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τῇ μείζονι σύμμετρος καὶ αὐτὴ μείζων ἐστίν.

ἔστω μείζων ΑΒ, καὶ τῇ ΑΒ σύμμετρος ἔστω ΓΔ: λέγω, ὅτι ΓΔ μείζων ἐστίν.

διῃρήσθω ΑΒ κατὰ τὸ Ε: αἱ ΑΕ, ΕΒ ἄρα δυνάμει
5εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ᾽ αὐτῶν τετραγώνων ῥητόν, τὸ δ᾽ ὑπ᾽ αὐτῶν μέσον: καὶ γεγονέτω τὰ αὐτὰ τοῖς πρότερον. καὶ ἐπεί ἐστιν ὡς ΑΒ πρὸς τὴν ΓΔ, οὕτως τε ΑΕ πρὸς τὴν ΓΖ καὶ ΕΒ
10πρὸς τὴν ΖΔ, καὶ ὡς ἄρα ΑΕ πρὸς τὴν ΓΖ, οὕτως ΕΒ πρὸς τὴν ΖΔ. σύμμετρος δὲ ΑΒ τῇ ΓΔ. σύμμετρος ἄρα καὶ ἑκατέρα τῶν ΑΕ, ΕΒ ἑκατέρᾳ τῶν ΓΖ, ΖΔ. καὶ ἐπεί ἐστιν ὡς ΑΕ πρὸς τὴν ΓΖ, οὕτως ΕΒ
15πρὸς τὴν ΖΔ, καὶ ἐναλλὰξ ὡς ΑΕ πρὸς ΕΒ, οὕτως ΓΖ πρὸς ΖΔ, καὶ συνθέντι ἄρα ἐστὶν ὡς ΑΒ πρὸς τὴν ΒΕ, οὕτως ΓΔ πρὸς τὴν ΔΖ: καὶ ὡς ἄρα τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΕ, οὕτως τὸ ἀπὸ τῆς ΓΔ πρὸς τὸ ἀπὸ τῆς ΔΖ. ὁμοίως δὴ δείξομεν, ὅτι καὶ
20ὡς τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΑΕ, οὕτως τὸ ἀπὸ τῆς ΓΔ πρὸς τὸ ἀπὸ τῆς ΓΖ. καὶ ὡς ἄρα τὸ ἀπὸ τῆς ΑΒ πρὸς τὰ ἀπὸ τῶν ΑΕ, ΕΒ, οὕτως τὸ ἀπὸ τῆς ΓΔ πρὸς τὰ ἀπὸ τῶν ΓΖ, ΖΔ: καὶ ἐναλλὰξ ἄρα ἐστὶν ὡς τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΓΔ, οὕτως τὰ ἀπὸ τῶν
25ΑΕ, ΕΒ πρὸς τὰ ἀπὸ τῶν ΓΖ, ΖΔ. σύμμετρον δὲ τὸ ἀπὸ τῆς ΑΒ τῷ ἀπὸ τῆς ΓΔ: σύμμετρα ἄρα καὶ τὰ ἀπὸ τῶν ΑΕ, ΕΒ τοῖς ἀπὸ τῶν ΓΖ, ΖΔ. καί ἐστι τὰ ἀπὸ τῶν ΑΕ, ΕΒ ἅμα ῥητόν, καὶ τὰ ἀπὸ τῶν ΓΖ, ΖΔ ἅμα ῥητόν ἐστιν. ὁμοίως δὲ καὶ τὸ δὶς ὑπὸ τῶν ΑΕ, ΕΒ σύμμετρόν
30ἐστι τῷ δὶς ὑπὸ τῶν ΓΖ, ΖΔ. καί ἐστι μέσον τὸ δὶς ὑπὸ τῶν ΑΕ, ΕΒ: μέσον ἄρα καὶ τὸ δὶς ὑπὸ τῶν ΓΖ, ΖΔ. αἱ ΓΖ, ΖΔ ἄρα δυνάμει ἀσύμμετροί εἰσι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ᾽ αὐτῶν τετραγώνων ἅμα ῥητόν, τὸ δὲ δὶς ὑπ᾽ αὐτῶν μέσον: ὅλη ἄρα ΓΔ ἄλογός
35ἐστιν καλουμένη μείζων.

ἄρα τῇ μείζονι σύμμετρος μείζων ἐστίν: ὅπερ ἔδει δεῖξαι.

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