§10.prop2.73ἐὰν ἀπὸ ῥητῆς ῥητὴ ἀφαιρεθῇ δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ, ἡ λοιπὴ ἄλογός ἐστιν· καλείσθω δὲ ἀποτομή.
If from a rational straight line there be subtracted a rational straight line which is commensurable in square only with the whole, the remainder is irrational; and let it be called an apotome.
ἀπὸ γὰρ ῥητῆς τῆς ΑΒ ῥητὴ ἀφῃρήσθω ἡ ΒΓ δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ· λέγω, ὅτι ἡ λοιπὴ ἡ ΑΓ ἄλογός ἐστιν ἡ καλουμένη ἀποτομή.
For from the rational straight line AB let there be subtracted the rational straight line BC which is commensurable in square only with the whole; I say that the remainder AC is the irrational straight line called an apotome.
ἐπεὶ γὰρ ἀσύμμετρός ἐστιν ἡ ΑΒ τῇ ΒΓ μήκει, καί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ὑπὸ τῶν ΑΒ, ΒΓ, ἀσύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΑΒ τῷ ὑπὸ τῶν ΑΒ, ΒΓ. ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΒ σύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ τετράγωνα, τῷ δὲ ὑπὸ τῶν ΑΒ, ΒΓ σύμμετρόν ἐστι τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ. καὶ ἐπειδήπερ τὰ ἀπὸ τῶν ΑΒ, ΒΓ ἴσα ἐστὶ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ μετὰ τοῦ ἀπὸ ΓΑ, καὶ λοιπῷ ἄρα τῷ ἀπὸ τῆς ΑΓ ἀσύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ. ῥητὰ δὲ τὰ ἀπὸ τῶν ΑΒ, ΒΓ· ἄλογος ἄρα ἐστὶν ἡ ΑΓ·
For since AB is incommensurable in length with BC, and as AB is to BC, so is the square on AB to the rectangle contained by AB, BC, therefore the square on AB is incommensurable with the rectangle contained by AB, BC. But to the square on AB the sum of the squares on AB, BC is commensurable, and to the rectangle contained by AB, BC twice the rectangle contained by AB, BC is commensurable. And since the sum of the squares on AB, BC is equal to twice the rectangle contained by AB, BC together with the square on CA, therefore the sum of the squares on AB, BC is also incommensurable with the remainder, the square on AC. But the sum of the squares on AB, BC is rational; therefore AC is irrational.
καλείσθω δὲ ἀποτομή.
And let it be called an apotome.
ὅπερ ἔδει δεῖξαι.
Which was to be proved.
§10.prop2.74ἐὰν ἀπὸ μέσης μέση ἀφαιρεθῇ δυνάμει μόνον σύμμετρος οὖσα τῇ ὅλῃ, μετὰ δὲ τῆς ὅλης ῥητὸν περιέχουσα, ἡ λοιπὴ ἄλογός ἐστιν· καλείσθω δὲ μέσης ἀποτομὴ πρώτη.
If from a medial straight line there be subtracted a medial straight line which is commensurable in square only with the whole, and which contains with the whole a rational area, the remainder is irrational; and let it be called a first apotome of a medial.
ἀπὸ γὰρ μέσης τῆς ΑΒ μέση ἀφῃρήσθω ἡ ΒΓ δυνάμει μόνον σύμμετρος οὖσα τῇ ΑΒ, μετὰ δὲ τῆς ΑΒ ῥητὸν ποιοῦσα τὸ ὑπὸ τῶν ΑΒ, ΒΓ· λέγω, ὅτι ἡ λοιπὴ ἡ ΑΓ ἄλογός ἐστιν· καλείσθω δὲ μέσης ἀποτομὴ πρώτη.
For from the medial straight line AB let there be subtracted the medial straight line BC which is commensurable in square only with AB, and which with AB makes the rectangle contained by AB, BC rational; I say that the remainder AC is irrational; and let it be called a first apotome of a medial.
ἐπεὶ γὰρ αἱ ΑΒ, ΒΓ μέσαι εἰσίν, μέσα ἐστὶ καὶ τὰ ἀπὸ τῶν ΑΒ, ΒΓ. ῥητὸν δὲ τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ· ἀσύμμετρα ἄρα τὰ ἀπὸ τῶν ΑΒ, ΒΓ τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ·
For since AB, BC are medial straight lines, the sum of the squares on AB, BC is also medial. But twice the rectangle contained by AB, BC is rational; therefore the sum of the squares on AB, BC is incommensurable with twice the rectangle contained by AB, BC.
καὶ λοιπῷ ἄρα τῷ ἀπὸ τῆς ΑΓ ἀσύμμετρόν ἐστι τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ, ἐπεὶ κἂν τὸ ὅλον ἑνὶ αὐτῶν ἀσύμμετρον ᾖ, καὶ τὰ ἐξ ἀρχῆς μεγέθη ἀσύμμετρα ἔσται.
Therefore twice the rectangle contained by AB, BC is also incommensurable with the remainder, the square on AC; since, if the whole be incommensurable with one of them, the original magnitudes will also be incommensurable.
ῥητὸν δὲ τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ· ἄλογον ἄρα τὸ ἀπὸ τῆς ΑΓ·
But twice the rectangle contained by AB, BC is rational; therefore the square on AC is irrational.
ἄλογος ἄρα ἐστὶν ἡ ΑΓ· καλείσθω δὲ μέσης ἀποτομὴ πρώτη.
Therefore AC is irrational; and let it be called a first apotome of a medial.