§10.prop2.77ἐὰν ἀπὸ εὐθείας εὐθεῖα ἀφαιρεθῇ δυνάμει ἀσύμμετρος οὖσα τῇ ὅλῃ, μετὰ δὲ τῆς ὅλης ποιοῦσα τὸ μὲν συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τετραγώνων μέσον, τὸ δὲ δὶς ὑπʼ αὐτῶν ῥητόν, ἡ λοιπὴ ἄλογός ἐστιν· καλείσθω δὲ ἡ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα.
If from a straight line there be subtracted a straight line which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial, but twice the rectangle contained by them rational, the remainder is irrational; and let it be called that which makes with a rational a medial whole.
ἀπὸ γὰρ εὐθείας τῆς ΑΒ εὐθεῖα ἀφῃρήσθω ἡ ΒΓ δυνάμει ἀσύμμετρος οὖσα τῇ ΑΒ ποιοῦσα τὰ προκείμενα· λέγω, ὅτι ἡ λοιπὴ ἡ ΑΓ ἄλογός ἐστιν ἡ προειρημένη.
For from the straight line AB let there be subtracted the straight line BC which is incommensurable in square with AB, and which makes the proposed properties; I say that the remainder AC is the aforesaid irrational straight line.
ἐπεὶ γὰρ τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΒ, ΒΓ τετραγώνων μέσον ἐστίν, τὸ δὲ δὶς ὑπὸ τῶν ΑΒ, ΒΓ ῥητόν, ἀσύμμετρα ἄρα ἐστὶ τὰ ἀπὸ τῶν ΑΒ, ΒΓ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ·
For since the sum of the squares on AB, BC is medial, but twice the rectangle contained by AB, BC is rational, the sum of the squares on AB, BC is incommensurable with twice the rectangle contained by AB, BC; and therefore the remainder, the square on AC, is also incommensurable with twice the rectangle contained by AB, BC.
καὶ λοιπὸν ἄρα τὸ ἀπὸ τῆς ΑΓ ἀσύμμετρόν ἐστι τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ. καί ἐστι τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ ῥητόν· τὸ ἄρα ἀπὸ τῆς ΑΓ ἄλογόν ἐστιν·
And twice the rectangle contained by AB, BC is rational; therefore the square on AC is irrational.
ἄλογος ἄρα ἐστὶν ἡ ΑΓ· καλείσθω δὲ ἡ μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσα.
Therefore AC is irrational; and let it be called that which makes with a rational a medial whole.
ὅπερ ἔδει δεῖξαι.
Which was to be proved.
§10.prop2.78ἐὰν ἀπὸ εὐθείας εὐθεῖα ἀφαιρεθῇ δυνάμει ἀσύμμετρος οὖσα τῇ ὅλῃ, μετὰ δὲ τῆς ὅλης ποιοῦσα τό τε συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τετραγώνων μέσον τό τε δὶς ὑπʼ αὐτῶν μέσον καὶ ἔτι τὰ ἀπʼ αὐτῶν τετράγωνα ἀσύμμετρα τῷ δὶς ὑπʼ αὐτῶν, ἡ λοιπὴ ἄλογός ἐστιν· καλείσθω δὲ ἡ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα.
If from a straight line there be subtracted a straight line which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and moreover the squares on them incommensurable with twice the rectangle contained by them, the remainder is irrational; and let it be called that which makes with a medial a medial whole.
ἀπὸ γὰρ εὐθείας τῆς ΑΒ εὐθεῖα ἀφῃρήσθω ἡ ΒΓ δυνάμει ἀσύμμετρος οὖσα τῇ ΑΒ ποιοῦσα τὰ προκείμενα· λέγω, ὅτι ἡ λοιπὴ ἡ ΑΓ ἄλογός ἐστιν ἡ καλουμένη ἡ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα.
For from the straight line AB let there be subtracted the straight line BC which is incommensurable in square with AB, and which makes the proposed properties; I say that the remainder AC is the irrational straight line called that which makes with a medial a medial whole.
Ἐκκείσθω γὰρ ῥητὴ ἡ ΔΙ, καὶ τοῖς μὲν ἀπὸ τῶν ΑΒ, ΒΓ ἴσον παρὰ τὴν ΔΙ παραβεβλήσθω τὸ ΔΕ πλάτος ποιοῦν τὴν ΔΗ, τῷ δὲ δὶς ὑπὸ τῶν ΑΒ, ΒΓ ἴσον ἀφῃρήσθω τὸ ΔΘ.
For let there be set out the rational straight line DI, and to DI let there be applied the rectangle DE equal to the sum of the squares on AB, BC, producing DH as breadth, and let there be subtracted the rectangle DΘ equal to twice the rectangle contained by AB, BC.
λοιπὸν ἄρα τὸ ΖΕ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΓ· ὥστε ἡ ΑΓ δύναται τὸ ΖΕ. καὶ ἐπεὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΒ, ΒΓ τετραγώνων μέσον ἐστὶ καί ἐστιν ἴσον τῷ ΔΕ, μέσον ἄρα τὸ ΔΕ. καὶ παρὰ ῥητὴν τὴν ΔΙ παράκειται πλάτος ποιοῦν τὴν ΔΗ·
therefore the remainder ZE is equal to the square on AC; so that AC is square-equal to ZE. And since the sum of the squares on AB, BC is medial and is equal to DE, therefore DE is medial.
ῥητὴ ἄρα ἐστὶν ἡ ΔΗ καὶ ἀσύμμετρος τῇ ΔΙ μήκει.
And it is applied to the rational straight line DI, producing DH as breadth; therefore DH is rational and incommensurable in length with DI.
πάλιν, ἐπεὶ τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ μέσον ἐστὶ καί ἐστιν ἴσον τῷ ΔΘ, τὸ ἄρα ΔΘ μέσον ἐστίν.
Again, since twice the rectangle contained by AB, BC is medial and is equal to DΘ, therefore DΘ is medial.
καὶ παρὰ ῥητὴν τὴν ΔΙ παράκειται πλάτος ποιοῦν τὴν ΔΖ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΔΖ καὶ ἀσύμμετρος τῇ ΔΙ μήκει.
And it is applied to the rational straight line DI, producing DZ as breadth; therefore DZ is also rational and incommensurable in length with DI.
καὶ ἐπεὶ ἀσύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ, ἀσύμμετρον ἄρα καὶ τὸ ΔΕ τῷ ΔΘ. ὡς δὲ τὸ ΔΕ πρὸς τὸ ΔΘ, οὕτως ἐστὶ καὶ ἡ ΔΗ πρὸς τὴν ΔΖ· ἀσύμμετρος ἄρα ἡ ΔΗ τῇ ΔΖ. καί εἰσιν ἀμφότεραι ῥηταί· αἱ ΗΔ, ΔΖ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι. ἀποτομὴ ἄρα ἐστὶν ἡ ΖΗ·
And since the sum of the squares on AB, BC is incommensurable with twice the rectangle contained by AB, BC, therefore DE is also incommensurable with DΘ. And as DE is to DΘ, so is also DH to DZ; therefore DH is incommensurable with DZ. And both are rational; therefore HD, DZ are rational straight lines commensurable in square only; therefore ZH is an apotome.
ῥητὴ δὲ ἡ ΖΘ. τὸ δὲ ὑπὸ ῥητῆς καὶ ἀποτομῆς περιεχόμενον ἄλογόν ἐστιν, καὶ ἡ δυναμένη αὐτὸ ἄλογός ἐστιν.
But ZΘ is rational. And the rectangle contained by a rational straight line and an apotome is irrational, and the side of the square equal to it is irrational.
καὶ δύναται τὸ ΖΕ ἡ ΑΓ· ἡ ΑΓ ἄρα ἄλογός ἐστιν· καλείσθω δὲ ἡ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα.
And AC is square-equal to ZE; therefore AC is irrational; and let it be called that which makes with a medial a medial whole.
ὅπερ ἔδει δεῖξαι.
Which was to be proved.