§10.prop3.111ἡ ἀποτομὴ οὐκ ἔστιν ἡ αὐτὴ τῇ ἐκ δύο ὀνομάτων. ἔστω ἀποτομὴ ἡ ΑΒ· λέγω, ὅτι ἡ ΑΒ οὐκ ἔστιν ἡ αὐτὴ τῇ ἐκ δύο ὀνομάτων.
An apotome is not the same with a binomial straight line. Let AB be an apotome; I say that AB is not the same with a binomial straight line.
εἰ γὰρ δυνατόν, ἔστω· καὶ ἐκκείσθω ῥητὴ ἡ ΔΓ, καὶ τῷ ἀπὸ τῆς ΑΒ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω ὀρθογώνιον τὸ ΓΕ πλάτος ποιοῦν τὴν ΔΕ. ἐπεὶ οὖν ἀποτομή ἐστιν ἡ ΑΒ, ἀποτομὴ πρώτη ἐστὶν ἡ ΔΕ. ἔστω αὐτῇ προσαρμόζουσα ἡ ΕΖ· αἱ ΔΖ, ΖΕ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ ἡ ΔΖ τῆς ΖΕ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ ἡ ΔΖ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΔΓ. πάλιν, ἐπεὶ ἐκ δύο ὀνομάτων ἐστὶν ἡ ΑΒ, ἐκ δύο ἄρα ὀνομάτων πρώτη ἐστὶν ἡ ΔΕ. διῃρήσθω εἰς τὰ ὀνόματα κατὰ τὸ Η, καὶ ἔστω μεῖζον ὄνομα τὸ ΔΗ· αἱ ΔΗ, ΗΕ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ ἡ ΔΗ τῆς ΗΕ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ τὸ μεῖζον ἡ ΔΗ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΔΓ. καὶ ἡ ΔΖ ἄρα τῇ ΔΗ σύμμετρός ἐστι μήκει· καὶ λοιπὴ ἄρα ἡ ΗΖ σύμμετρός ἐστι τῇ ΔΖ μήκει. ἀσύμμετρος δὲ ἡ ΔΖ τῇ ΕΖ μήκει· ἀσύμμετρος ἄρα ἐστὶ καὶ ἡ ΖΗ τῇ ΕΖ μήκει. αἱ ΗΖ, ΖΕ ἄρα ῥηταί δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΕΗ. ἀλλὰ καὶ ῥητή· ὅπερ ἐστὶν ἀδύνατον. ἡ ἄρα ἀποτομὴ οὐκ ἔστιν ἡ αὐτὴ τῇ ἐκ δύο ὀνομάτων· ὅπερ ἔδει δεῖξαι.
For, if possible, let it be so; and let the rational straight line DΓ be set out, and let the rectangle ΓE, equal to the square on AB, be applied to ΓD, producing DE as breadth. Since then AB is an apotome, DE is a first apotome. Let EZ be its annex; therefore DZ, ZE are rational straight lines commensurable in square only, DZ is greater in square than ZE by the square on a straight line commensurable in length with itself, and DZ is commensurable in length with the rational straight line DΓ set out. Again, since AB is a binomial straight line, DE is therefore a first binomial straight line. Let it be divided into its terms at H, and let DH be the greater term; therefore DH, HE are rational straight lines commensurable in square only, DH is greater in square than HE by the square on a straight line commensurable with itself, and the greater term DH is commensurable in length with the rational straight line DΓ set out. Therefore DZ is also commensurable in length with DH; therefore the remainder HZ is also commensurable in length with DZ. But DZ is incommensurable in length with EZ; therefore ZH is also incommensurable in length with EZ. Therefore HZ, ZE are rational straight lines commensurable in square only; therefore EH is an apotome. But it is also rational, which is impossible. For, if possible, let it be so; and let the rational straight line DΓ be set out, and let the rectangle ΓE, equal to the square on AB, be applied to ΓD, producing DE as breadth. Since then AB is an apotome, DE is a first apotome. Let EZ be its annex; therefore DZ, ZE are rational straight lines commensurable in square only, DZ is greater in square than ZE by the square on a straight line commensurable in length with itself, and DZ is commensurable in length with the rational straight line DΓ set out. Again, since AB is a binomial straight line, DE is therefore a first binomial straight line. Let it be divided into its terms at H, and let DH be the greater term; therefore DH, HE are rational straight lines commensurable in square only, DH is greater in square than HE by the square on a straight line commensurable with itself, and the greater term DH is commensurable in length with the rational straight line DΓ set out. Therefore DZ is also commensurable in length with DH; therefore the remainder HZ is also commensurable in length with DZ. But DZ is incommensurable in length with EZ; therefore ZH is also incommensurable in length with EZ. Therefore HZ, ZE are rational straight lines commensurable in square only; therefore EH is an apotome. But it is also rational, which is impossible. Therefore an apotome is not the same with a binomial straight line; which was to be proved. Therefore an apotome is not the same with a binomial straight line; which was to be proved.
ἡ ἀποτομὴ καὶ αἱ μετʼ αὐτὴν ἄλογοι οὔτε τῇ μέσῃ οὔτε ἀλλήλαις εἰσὶν αἱ αὐταί. τὸ μὲν γὰρ ἀπὸ μέσης παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ῥητὴν καὶ ἀσύμμετρον τῇ, παρʼ ἣν παράκειται, μήκει, τὸ δὲ ἀπὸ ἀποτομῆς παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν πρώτην, τὸ δὲ ἀπὸ μέσης ἀποτομῆς πρώτης παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν δευτέραν, τὸ δὲ ἀπὸ μέσης ἀποτομῆς δευτέρας παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν τρίτην, τὸ δὲ ἀπὸ ἐλάσσονος παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν τετάρτην, τὸ δὲ ἀπὸ τῆς μετὰ ῥητοῦ μέσον τὸ ὅλον ποιούσης παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν πέμπτην, τὸ δὲ ἀπὸ τῆς μετὰ μέσου μέσον τὸ ὅλον ποιούσης παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ἀποτομὴν ἕκτην.
The apotome and the irrational straight lines after it are neither the same with the medial nor with one another. The apotome and the irrational straight lines after it are neither the same with the medial nor with one another. For the square on the medial straight line applied to a rational straight line produces as breadth a straight line rational and incommensurable in length with that to which it is applied, but that on an apotome applied to a rational straight line produces as breadth a first apotome, that on a first apotome of a medial applied to a rational straight line produces as breadth a second apotome, that on a second apotome of a medial applied to a rational straight line produces as breadth a third apotome, that on a minor applied to a rational straight line produces as breadth a fourth apotome, that on that which produces with a rational area a medial whole applied to a rational straight line produces as breadth a fifth apotome, and that on that which produces with a medial area a medial whole applied to a rational straight line produces as breadth a sixth apotome.
ἐπεὶ οὖν τὰ εἰρημένα πλάτη διαφέρει τοῦ τε πρώτου καὶ ἀλλήλων, τοῦ μὲν πρώτου, ὅτι ῥητή ἐστιν, ἀλλήλων δέ, ἐπεὶ τῇ τάξει οὐκ εἰσὶν αἱ αὐταί, δῆλον, ὡς καὶ αὐταὶ αἱ ἄλογοι διαφέρουσιν ἀλλήλων. καὶ ἐπεὶ δέδεικται ἡ ἀποτομὴ οὐκ οὖσα ἡ αὐτὴ τῇ ἐκ δύο ὀνομάτων, ποιοῦσι δὲ πλάτη παρὰ ῥητὴν παραβαλλόμεναι αἱ μετὰ τὴν ἀποτομὴν ἀποτομὰς ἀκολούθως ἑκάστη τῇ τάξει τῇ καθʼ αὑτήν, αἱ δὲ μετὰ τὴν ἐκ δύο ὀνομάτων τὰς ἐκ δύο ὀνομάτων καὶ αὐταὶ τῇ τάξει ἀκολούθως, ἕτεραι ἄρα εἰσὶν αἱ μετὰ τὴν ἀποτομὴν καὶ ἕτεραι αἱ μετὰ τὴν ἐκ δύο ὀνομάτων, ὡς εἶναι τῇ τάξει πάσας ἀλόγους ιγ,
μέσην,
ἐκ δύο ὀνομάτων,
ἐκ δύο μέσων πρώτην,
ἐκ δύο μέσων δευτέραν,
μείζονα,
ῥητὸν καὶ μέσον δυναμένην,
δύο μέσα δυναμένην,
ἀποτομήν,
μέσης ἀποτομὴν πρώτην,
μέσης ἀποτομὴν δευτέραν,
ἐλάσσονα,
μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσαν,
μετὰ μέσου μέσον τὸ ὅλον ποιοῦσαν.
Since then the said breadths differ from the first breadth and from one another, from the first because it is rational, and from one another because they are not the same in order, it is manifest that the irrational straight lines themselves also differ from one another. And since the apotome has been proved not to be the same with the binomial straight line, and the irrational straight lines after the apotome applied to a rational straight line produce as breadths apotomes each in its own order, while those after the binomial straight line produce binomial straight lines each in its own order, therefore those after the apotome are different from those after the binomial straight line, so that there are in order thirteen irrational straight lines in all: medial, binomial, first bimedial, second bimedial, major, that which produces a rational and a medial area, that which produces two medial areas, apotome, first apotome of a medial, second apotome of a medial, minor, that which produces with a rational area a medial whole, that which produces with a medial area a medial whole. Since then the said breadths differ from the first breadth and from one another, from the first because it is rational, and from one another because they are not the same in order, it is manifest that the irrational straight lines themselves also differ from one another. And since the apotome has been proved not to be the same with the binomial straight line, and the irrational straight lines after the apotome applied to a rational straight line produce as breadths apotomes each in its own order, while those after the binomial straight line produce binomial straight lines each in its own order, therefore those after the apotome are different from those after the binomial straight line, so that there are in order thirteen irrational straight lines in all: μέσην, medial, ἐκ δύο ὀνομάτων, binomial, ἐκ δύο μέσων πρώτην, first bimedial, ἐκ δύο μέσων δευτέραν, second bimedial, μείζονα, major, ῥητὸν καὶ μέσον δυναμένην, that which produces a rational and a medial area, δύο μέσα δυναμένην, that which produces two medial areas, ἀποτομήν, apotome, μέσης ἀποτομὴν πρώτην, first apotome of a medial, μέσης ἀποτομὴν δευτέραν, second apotome of a medial, ἐλάσσονα, minor, μετὰ ῥητοῦ μέσον τὸ ὅλον ποιοῦσαν, that which produces with a rational area a medial whole, μετὰ μέσου μέσον τὸ ὅλον ποιοῦσαν. that which produces with a medial area a medial whole.