§10.prop3.88εὑρεῖν τὴν τετάρτην ἀποτομήν.
To find the fourth apotome.
Ἐκκείσθω ῥητὴ ἡ Α καὶ τῇ Α μήκει σύμμετρος ἡ ΒΗ·
Let the rational straight line A be set out, and BH commensurable in length with A; therefore BH is also rational.
ῥητὴ ἄρα ἐστὶ καὶ ἡ ΒΗ. καὶ ἐκκείσθωσαν δύο ἀριθμοὶ οἱ ΔΖ, ΖΕ, ὥστε τὸν ΔΕ ὅλον πρὸς ἑκάτερον τῶν ΔΖ, ΕΖ λόγον μὴ ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.
And let two numbers ΔΖ, ΖΕ be set out, so that the whole ΔΕ does not have to each of ΔΖ, ΕΖ the ratio which a square number has to a square number.
καὶ πεποιήσθω ὡς ὁ ΔΕ πρὸς τὸν ΕΖ, οὕτως τὸ ἀπὸ τῆς ΒΗ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΗΓ. σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΒΗ τῷ ἀπὸ τῆς ΗΓ. ῥητὸν δὲ τὸ ἀπὸ τῆς ΒΗ· ῥητὸν ἄρα καὶ τὸ ἀπὸ τῆς ΗΓ· ῥητὴ ἄρα ἐστὶν ἡ ΗΓ. καὶ ἐπεὶ ὁ ΔΕ πρὸς τὸν ΕΖ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδʼ ἄρα τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ ΗΓ μήκει.
And let it be made that, as ΔΕ is to ΕΖ, so is the square on ΒΗ to the square on ΗΓ. Therefore the square on ΒΗ is commensurable with the square on ΗΓ. And the square on ΒΗ is rational; therefore the square on ΗΓ is also rational; therefore ΗΓ is rational. And since ΔΕ does not have to ΕΖ the ratio which a square number has to a square number, therefore the square on ΒΗ also does not have to the square on ΗΓ the ratio which a square number has to a square number; therefore ΒΗ is incommensurable in length with ΗΓ.
καί εἰσιν ἀμφότεραι ῥηταί· αἱ ΒΗ, ΗΓ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΒΓ.
.
And both are rational; therefore ΒΗ, ΗΓ are rational straight lines commensurable in square only; therefore ΒΓ is an apotome..
ὧι οὖν μεῖζόν ἐστι τὸ ἀπὸ τῆς ΒΗ τοῦ ἀπὸ τῆς ΗΓ, ἔστω τὸ ἀπὸ τῆς Θ. ἐπεὶ οὖν ἐστιν ὡς ὁ ΔΕ πρὸς τὸν ΕΖ, οὕτως τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ, καὶ ἀναστρέψαντι ἄρα ἐστὶν ὡς ὁ ΕΔ πρὸς τὸν ΔΖ, οὕτως τὸ ἀπὸ τῆς ΗΒ πρὸς τὸ ἀπὸ τῆς Θ. ὁ δὲ ΕΔ πρὸς τὸν ΔΖ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· οὐδʼ ἄρα τὸ ἀπὸ τῆς ΗΒ πρὸς τὸ ἀπὸ τῆς Θ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ Θ μήκει.
And let the square on Θ be that by which the square on ΒΗ is greater than the square on ΗΓ. Since, then, as ΔΕ is to ΕΖ, so is the square on ΒΗ to the square on ΗΓ, therefore, convertendo, as ΕΔ is to ΔΖ, so is the square on ΗΒ to the square on Θ. And ΕΔ does not have to ΔΖ the ratio which a square number has to a square number; therefore the square on ΗΒ also does not have to the square on Θ the ratio which a square number has to a square number; therefore ΒΗ is incommensurable in length with Θ.
καὶ δύναται ἡ ΒΗ τῆς ΗΓ μεῖζον τῷ ἀπὸ τῆς Θ· ἡ ἄρα ΒΗ τῆς ΗΓ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ.
And the square on ΒΗ is greater than the square on ΗΓ by the square on Θ; therefore ΒΗ is greater in square than ΗΓ by the square on a straight line incommensurable in length with itself.
καί ἐστιν ὅλη ἡ ΒΗ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ α. ἡ ἄρα ΒΓ ἀποτομή ἐστι τετάρτη.
And the whole ΒΗ is commensurable in length with the set-out rational straight line A; therefore ΒΓ is a fourth apotome.
εὕρηται ἄρα ἡ τετάρτη ἀποτομή· ὅπερ ἔδει δεῖξαι.
Therefore the fourth apotome has been found; which was to be demonstrated.
§10.prop3.89εὑρεῖν τὴν πέμπτην ἀποτομήν.
To find the fifth apotome.
Ἐκκείσθω ῥητὴ ἡ Α, καὶ τῇ Α μήκει σύμμετρος ἔστω ἡ ΓΗ·
Let the rational straight line A be set out, and let ΓΗ be commensurable in length with A; therefore ΓΗ is rational.
ῥητὴ ἄρα ἡ ΓΗ. καὶ ἐκκείσθωσαν δύο ἀριθμοὶ οἱ ΔΖ, ΖΕ, ὥστε τὸν ΔΕ πρὸς ἑκάτερον τῶν ΔΖ, ΖΕ λόγον πάλιν μὴ ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν·
And let two numbers ΔΖ, ΖΕ be set out, so that the whole ΔΕ again does not have to each of ΔΖ, ΖΕ the ratio which a square number has to a square number.
καὶ πεποιήσθω ὡς ὁ ΖΕ πρὸς τὸν ΕΔ, οὕτως τὸ ἀπὸ τῆς ΓΗ πρὸς τὸ ἀπὸ τῆς ΗΒ. ῥητὸν ἄρα καὶ τὸ ἀπὸ τῆς ΗΒ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΒΗ. καὶ ἐπεί ἐστιν ὡς ὁ ΔΕ πρὸς τὸν ΕΖ, οὕτως τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ, ὁ δὲ ΔΕ πρὸς τὸν ΕΖ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδʼ ἄρα τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ ΗΓ μήκει.
And let it be made that, as ΖΕ is to ΕΔ, so is the square on ΓΗ to the square on ΗΒ. Therefore the square on ΗΒ is also rational; therefore ΒΗ is also rational. And since, as ΔΕ is to ΕΖ, so is the square on ΒΗ to the square on ΗΓ, and ΔΕ does not have to ΕΖ the ratio which a square number has to a square number, therefore the square on ΒΗ also does not have to the square on ΗΓ the ratio which a square number has to a square number; therefore ΒΗ is incommensurable in length with ΗΓ.
καί εἰσιν ἀμφότεραι ῥηταί· αἱ ΒΗ, ΗΓ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἡ ΒΓ ἄρα ἀποτομή ἐστιν.
And both are rational; therefore ΒΗ, ΗΓ are rational straight lines commensurable in square only; therefore ΒΓ is an apotome.
λέγω δή, ὅτι καὶ πέμπτη.
¦20 I say then, that it is also a fifth apotome.
ὧι γὰρ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΒΗ τοῦ ἀπὸ τῆς ΗΓ, ἔστω τὸ ἀπὸ τῆς Θ. ἐπεὶ οὖν ἐστιν ὡς τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ, οὕτως ὁ ΔΕ πρὸς τὸν ΕΖ, ἀναστρέψαντι ἄρα ἐστὶν ὡς ὁ ΕΔ πρὸς τὸν ΔΖ, οὕτως τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς Θ. ὁ δὲ ΕΔ πρὸς τὸν ΔΖ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· οὐδʼ ἄρα τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς Θ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ Θ μήκει.
For let the square on Θ be that by which the square on ΒΗ is greater than the square on ΗΓ. Since, then, as the square on ΒΗ is to the square on ΗΓ, so is ΔΕ to ΕΖ, therefore, convertendo, as ΕΔ is to ΔΖ, so is the square on ΒΗ to the square on Θ. And ΕΔ does not have to ΔΖ the ratio which a square number has to a square number; therefore the square on ΒΗ also does not have to the square on Θ the ratio which a square number has to a square number; therefore ΒΗ is incommensurable in length with Θ.
καὶ δύναται ἡ ΒΗ τῆς ΗΓ μεῖζον τῷ ἀπὸ τῆς Θ· ἡ ΗΒ ἄρα τῆς ΗΓ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ μήκει.
And the square on ΒΗ is greater than the square on ΗΓ by the square on Θ; therefore ΗΒ is greater in square than ΗΓ by the square on a straight line incommensurable in length with itself.
καί ἐστιν ἡ προσαρμόζουσα ἡ ΓΗ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ τῇ Α μήκει· ἡ ἄρα ΒΓ ἀποτομή ἐστι πέμπτη.
And the annex ΓΗ is commensurable in length with the set-out rational straight line A; therefore ΒΓ is a fifth apotome.
εὕρηται ἄρα ἡ πέμπτη ἀποτομὴ ἡ ΒΓ· ὅπερ ἔδει δεῖξαι.
¦35 Therefore the fifth apotome ΒΓ has been found; which was to be demonstrated.