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τὴν δοθεῖσαν εὐθεῖαν ἄτμητον τῇ δοθείσῃ τετμημένῃ ὁμοίως τεμεῖν.

ἔστω μὲν δοθεῖσα εὐθεῖα ἄτμητος ΑΒ, δὲ τετμημένη ΑΓ κατὰ τὰ Δ, Ε σημεῖα, καὶ κείσθωσαν
5ὥστε γωνίαν τυχοῦσαν περιέχειν, καὶ ἐπεζεύχθω ΓΒ, καὶ διὰ τῶν Δ, Ε τῇ ΒΓ παράλληλοι ἤχθωσαν αἱ ΔΖ, ΕΗ, διὰ δὲ τοῦ Δ τῇ ΑΒ παράλληλος ἤχθω ΔΘΚ.
10

παραλληλόγραμμον ἄρα ἐστὶν ἑκάτερον τῶν ΖΘ, ΘΒ: ἴση ἄρα μὲν ΔΘ τῇ ΖΗ, δὲ ΘΚ τῇ ΗΒ. καὶ ἐπεὶ τριγώνου τοῦ ΔΚΓ παρὰ μίαν τῶν πλευρῶν τὴν ΚΓ
15εὐθεῖα ἦκται ΘΕ, ἀνάλογον ἄρα ἐστὶν ὡς ΓΕ πρὸς τὴν ΕΔ, οὕτως ΚΘ πρὸς τὴν ΘΔ. ἴση δὲ μὲν ΚΘ τῇ ΒΗ, δὲ ΘΔ τῇ ΗΖ. ἔστιν ἄρα ὡς ΓΕ πρὸς τὴν ΕΔ, οὕτως ΒΗ πρὸς τὴν ΗΖ. πάλιν, ἐπεὶ τριγώνου τοῦ ΑΗΕ παρὰ μίαν τῶν πλευρῶν τὴν ΗΕ ἦκται ΖΔ, ἀνάλογον
20ἄρα ἐστὶν ὡς ΕΔ πρὸς τὴν ΔΑ, οὕτως ΗΖ πρὸς τὴν ΖΑ. ἐδείχθη δὲ καὶ ὡς ΓΕ πρὸς τὴν ΕΔ, οὕτως ΒΗ πρὸς τὴν ΗΖ: ἔστιν ἄρα ὡς μὲν ΓΕ πρὸς τὴν ΕΔ, οὕτως ΒΗ πρὸς τὴν ΗΖ, ὡς δὲ ΕΔ πρὸς τὴν ΔΑ, οὕτως ΗΖ πρὸς τὴν ΖΑ.
25

ἄρα δοθεῖσα εὐθεῖα ἄτμητος ΑΒ τῇ δοθείσῃ εὐθείᾳ τετμημένῃ τῇ ΑΓ ὁμοίως τέτμηται: ὅπερ ἔδει ποιῆσαι.

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