§10.prop1.44ἡ ἐκ μέσων δευτέρα καθʼ ἓν μόνον σημεῖον διαιρεῖται.
A second bimedial straight line is divided at one point only.
ἔστω ἐκ δύο μέσων δευτέρα ἡ ΑΒ διῃρημένη κατὰ τὸ Γ, ὥστε τὰς ΑΓ, ΓΒ μέσας εἶναι δυνάμει μόνον συμμέτρους μέσον περιεχούσας·
For let the second bimedial straight line AB be divided at C, so that AC, CB are medial straight lines commensurable in square only containing a medial area.
φανερὸν δή, ὅτι τὸ Γ οὐκ ἔστι κατὰ τῆς διχοτομίας, ὅτι οὐκ εἰσὶ μήκει σύμμετροι.
It is then manifest that C is not at the point of bisection, because they are not commensurable in length.
λέγω, ὅτι ἡ ΑΒ κατʼ ἄλλο σημεῖον οὐ διαιρεῖται.
I say that AB is not divided at another point.
εἰ γὰρ δυνατόν, διῃρήσθω καὶ κατὰ τὸ Δ, ὥστε τὴν ΑΓ τῇ ΔΒ μὴ εἶναι τὴν αὐτήν, ἀλλὰ μείζονα καθʼ ὑπόθεσιν τὴν ΑΓ·
For, if possible, let it also be divided at D, so that AC is not the same with DB, but let AC be greater by hypothesis.
δῆλον δή, ὅτι καὶ τὰ ἀπὸ τῶν ΑΔ, ΔΒ, ὡς ἐπάνω ἐδείξαμεν, ἐλάσσονα τῶν ἀπὸ τῶν ΑΓ, ΓΒ·
It is then manifest that the sum of the squares on AD, DB is also, as we proved above, less than the sum of the squares on AC, CB.
καὶ τὰς ΑΔ, ΔΒ μέσας εἶναι δυνάμει μόνον συμμέτρους μέσον περιεχούσας.
And let AD, DB also be medial straight lines commensurable in square only containing a medial area.
καὶ ἐκκείσθω ῥητὴ ἡ ΕΖ, καὶ τῷ μὲν ἀπὸ τῆς ΑΒ ἴσον παρὰ τὴν ΕΖ παραλληλόγραμμον ὀρθογώνιον παραβεβλήσθω τὸ ΕΚ, τοῖς δὲ ἀπὸ τῶν ΑΓ, ΓΒ ἴσον ἀφῃρήσθω τὸ ΕΗ· λοιπὸν ἄρα τὸ ΘΚ ἴσον ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ. πάλιν δὴ τοῖς ἀπὸ τῶν ΑΔ, ΔΒ, ἅπερ ἐλάσσονα ἐδείχθη τῶν ἀπὸ τῶν ΑΓ, ΓΒ, ἴσον ἀφῃρήσθω τὸ ΕΛ· καὶ λοιπὸν ἄρα τὸ ΜΚ ἴσον τῷ δὶς ὑπὸ τῶν ΑΔ, ΔΒ. καὶ ἐπεὶ μέσα ἐστὶ τὰ ἀπὸ τῶν ΑΓ, ΓΒ, μέσον ἄρα τὸ ΕΗ. καὶ παρὰ ῥητὴν τὴν ΕΖ παράκειται· ῥητὴ ἄρα ἐστὶν ἡ ΕΘ καὶ ἀσύμμετρος τῇ ΕΖ μήκει.
And let the rational straight line EF be set out, and let there be applied to EF the rectangle EK equal to the square on AB, and let there be cut off EH equal to the sum of the squares on AC, CB; therefore the remainder HK is equal to twice the rectangle contained by AC, CB. Again, let there be cut off EL equal to the sum of the squares on AD, DB, which was proved to be less than the sum of the squares on AC, CB; and therefore the remainder MK is equal to twice the rectangle contained by AD, DB. And, since the sum of the squares on AC, CB is medial, EH is also medial. And it is applied to the rational straight line EF; therefore EG is rational and incommensurable in length with EF.
διὰ τὰ αὐτὰ δὴ καὶ ἡ ΘΝ ῥητή ἐστι καὶ ἀσύμμετρος τῇ ΕΖ μήκει.
For the same reason then, GN is also rational and incommensurable in length with EF.
καὶ ἐπεὶ αἱ ΑΓ, ΓΒ μέσαι εἰσὶ δυνάμει μόνον σύμμετροι, ἀσύμμετρος ἄρα ἐστὶν ἡ ΑΓ τῇ ΓΒ μήκει.
And, since AC, CB are medial straight lines commensurable in square only, AC is incommensurable in length with CB.
ὡς δὲ ἡ ΑΓ πρὸς τὴν ΓΒ, οὕτως τὸ ἀπὸ τῆς ΑΓ πρὸς τὸ ὑπὸ τῶν ΑΓ, ΓΒ· ἀσύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΑΓ τῷ ὑπὸ τῶν ΑΓ, ΓΒ. ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΓ σύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΓ, ΓΒ· δυνάμει γάρ εἰσι σύμμετροι αἱ ΑΓ, ΓΒ. τῷ δὲ ὑπὸ τῶν ΑΓ, ΓΒ σύμμετρόν ἐστι τὸ δὶς ὑπὸ τῶν ΑΓ, ΓΒ. καὶ τὰ ἀπὸ τῶν ΑΓ, ΓΒ ἄρα ἀσύμμετρά ἐστι τῷ δὶς ὑπὸ τῶν ΑΓ, ΓΒ. ἀλλὰ τοῖς μὲν ἀπὸ τῶν ΑΓ, ΓΒ ἴσον ἐστὶ τὸ ΕΗ, τῷ δὲ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ἴσον τὸ ΘΚ·
But as AC is to CB, so is the square on AC to the rectangle contained by AC, CB; therefore the square on AC is incommensurable with the rectangle contained by AC, CB. But the square on AC is commensurable with the sum of the squares on AC, CB; for AC, CB are commensurable in square. And the rectangle contained by AC, CB is commensurable with twice the rectangle contained by AC, CB. Therefore the sum of the squares on AC, CB is also incommensurable with twice the rectangle contained by AC, CB.
ἀσύμμετρον ἄρα ἐστὶ τὸ ΕΗ τῷ ΘΚ· ὥστε καὶ ἡ ΕΘ τῇ ΘΝ ἀσύμμετρός ἐστι μήκει.
But EH is equal to the sum of the squares on AC, CB, and HK is equal to twice the rectangle contained by AC, CB; therefore EH is incommensurable with HK; so that EG is also incommensurable in length with GN.
καί εἰσι ῥηταί· αἱ ΕΘ, ΘΝ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι.
And they are rational; therefore EG, GN are rational straight lines commensurable in square only.
ἐὰν δὲ δύο ῥηταὶ δυνάμει μόνον σύμμετροι συντεθῶσιν, ἡ ὅλη ἄλογός ἐστιν ἡ καλουμένη ἐκ δύο ὀνομάτων· ἡ ΕΝ ἄρα ἐκ δύο ὀνομάτων ἐστὶ διῃρημένη κατὰ τὸ Θ. κατὰ τὰ αὐτὰ δὴ δειχθήσονται καὶ αἱ ΕΜ, ΜΝ ῥηταὶ δυνάμει μόνον σύμμετροι· καὶ ἔσται ἡ ΕΝ ἐκ δύο ὀνομάτων κατʼ ἄλλο καὶ ἄλλο διῃρημένη τό τε Θ καὶ τὸ Μ, καὶ οὐκ ἔστιν ἡ ΕΘ τῇ ΜΝ ἡ αὐτή, ὅτι τὰ ἀπὸ τῶν ΑΓ, ΓΒ μείζονά ἐστι τῶν ἀπὸ τῶν ΑΔ, ΔΒ. ἀλλὰ τὰ ἀπὸ τῶν ΑΔ, ΔΒ μείζονά ἐστι τοῦ δὶς ὑπὸ ΑΔ, ΔΒ·
And if two rational straight lines commensurable in square only be added together, the whole is an irrational straight line called binomial; therefore EN is a binomial straight line divided at G. In the same way then, EM, MN will also be proved to be rational straight lines commensurable in square only; and EN, which is a binomial straight line, will be divided at different points, both G and M, and EG is not the same with MN, because the sum of the squares on AC, CB is greater than the sum of the squares on AD, DB.
πολλῷ ἄρα καὶ τὰ ἀπὸ τῶν ΑΓ, ΓΒ, τουτέστι τὸ ΕΗ, μεῖζόν ἐστι τοῦ δὶς ὑπὸ τῶν ΑΔ, ΔΒ, τουτέστι τοῦ ΜΚ· ὥστε καὶ ἡ ΕΘ τῆς ΜΝ μείζων ἐστίν.
But the sum of the squares on AD, DB is greater than twice the rectangle contained by AD, DB; therefore much more is the sum of the squares on AC, CB, that is, EH, greater than twice the rectangle contained by AD, DB, that is, MK; so that EG is also greater than MN.
ἡ ἄρα ΕΘ τῇ ΜΝ οὐκ ἔστιν ἡ αὐτή· ὅπερ ἔδει δεῖξαι.
Therefore EG is not the same with MN; which was to be proved.