§10.prop1.42ἡ ἐκ δύο ὀνομάτων κατὰ ἓν μόνον σημεῖον διαιρεῖται εἰς τὰ ὀνόματα.
A binomial straight line is divided into its terms at one point only.
ἔστω ἐκ δύο ὀνομάτων ἡ ΑΒ διῃρημένη εἰς τὰ ὀνόματα κατὰ τὸ Γ· αἱ ΑΓ, ΓΒ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι.
For let the binomial straight line AB be divided into its terms at C; therefore AC, CB are rational straight lines commensurable in square only.
λέγω, ὅτι ἡ ΑΒ κατʼ ἄλλο σημεῖον οὐ διαιρεῖται εἰς δύο ῥητὰς δυνάμει μόνον συμμέτρους.
I say that AB is not divided at another point into two rational straight lines commensurable in square only.
εἰ γὰρ δυνατόν, διῃρήσθω καὶ κατὰ τὸ Δ, ὥστε καὶ τὰς ΑΔ, ΔΒ ῥητὰς εἶναι δυνάμει μόνον συμμέτρους.
For, if possible, let it also be divided at D, so that AD, DB are also rational straight lines commensurable in square only.
φανερὸν δή, ὅτι ἡ ΑΓ τῇ ΔΒ οὐκ ἔστιν ἡ αὐτή.
It is then manifest that AC is not the same with DB.
εἰ γὰρ δυνατόν, ἔστω.
For, if possible, let it be so.
ἔσται δὴ καὶ ἡ ΑΔ τῇ ΓΒ ἡ αὐτή· καὶ ἔσται ὡς ἡ ΑΓ πρὸς τὴν ΓΒ, οὕτως ἡ ΒΔ πρὸς τὴν ΔΑ, καὶ ἔσται ἡ ΑΒ κατὰ τὸ αὐτὸ τῇ κατὰ τὸ Γ διαιρέσει διαιρεθεῖσα καὶ κατὰ τὸ Δ· ὅπερ οὐχ ὑπόκειται.
Then AD will also be the same with CB; and as AC is to CB, so will BD be to DA, and AB will be divided at D by the same division as at C; which is not assumed.
οὐκ ἄρα ἡ ΑΓ τῇ ΔΒ ἐστιν ἡ αὐτή.
Therefore AC is not the same with DB.
διὰ δὴ τοῦτο καὶ τὰ Γ, Δ σημεῖα οὐκ ἴσον ἀπέχουσι τῆς διχοτομίας.
For this reason then, the points C, D do not lie at an equal distance from the point of bisection.
ᾧ ἄρα διαφέρει τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ, τούτῳ διαφέρει καὶ τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ διὰ τὸ καὶ τὰ ἀπὸ τῶν ΑΓ, ΓΒ μετὰ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ καὶ τὰ ἀπὸ τῶν ΑΔ, ΔΒ μετὰ τοῦ δὶς ὑπὸ τῶν ΑΔ, ΔΒ ἴσα εἶναι τῷ ἀπὸ τῆς ΑΒ. ἀλλὰ τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ διαφέρει ῥητῷ·
Therefore, by how much the sum of the squares on AC, CB differs from the sum of the squares on AD, DB, by so much also twice the rectangle contained by AD, DB differs from twice the rectangle contained by AC, CB, because both the sum of the squares on AC, CB with twice the rectangle contained by AC, CB, and the sum of the squares on AD, DB with twice the rectangle contained by AD, DB are equal to the square on AB.
ῥητὰ γὰρ ἀμφότερα·
But the sum of the squares on AC, CB differs from the sum of the squares on AD, DB by a rational area; for both are rational.
καὶ τὸ δὶς ἄρα ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ διαφέρει ῥητῷ μέσα ὄντα· ὅπερ ἄτοπον· μέσον γὰρ μέσου οὐχ ὑπερέχει ῥητῷ.
Therefore also twice the rectangle contained by AD, DB differs from twice the rectangle contained by AC, CB by a rational area, though they are medial; which is absurd; for a medial area does not exceed a medial area by a rational area.
οὐκ ἄρα ἡ ἐκ δύο ὀνομάτων κατʼ ἄλλο καὶ ἄλλο σημεῖον διαιρεῖται· καθʼ ἓν ἄρα μόνον· ὅπερ ἔδει δεῖξαι.
Therefore a binomial straight line is not divided at different points; therefore [it is divided] at one point only; which was to be proved.
§10.prop1.43ἡ ἐκ δύο μέσων πρώτη καθʼ ἓν μόνον σημεῖον διαιρεῖται.
A first bimedial straight line is divided at one point only.
ἔστω ἐκ δύο μέσων πρώτη ἡ ΑΒ διῃρημένη κατὰ τὸ Γ, ὥστε τὰς ΑΓ, ΓΒ μέσας εἶναι δυνάμει μόνον συμμέτρους ῥητὸν περιεχούσας·
For let the first bimedial straight line AB be divided at C, so that AC, CB are medial straight lines commensurable in square only containing a rational area.
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 καὶ τὰς ΑΔ, ΔΒ μέσας εἶναι δυνάμει μόνον συμμέτρους ῥητὸν περιεχούσας.
For, if possible, let it also be divided at D, so that AD, DB are also medial straight lines commensurable in square only containing a rational area.
ἐπεὶ οὖν, ᾧ διαφέρει τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ, τούτῳ διαφέρει τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ, ῥητῷ δὲ διαφέρει τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ· ῥητὰ γὰρ ἀμφότερα· ῥητῷ ἄρα διαφέρει καὶ τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ μέσα ὄντα· ὅπερ ἄτοπον.
Since then, by how much twice the rectangle contained by AD, DB differs from twice the rectangle contained by AC, CB, by so much also the sum of the squares on AC, CB differs from the sum of the squares on AD, DB, and twice the rectangle contained by AD, DB differs from twice the rectangle contained by AC, CB by a rational area, for both are rational; therefore the sum of the squares on AC, CB also differs from the sum of the squares on AD, DB by a rational area, though they are medial; which is absurd.
οὐκ ἄρα ἡ ἐκ δύο μέσων πρώτη κατʼ ἄλλο καὶ ἄλλο σημεῖον διαιρεῖται εἰς τὰ ὀνόματα· καθʼ ἓν ἄρα μόνον· ὅπερ ἔδει δεῖξαι.
Therefore a first bimedial straight line is not divided at different points into its terms; therefore [it is divided] at one point only; which was to be proved.