶ πυραμὶς ἄρα, ἧς βάσις μέν ἐστι τὸ ΑΒΓ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον, ὁμοία ἐστὶ πυραμίδι, ἧς βάσις μέν ἐστι τὸ ΘΚΛ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον.

Therefore also the pyramid of which the base is the triangle ABG, and the vertex the point D, is similar to the pyramid of which the base is the triangle QKL, and the vertex the point D.

ἀλλὰ πυραμίς, ἧς βάσις μὲν τὸ ΘΚΛ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον, ὁμοία ἐδείχθη πυραμίδι, ἧς βάσις μέν ἐστι τὸ ΑΕΗ τρίγωνον, κορυφὴ δὲ τὸ Θ σημεῖον.

But the pyramid of which the base is the triangle QKL, and the vertex the point D, was shown to be similar to the pyramid of which the base is the triangle AEH, and the vertex the point Q.

ἑκατέρα ἄρα τῶν ΑΕΗΘ, ΘΚΛΔ πυραμίδων ὁμοία ἐστὶ τῇ ὅλῃ τῇ ΑΒΓΔ πυραμίδι.

Therefore each of the pyramids AEHQ, QKLD is similar to the whole pyramid ABGD.

¯Καὶ ἐπεὶ ἴση ἐστὶν ἡ ΒΖ τῇ ΖΓ, διπλάσιόν ἐστι τὸ ΕΒΖΗ παραλληλόγραμμον τοῦ ΗΖΓ τριγώνου.

And since BZ is equal to ZG, the parallelogram EBZH is double of the triangle HZG.

καὶ ἐπεί, ἐὰν ᾖ δύο πρίσματα ἰσοϋψῆ, καὶ τὸ μὲν ἔχῃ βάσιν παραλληλόγραμμον, τὸ δὲ τρίγωνον, διπλάσιον δὲ ᾖ τὸ παραλληλόγραμμον τοῦ τριγώνου, ἴσα ἐστὶ τὰ πρίσματα, ἴσον ἄρα ἐστὶ τὸ πρίσμα τὸ περιεχόμενον ὑπὸ δύο μὲν τριγώνων τῶν ΒΚΖ, ΕΘΗ, τριῶν δὲ παραλληλογράμμων τῶν ΕΒΖΗ, ΕΒΚΘ, ΘΚΖΗ τῷ πρίσματι τῷ περιεχομένῳ ὑπὸ δύο μὲν τριγώνων τῶν ΗΖΓ, ΘΚΛ, τριῶν δὲ παραλληλογράμμων τῶν ΚΖΓΛ, ΛΓΗΘ, ΘΚΖΗ. καὶ φανερόν, ὅτι ἑκάτερον τῶν πρισμάτων, οὗ τε βάσις τὸ ΕΒΖΗ παραλληλόγραμμον, ἀπεναντίον δὲ ἡ ΘΚ εὐθεῖα, καὶ οὗ βάσις τὸ ΗΖΓ τρίγωνον, ἀπεναντίον δὲ τὸ ΘΚΛ τρίγωνον, μεῖζόν ἐστιν ἑκατέρας τῶν πυραμίδων, ὧν βάσεις μὲν τὰ ΑΕΗ, ΘΚΛ τρίγωνα, κορυφαὶ δὲ τὰ Θ, Δ σημεῖα, ἐπειδήπερ ἐὰν ἐπιζεύξωμεν τὰς ΕΖ, ΕΚ εὐθείας, τὸ μὲν πρίσμα, οὗ βάσις τὸ ΕΒΖΗ παραλληλόγραμμον, ἀπεναντίον δὲ ἡ ΘΚ εὐθεῖα, μεῖζόν ἐστι τῆς πυραμίδος, ἧς βάσις τὸ ΕΒΖ τρίγωνον, κορυφὴ δὲ τὸ Κ σημεῖον.

And since, if there are two prisms of equal height, and one has a parallelogram as base, and the other a triangle, and the parallelogram is double of the triangle, the prisms are equal, therefore the prism contained by two triangles BKZ, EQH, and three parallelograms EBZH, EBKQ, QKZH, is equal to the prism contained by two triangles HZG, QKL, and three parallelograms KZGL, LGHQ, QKZH. And it is manifest that each of the prisms, both that of which the base is the parallelogram EBZH and the opposite the straight line QK, and that of which the base is the triangle HZG and the opposite the triangle QKL, is greater than each of the pyramids of which the bases are the triangles AEH, QKL and the vertices the points Q, D, because if we join the straight lines EZ, EK, the prism of which the base is the parallelogram EBZH and the opposite the straight line QK is greater than the pyramid of which the base is the triangle EBZ and the vertex the point K.

ἀλλʼ ἡ πυραμίς, ἧς βάσις τὸ ΕΒΖ τρίγωνον, κορυφὴ δὲ τὸ Κ σημεῖον, ἴση ἐστὶ πυραμίδι, ἧς βάσις τὸ ΑΕΗ τρίγωνον, κορυφὴ δὲ τὸ Θ σημεῖον· ὑπὸ γὰρ ἴσων καὶ ὁμοίων ἐπιπέδων περιέχονται.

But the pyramid of which the base is the triangle EBZ and the vertex the point K is equal to the pyramid of which the base is the triangle AEH and the vertex the point Q; for they are contained by equal and similar planes.

ὥστε καὶ τὸ πρίσμα, οὗ βάσις μὲν τὸ ΕΒΖΗ παραλληλόγραμμον, ἀπεναντίον δὲ ἡ ΘΚ εὐθεῖα, μεῖζόν ἐστι πυραμίδος, ἧς βάσις μὲν τὸ ΑΕΗ τρίγωνον, κορυφὴ δὲ τὸ Θ σημεῖον.

So that the prism also of which the base is the parallelogram EBZH and the opposite the straight line QK, is greater than the pyramid of which the base is the triangle AEH and the vertex the point Q.

ἴσον δὲ τὸ μὲν πρίσμα, οὗ βάσις τὸ ΕΒΖΗ παραλληλόγραμμον, ἀπεναντίον δὲ ἡ ΘΚ εὐθεῖα, τῷ πρίσματι, οὗ βάσις μὲν τὸ ΗΖΓ τρίγωνον, ἀπεναντίον δὲ τὸ ΘΚΛ τρίγωνον· ἡ δὲ πυραμίς, ἧς βάσις τὸ ΑΕΗ τρίγωνον, κορυφὴ δὲ τὸ Θ σημεῖον, ἴση ἐστὶ πυραμίδι, ἧς βάσις τὸ ΘΚΛ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον.

And the prism of which the base is the parallelogram EBZH and the opposite the straight line QK is equal to the prism of which the base is the triangle HZG and the opposite the triangle QKL; and the pyramid of which the base is the triangle AEH and the vertex the point Q is equal to the pyramid of which the base is the triangle QKL and the vertex the point D.

τὰ ἄρα εἰρημένα δύο πρίσματα μείζονά ἐστι τῶν εἰρημένων δύο πυραμίδων, ὧν βάσεις μὲν τὰ ΑΕΗ, ΘΚΛ τρίγωνα, κορυφαὶ δὲ τὰ Θ, Δ σημεῖα.

Therefore the two aforesaid prisms are greater than the two aforesaid pyramids of which the bases are the triangles AEH, QKL and the vertices the points Q, D.

ἡ ἄρα ὅλη πυραμίς, ἧς βάσις τὸ ΑΒΓ τρίγωνον, κορυφὴ δὲ τὸ Δ σημεῖον, διῄρηται εἴς τε δύο πυραμίδας ἴσας ἀλλήλαις καὶ εἰς δύο πρίσματα ἴσα, καὶ τὰ δύο πρίσματα μείζονά ἐστιν ἢ τὸ ἥμισυ τῆς ὅλης πυραμίδος· ὅπερ ἔδει δεῖξαι.

Therefor the whole pyramid of which the base is the triangle ABG and the vertex the point D has been divided into two pyramids equal to one another and into two equal prisms, and the two prisms are greater than half of the whole pyramid; which was to be proved.

Notes

65καὶ ἐπεί, ἐὰν ᾖ δύο πρίσματα... ἴσα ἐστὶ τὰ πρίσματα, — The clause introduced by the causal conjunction ἐπεί contains a general conditional sentence (ἐὰν + subjunctive ᾖ, ἔχῃ). This entire conditional structure ("if there are two prisms of equal height... the prisms are equal") forms the premise of ἐπεί, with the subsequent ἴσον ἄρα ἐστὶ τὸ πρίσμα... serving as the main clause.
75τῷ πρίσματι τῷ περιεχομένῳ ὑπὸ... — This is a dative governed by the preceding adjective ἴσον ("equal to..."). Two extremely long noun phrases modified by participles form the parallel comparative structure of "A (nominative: τὸ πρίσμα τὸ περιεχόμενον...) is equal to B (dative: τῷ πρίσματι τῷ περιεχομένῳ...)".
80καὶ φανερόν, ὅτι ἑκάτερον τῶν πρισμάτων... μεῖζόν ἐστιν ἑκατέρας τῶν πυραμίδων... — The subject of the ὅτι clause introduced by the impersonal expression φανερόν [ἐστιν] ("it is manifest that...") is ἑκάτερον τῶν πρισμάτων ("each of the prisms"), which is modified by parallel relative clauses. The predicate for this subject is μεῖζόν ἐστιν ("is greater than"), followed by the genitive of comparison ἑκατέρας τῶν πυραμίδων ("each of the pyramids").

Cite this passage

Euclid, Elements §12.prop.3#2. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:12.prop.3%232

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