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ἐὰν δύο εὐθεῖαι ὑπὸ παραλλήλων ἐπιπέδων τέμνωνται εἰς τοὺς αὐτοὺς λόγους τμηθήσονται.

δύο γὰρ εὐθεῖαι αἱ ΑΒ, ΓΔ ὑπὸ παραλλήλων ἐπιπέδων τῶν ΗΘ, ΚΛ, ΜΝ τεμνέσθωσαν κατὰ τὰ Α, Ε, Β,
5Γ, Ζ, Δ σημεῖα: λέγω, ὅτι ἐστὶν ὡς ΑΕ εὐθεῖα πρὸς τὴν ΕΒ, οὕτως ΓΖ πρὸς τὴν ΖΔ.

ἐπεζεύχθωσαν γὰρ αἱ ΑΓ, ΒΔ, ΑΔ, καὶ συμβαλλέτω ΑΔ τῷ ΚΛ ἐπιπέδῳ κατὰ τὸ Ξ σημεῖον, καὶ ἐπεζεύχθωσαν αἱ ΕΞ, ΞΖ. καὶ ἐπεὶ δύο ἐπίπεδα παράλληλα τὰ ΚΛ,
10ΜΝ ὑπὸ ἐπιπέδου τοῦ ΕΒΔΞ τέμνεται, αἱ κοιναὶ αὐτῶν τομαὶ αἱ ΕΞ, ΒΔ παράλληλοί εἰσιν. διὰ τὰ αὐτὰ δὴ ἐπεὶ δύο ἐπίπεδα παράλληλα τὰ ΗΘ, ΚΛ ὑπὸ ἐπιπέδου τοῦ ΑΞΖΓ τέμνεται, αἱ κοιναὶ αὐτῶν τομαὶ αἱ ΑΓ, ΞΖ παράλληλοί εἰσιν. καὶ ἐπεὶ τριγώνου τοῦ ΑΒΔ παρὰ μίαν
15τῶν πλευρῶν τὴν ΒΔ εὐθεῖα ἦκται ΕΞ, ἀνάλογον ἄρα ἐστὶν ὡς ΑΕ πρὸς ΕΒ, οὕτως ΑΞ πρὸς ΞΔ. πάλιν ἐπεὶ τριγώνου τοῦ ΑΔΓ παρὰ μίαν τῶν πλευρῶν τὴν ΑΓ εὐθεῖα ἦκται ΞΖ, ἀνάλογόν ἐστιν ὡς ΑΞ πρὸς ΞΔ, οὕτως ΓΖ πρὸς ΖΔ. ἐδείχθη δὲ καὶ ὡς ΑΞ πρὸς ΞΔ,
20οὕτως ΑΕ πρὸς ΕΒ: καὶ ὡς ἄρα ΑΕ πρὸς ΕΒ, οὕτως
20 ΓΖ πρὸς ΖΔ.

ἐὰν ἄρα δύο εὐθεῖαι ὑπὸ παραλλήλων ἐπιπέδων τέμνωνται, εἰς τοὺς αὐτοὺς λόγους τμηθήσονται: ὅπερ ἔδει δεῖξαι.

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load focus English (Thomas L. Heath, Sir Thomas Little Heath, 1956)
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