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ἐὰν ἀπὸ ῥητῆς ῥητὴ ἀφαιρεθῇ δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ, λοιπὴ ἄλογός ἐστιν: καλείσθω δὲ ἀποτομή.

ἀπὸ γὰρ ῥητῆς τῆς ΑΒ ῥητὴ ἀφῃρήσθω ΒΓ δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ: λέγω, ὅτι λοιπὴ ΑΓ
5ἄλογός ἐστιν καλουμένη ἀποτομή.

ἐπεὶ γὰρ ἀσύμμετρός ἐστιν ΑΒ τῇ ΒΓ μήκει, καί ἐστιν ὡς ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ὑπὸ τῶν ΑΒ, ΒΓ, ἀσύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΑΒ τῷ ὑπὸ τῶν ΑΒ, ΒΓ. ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΒ
10σύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ τετράγωνα, τῷ δὲ ὑπὸ τῶν ΑΒ, ΒΓ σύμμετρόν ἐστι τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ. καὶ ἐπειδήπερ τὰ ἀπὸ τῶν ΑΒ, ΒΓ ἴσα ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ μετὰ τοῦ ἀπὸ ΓΑ, καὶ λοιπῷ ἄρα
10τῷ ἀπὸ τῆς ΑΓ ἀσύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ.
15ῥητὰ δὲ τὰ ἀπὸ τῶν ΑΒ, ΒΓ: ἄλογος ἄρα ἐστὶν ΑΓ: καλείσθω δὲ ἀποτομή. ὅπερ ἔδει δεῖξαι.

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