§10.prop2.69ἡ τῇ ῥητὸν καὶ μέσον δυναμένῃ σύμμετρος ῥητὸν καὶ μέσον δυναμένη ἐστίν.
A straight line commensurable with that which produces a rational and a medial area is itself also that which produces a rational and a medial area.
ἔστω ῥητὸν καὶ μέσον δυναμένη ἡ ΑΒ, καὶ τῇ ΑΒ σύμμετρος ἔστω ἡ ΓΔ· δεικτέον, ὅτι καὶ ἡ ΓΔ ῥητὸν καὶ μέσον δυναμένη ἐστίν.
Let AB be that which produces a rational and a medial area, and let CD be commensurable with AB; it is to be proved that CD is also that which produces a rational and a medial area.
διῃρήσθω ἡ ΑΒ εἰς τὰς εὐθείας κατὰ τὸ Ε· αἱ ΑΕ, ΕΒ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τετραγώνων μέσον, τὸ δʼ ὑπʼ αὐτῶν ῥητόν·
Let AB be divided into its component straight lines at E; therefore AE, EB are incommensurable in square, making the sum of the squares on them medial, and the rectangle contained by them rational.
καὶ τὰ αὐτὰ κατεσκευάσθω τοῖς πρότερον.
And let the same construction be made as in what was proved before.
ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ ΓΖ, ΖΔ δυνάμει εἰσὶν ἀσύμμετροι, καὶ σύμμετρον τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ, τὸ δὲ ὑπὸ ΑΕ, ΕΒ τῷ ὑπὸ ΓΖ, ΖΔ· ὥστε καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων ἐστὶ μέσον, τὸ δʼ ὑπὸ τῶν ΓΖ, ΖΔ ῥητόν.
Similarly we will prove that CZ, ZD are also incommensurable in square, and the sum of the squares on AE, EB is commensurable with the sum of the squares on CZ, ZD, and the rectangle contained by AE, EB with that contained by CZ, ZD; so that the sum of the squares on CZ, ZD is also medial, and the rectangle contained by CZ, ZD is rational.
ῥητὸν ἄρα καὶ μέσον δυναμένη ἐστὶν ἡ ΓΔ· ὅπερ ἔδει δεῖξαι.
Therefore CD is that which produces a rational and a medial area; which was to be proved.
§10.prop2.70ἡ τῇ δύο μέσα δυναμένῃ σύμμετρος δύο μέσα δυναμένη ἐστίν.
A straight line commensurable with that which produces two medial areas is itself also that which produces two medial areas.
ἔστω δύο μέσα δυναμένη ἡ ΑΒ, καὶ τῇ ΑΒ σύμμετρος ἡ ΓΔ· δεικτέον, ὅτι καὶ ἡ ΓΔ δύο μέσα δυναμένη ἐστίν.
Let AB be that which produces two medial areas, and let CD be commensurable with AB; it is to be proved that CD is also that which produces two medial areas.
ʼἐπεὶ γὰρ δύο μέσα δυναμένη ἐστὶν ἡ ΑΒ, διῃρήσθω εἰς τὰς εὐθείας κατὰ τὸ Ε· αἱ ΑΕ, ΕΒ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν μέσον καὶ τὸ ὑπʼ αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων τῷ ὑπὸ τῶν ΑΕ, ΕΒ·
For since AB produces two medial areas, let it be divided into its component straight lines at E; therefore AE, EB are incommensurable in square, making the sum of the squares on them medial, and the rectangle contained by them medial, and furthermore the sum of the squares on AE, EB incommensurable with the rectangle contained by AE, EB.
καὶ κατεσκευάσθω τὰ αὐτὰ τοῖς πρότερον.
And let the same construction be made as in what was proved before.
ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ ΓΖ, ΖΔ δυνάμει εἰσὶν ἀσύμμετροι καὶ σύμμετρον τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ, τὸ δὲ ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν ΓΖ, ΖΔ· ὥστε καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων μέσον ἐστὶ καὶ τὸ ὑπὸ τῶν ΓΖ, ΖΔ μέσον καὶ ἔτι ἀσύμμετρον τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων τῷ ὑπὸ τῶν ΓΖ, ΖΔ.
ἡ ἄρα ΓΔ δύο μέσα δυναμένη ἐστίν·
Similarly we will prove that CZ, ZD are also incommensurable in square, and the sum of the squares on AE, EB is commensurable with the sum of the squares on CZ, ZD, and the rectangle contained by AE, EB with that contained by CZ, ZD; so that the sum of the squares on CZ, ZD is also medial, and the rectangle contained by CZ, ZD is medial, and furthermore the sum of the squares on CZ, ZD is incommensurable with the rectangle contained by CZ, ZD.
ὅπερ ἔδει δεῖξαι.
Therefore CD is that which produces two medial areas; which was to be proved.