§10.prop3.105ἡ τῇ ἐλάσσονι σύμμετρος ἐλάσσων ἐστίν. ἔστω γὰρ ἐλάσσων ἡ ΑΒ καὶ τῇ ΑΒ σύμμετρος ἡ ΓΔ· λέγω, ὅτι καὶ ἡ ΓΔ ἐλάσσων ἐστίν.
A straight line commensurable with a minor straight line is minor. For let AB be a minor straight line, and let GD be commensurable with AB; I say that GD is also minor.
γεγονέτω γὰρ τὰ αὐτά· καὶ ἐπεὶ αἱ ΑΕ, ΕΒ δυνάμει εἰσὶν ἀσύμμετροι, καὶ αἱ ΓΖ, ΖΔ ἄρα δυνάμει εἰσὶν ἀσύμμετροι.
For let the same construction be made; and since AE, EB are incommensurable in square, therefore GZ, ZD are also incommensurable in square.
ἐπεὶ οὖν ἐστιν ὡς ἡ ΑΕ πρὸς τὴν ΕΒ, οὕτως ἡ ΓΖ πρὸς τὴν ΖΔ, ἔστιν ἄρα καὶ ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ τῆς ΕΒ, οὕτως τὸ ἀπὸ τῆς ΓΖ πρὸς τὸ ἀπὸ τῆς ΖΔ. συνθέντι ἄρα ἐστὶν ὡς τὰ ἀπὸ τῶν ΑΕ, ΕΒ πρὸς τὸ ἀπὸ τῆς ΕΒ, οὕτως τὰ ἀπὸ τῶν ΓΖ, ΖΔ πρὸς τὸ ἀπὸ τῆς ΖΔ· σύμμετρον δέ ἐστι τὸ ἀπὸ τῆς ΒΕ τῷ ἀπὸ τῆς ΔΖ· σύμμετρον ἄρα καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων. ῥητὸν δέ ἐστι τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων· ῥητὸν ἄρα ἐστὶ καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων.
Since then as AE is to EB, so is GZ to ZD, therefore also as the square on AE is to the square on EB, so is the square on GZ to the square on ZD. Therefore, by addition, as the sum of the squares on AE, EB is to the square on EB, so is the sum of the squares on GZ, ZD to the square on ZD. Since then as AE is to EB, so is GZ to ZD, therefore also as the square on AE is to the square on EB, so is the square on GZ to the square on ZD. Therefore, by addition, as the sum of the squares on AE, EB is to the square on EB, so is the sum of the squares on GZ, ZD to the square on ZD. But the square on BE is commensurable with the square on DZ; therefore the sum of the squares on AE, EB is also commensurable with the sum of the squares on GZ, ZD. Therefore, by addition, as the sum of the squares on AE, EB is to the square on EB, so is the sum of the squares on GZ, ZD to the square on ZD. But the square on BE is commensurable with the square on DZ; therefore the sum of the squares on AE, EB is also commensurable with the sum of the squares on GZ, ZD. But the sum of the squares on AE, EB is rational; therefore the sum of the squares on GZ, ZD is also rational.
πάλιν, ἐπεί ἐστιν ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ὑπὸ τῶν ΑΕ, ΕΒ, οὕτως τὸ ἀπὸ τῆς ΓΖ πρὸς τὸ ὑπὸ τῶν ΓΖ, ΖΔ, σύμμετρον δὲ τὸ ἀπὸ τῆς ΑΕ τετράγωνον τῷ ἀπὸ τῆς ΓΖ τετραγώνῳ, σύμμετρον ἄρα ἐστὶ καὶ τὸ ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν ΓΖ, ΖΔ. μέσον δὲ τὸ ὑπὸ τῶν ΑΕ, ΕΒ· μέσον ἄρα καὶ τὸ ὑπὸ τῶν ΓΖ, ΖΔ· αἱ ΓΖ, ΖΔ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τετραγώνων ῥητόν, τὸ δʼ ὑπʼ αὐτῶν μέσον. ἐλάσσων ἄρα ἐστὶν ἡ ΓΔ· ὅπερ ἔδει δεῖξαι.
Again, since as the square on AE is to the rectangle contained by AE, EB, so is the square on GZ to the rectangle contained by GZ, ZD, and the square on AE is commensurable with the square on GZ, therefore the rectangle contained by AE, EB is also commensurable with the rectangle contained by GZ, ZD. But the rectangle contained by AE, EB is medial; therefore the rectangle contained by GZ, ZD is also medial. Therefore GZ, ZD are incommensurable in square, producing the sum of the squares on them rational, but the rectangle contained by them medial. But the rectangle contained by AE, EB is medial; therefore the rectangle contained by GZ, ZD is also medial. Therefore GZ, ZD are incommensurable in square, producing the sum of the squares on them rational, but the rectangle contained by them medial. Therefore GD is minor; which was to be proved. Therefore GD is minor; which was to be proved.
§10.prop3.106ἡ τῇ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιούσῃ σύμμετρος μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσά ἐστιν. ἔστω μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα ἡ ΑΒ καὶ τῇ ΑΒ σύμμετρος ἡ ΓΔ· λέγω, ὅτι καὶ ἡ ΓΔ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσά ἐστιν. ἔστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΕ· αἱ ΑΕ, ΕΒ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων μέσον, τὸ δʼ ὑπʼ αὐτῶν ῥητόν. καὶ τὰ αὐτὰ κατεσκευάσθω.
A straight line commensurable with that which produces with a rational area a medial whole is that which produces with a rational area a medial whole. For let AB be that which produces with a rational area a medial whole, and let GD be commensurable with AB; I say that GD is also that which produces with a rational area a medial whole. For let AB be that which produces with a rational area a medial whole, and let GD be commensurable with AB; I say that GD is also that which produces with a rational area a medial whole. For let BE be the annex to AB; therefore AE, EB are incommensurable in square, producing the sum of the squares on AE, EB medial, but the rectangle contained by them rational. And let the same construction be made. For let BE be the annex to AB; therefore AE, EB are incommensurable in square, producing the sum of the squares on AE, EB medial, but the rectangle contained by them rational. And let the same construction be made.
ὁμοίως δὴ δείξομεν τοῖς πρότερον, ὅτι αἱ ΓΖ, ΖΔ ἐν τῷ αὐτῷ λόγῳ εἰσὶ ταῖς ΑΕ, ΕΒ, καὶ σύμμετρόν ἐστι τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων, τὸ δὲ ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν ΓΖ, ΖΔ· ὥστε καὶ αἱ ΓΖ, ΖΔ δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων μέσον, τὸ δʼ ὑπʼ αὐτῶν ῥητόν.
Similarly then we shall prove as before that GZ, ZD are in the same ratio as AE, EB, and the sum of the squares on AE, EB is commensurable with the sum of the squares on GZ, ZD, and the rectangle contained by AE, EB with the rectangle contained by GZ, ZD; so that GZ, ZD are also incommensurable in square, producing the sum of the squares on GZ, ZD medial, but the rectangle contained by them rational. so that GZ, ZD are also incommensurable in square, producing the sum of the squares on GZ, ZD medial, but the rectangle contained by them rational.
ἡ ΓΔ ἄρα μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσά ἐστιν· ὅπερ ἔδει δεῖξαι.
Therefore GD is that which produces with a rational area a medial whole; which was to be proved. Therefore GD is that which produces with a rational area a medial whole; which was to be proved.