§10.prop3.113τὸ ἀπὸ ῥητῆς παρὰ ἀποτομὴν παραβαλλόμενον πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων, ἧς τὰ ὀνόματα σύμμετρά ἐστι τοῖς τῆς ἀποτομῆς ὀνόμασι καὶ ἐν τῷ αὐτῷ λόγῳ, ἔτι δὲ ἡ γινομένη ἐκ δύο ὀνομάτων τὴν αὐτὴν τάξιν ἔχει τῇ ἀποτομῇ.
The square on a rational straight line applied to an apotome produces as breadth a binomial straight line, whose terms are commensurable with the terms of the apotome and further in the same ratio, and further the binomial straight line so created will have the same order as the apotome.
ἔστω ῥητὴ μὲν ἡ Α, ἀποτομὴ δὲ ἡ ΒΔ, καὶ τῷ ἀπὸ τῆς Α ἴσον ἔστω τὸ ὑπὸ τῶν ΒΔ, ΚΘ, ὥστε τὸ ἀπὸ τῆς Α ῥητῆς παρὰ τὴν ΒΔ ἀποτομὴν παραβαλλόμενον πλάτος ποιεῖ τὴν ΚΘ· λέγω, ὅτι ἐκ δύο ὀνομάτων ἐστὶν ἡ ΚΘ, ἧς τὰ ὀνόματα σύμμετρά ἐστι τοῖς τῆς ΒΔ ὀνόμασι καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ ἔτι ἡ ΚΘ τὴν αὐτὴν ἔχει τάξιν τῇ ΒΔ.
ἔστω γὰρ τῇ ΒΔ προσαρμόζουσα ἡ ΔΓ·
Let A be a rational straight line, and let BΔ be an apotome, and let the rectangle contained by BΔ, KΘ be equal to the square on A, so that the square on the rational straight line A applied to the apotome BΔ produces KΘ as breadth; I say that KΘ is a binomial straight line, whose terms are commensurable with the terms of BΔ, and in the same ratio, and further KΘ will have the same order as BΔ.
αἱ ΒΓ, ΓΔ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι.
For let ΔΓ be the annex to BΔ; therefore BΓ, ΓΔ are rational straight lines commensurable in square only.
καὶ τῷ ἀπὸ τῆς Α ἴσον ἔστω καὶ τὸ ὑπὸ τῶν ΒΓ, Η. ῥητὸν δὲ τὸ ἀπὸ τῆς Α· ῥητὸν ἄρα καὶ τὸ ὑπὸ τῶν ΒΓ, Η. καὶ παρὰ ῥητὴν τὴν ΒΓ παραβέβληται· ῥητὴ ἄρα ἐστὶν ἡ Η καὶ σύμμετρος τῇ ΒΓ μήκει.
And let the rectangle contained by BΓ, H be also equal to the square on A. And the square on A is rational; therefore the rectangle contained by BΓ, H is also rational. And it is applied to the rational straight line BΓ; therefore H is rational and commensurable in length with BΓ.
ἐπεὶ οὖν τὸ ὑπὸ τῶν ΒΓ, Η ἴσον ἐστὶ τῷ ὑπὸ τῶν ΒΔ, ΚΘ, ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΓΒ πρὸς ΒΔ, οὕτως ἡ ΚΘ πρὸς Η. μείζων δὲ ἡ ΒΓ τῆς ΒΔ· μείζων ἄρα καὶ ἡ ΚΘ τῆς Η. κείσθω τῇ Η ἴση ἡ ΚΕ· σύμμετρος ἄρα ἐστὶν ἡ ΚΕ τῇ ΒΓ μήκει.
Since then the rectangle contained by BΓ, H is equal to the rectangle contained by BΔ, KΘ, therefore in proportion, as ΓB is to BΔ, so is KΘ to H. But BΓ is greater than BΔ; therefore KΘ is also greater than H. Let KE be equal to H; therefore KE is commensurable in length with BΓ.
καὶ ἐπεί ἐστιν ὡς ἡ ΓΒ πρὸς ΒΔ, οὕτως ἡ ΘΚ πρὸς ΚΕ, ἀναστρέψαντι ἄρα ἐστὶν ὡς ἡ ΒΓ πρὸς τὴν ΓΔ, οὕτως ἡ ΚΘ πρὸς ΘΕ. γεγονέτω ὡς ἡ ΚΘ πρὸς ΘΕ, οὕτως ἡ ΘΖ πρὸς ΖΕ· καὶ λοιπὴ ἄρα ἡ ΚΖ πρὸς ΖΘ ἐστιν, ὡς ἡ ΚΘ πρὸς ΘΕ, τουτέστιν ἡ ΒΓ πρὸς ΓΔ. αἱ δὲ ΒΓ, ΓΔ δυνάμει μόνον σύμμετροι· καὶ αἱ ΚΖ, ΖΘ ἄρα δυνάμει μόνον εἰσὶ σύμμετροι.
And since, as ΓB is to BΔ, so is ΘK to KE, therefore, by conversion of ratio, as BΓ is to ΓΔ, so is KΘ to ΘE. Let it be contrived that, as KΘ is to ΘE, so is ΘΖ to ΖΕ; therefore also the remainder KΖ is to ΖΘ, as KΘ is to ΘE, that is, as BΓ is to ΓΔ. But BΓ, ΓΔ are commensurable in square only; therefore KΖ, ΖΘ are also commensurable in square only.
καὶ ἐπεί ἐστιν ὡς ἡ ΚΘ πρὸς ΘΕ, ἡ ΚΖ πρὸς ΖΘ, ἀλλʼ ὡς ἡ ΚΘ πρὸς ΘΕ, ἡ ΘΖ πρὸς ΖΕ, καὶ ὡς ἄρα ἡ ΚΖ πρὸς ΖΘ, ἡ ΘΖ πρὸς ΖΕ· ὥστε καὶ ὡς ἡ πρώτη πρὸς τὴν τρίτην, τὸ ἀπὸ τῆς πρώτης πρὸς τὸ ἀπὸ τῆς δευτέρας· καὶ ὡς ἄρα ἡ ΚΖ πρὸς ΖΕ, οὕτως τὸ ἀπὸ τῆς ΚΖ πρὸς τὸ ἀπὸ τῆς ΖΘ. σύμμετρον δέ ἐστι τὸ ἀπὸ τῆς ΚΖ τῷ ἀπὸ τῆς ΖΘ·
And since, as KΘ is to ΘE, so is KΖ to ΖΘ, but, as KΘ is to ΘE, so is ΘΖ to ΖΕ, therefore also, as KΖ is to ΖΘ, so is ΘΖ to ΖΕ; so that, as the first is to the third, so is the square on the first to the square on the second; therefore also, as KΖ is to ΖΕ, so is the square on KΖ to the square on ΖΘ.
αἱ γὰρ ΚΖ, ΖΘ δυνάμει εἰσὶ σύμμετροι· σύμμετρος ἄρα ἐστὶ καὶ ἡ ΚΖ τῇ ΖΕ μήκει· ὥστε ἡ ΚΖ καὶ τῇ ΚΕ σύμμετρός μήκει.
But the square on KΖ is commensurable with the square on ΖΘ; for KΖ, ΖΘ are commensurable in square; therefore KΖ is also commensurable in length with ΖΕ; so that KΖ is also commensurable in length with KE.
ῥητὴ δέ ἐστιν ἡ ΚΕ καὶ σύμμετρος τῇ ΒΓ μήκει· ῥητὴ ἄρα καὶ ἡ ΚΖ καὶ σύμμετρος τῇ ΒΓ μήκει.
And KE is rational and commensurable in length with BΓ; therefore KΖ is also rational and commensurable in length with BΓ.
καὶ ἐπεί ἐστιν ὡς ἡ ΒΓ πρὸς ΓΔ, οὕτως ἡ ΚΖ πρὸς ΖΘ, ἐναλλὰξ ὡς ἡ ΒΓ πρὸς ΚΖ, οὕτως ἡ ΔΓ πρὸς ΖΘ. σύμμετρος δὲ ἡ ΒΓ τῇ ΚΖ· σύμμετρος ἄρα καὶ ἡ ΖΘ τῇ ΓΔ μήκει.
And since, as BΓ is to ΓΔ, so is KΖ to ΖΘ, therefore, alternately, as BΓ is to KΖ, so is ΔΓ to ΖΘ. But BΓ is commensurable with KΖ; therefore ΖΘ is also commensurable in length with ΓΔ.
αἱ ΒΓ, ΓΔ δὲ ῥηταί εἰσι δυνάμει μόνον σύμμετροι· καὶ αἱ ΚΖ, ΖΘ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἐκ δύο ὀνομάτων ἐστὶν ἄρα ἡ ΚΘ.
εἰ μὲν οὖν ἡ ΒΓ τῆς ΓΔ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ ἡ ΚΖ τῆς ΖΘ μεῖζον δυνήσεται τῷ ἀπὸ συμμέτρου ἑαυτῇ.
But BΓ, ΓΔ are rational straight lines commensurable in square only; therefore KΖ, ΖΘ are also rational straight lines commensurable in square only; therefore KΘ is a binomial straight line. If then BΓ is greater in square than ΓΔ by the square on a straight line commensurable with itself, KΖ will also be greater in square than ΖΘ by the square on a straight line commensurable with itself.
καὶ εἰ μὲν σύμμετρός ἐστιν ἡ ΒΓ τῇ ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΚΖ, εἰ δὲ ἡ ΓΔ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΖΘ, εἰ δὲ οὐδετέρα τῶν ΒΓ, ΓΔ, οὐδετέρα τῶν ΚΖ, ΖΘ.
εἰ δὲ ἡ ΒΓ τῆς ΓΔ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, καὶ ἡ ΚΖ τῆς ΖΘ μεῖζον δυνήσεται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ.
And if BΓ is commensurable in length with the rational straight line set out, so is KΖ; if ΓΔ is commensurable in length with the rational straight line set out, so is ΖΘ; and if neither of BΓ, ΓΔ is, neither of KΖ, ΖΘ is. But if BΓ is greater in square than ΓΔ by the square on a straight line incommensurable with itself, KΖ will also be greater in square than ΖΘ by the square on a straight line incommensurable with itself.
καὶ εἰ μὲν σύμμετρός ἐστιν ἡ ΒΓ τῇ ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΚΖ, εἰ δὲ ἡ ΓΔ, καὶ ἡ ΖΘ, εἰ δὲ οὐδετέρα τῶν ΒΓ, ΓΔ, οὐδετέρα τῶν ΚΖ, ΖΘ.
ἐκ δύο ἄρα ὀνομάτων ἐστὶν ἡ ΚΘ, ἧς τὰ ὀνόματα τὰ ΚΖ, ΖΘ σύμμετρά τοῖς τῆς ἀποτομῆς ὀνόμασι τοῖς ΒΓ, ΓΔ καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ ἔτι ἡ ΚΘ τῇ ΒΓ τὴν αὐτὴν ἕξει τάξιν· ὅπερ ἔδει δεῖξαι.
And if BΓ is commensurable in length with the rational straight line set out, so is KΖ; if ΓΔ, so is ΖΘ; and if neither of the two BΓ, ΓΔ is, neither of the two KΖ, ΖΘ is; therefore KΘ is a binomial straight line, whose terms KΖ, ΖΘ are commensurable with the terms BΓ, ΓΔ of the apotome, and in the same ratio, and further KΘ will have the same order as BΓ; which was to be proved.