Humanitext Reader

Euclid · Elements §10.prop3.107-10.prop3.108

Straight Line Producing Medial Whole and the Remaining Area

Passage 234 of 316 · Greek

Summary

Proposition 107 proves that a straight line commensurable with that which produces with a medial area a medial whole is of the same type. Proposition 108 proves that the side of the area remaining when a medial area is subtracted from a rational area is either an apotome or a minor straight line.

§10.prop3.107ἡ τῇ μετὰ μέσου μέσον τὸ ὅλον ποιούσῃ σύμμετρος καὶ αὐτὴ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν. ἔστω μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα ἡ ΑΒ, καὶ τῇ ΑΒ ἔστω σύμμετρος ἡ ΓΔ·
A straight line commensurable with that which produces with a medial area a medial whole is itself also that which produces with a medial area a medial whole. ἔστω μετὰ μέσου μέσον τὸ ὅλον ποιοῦσα ἡ ΑΒ, καὶ τῇ ΑΒ ἔστω σύμμετρος ἡ ΓΔ· λέγω, ὅτι καὶ ἡ ΓΔ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν.
λέγω, ὅτι καὶ ἡ ΓΔ μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν. ἔστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΕ, καὶ τὰ αὐτὰ κατεσκευάσθω· αἱ ΑΕ, ΕΒ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τετραγώνων μέσον καὶ τὸ ὑπʼ αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τὸ συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τετραγώνων τῷ ὑπʼ αὐτῶν.
For let AB be that which produces with a medial area a medial whole, and let GD be commensurable with AB; I say that GD is also that which produces with a medial area a medial whole. ἔστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΕ, καὶ τὰ αὐτὰ κατεσκευάσθω· αἱ ΑΕ, ΕΒ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τετραγώνων μέσον καὶ τὸ ὑπʼ αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τὸ συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τετραγώνων τῷ ὑπʼ αὐτῶν.
καί εἰσιν, ὡς ἐδείχθη, αἱ ΑΕ, ΕΒ σύμμετροι ταῖς ΓΖ, ΖΔ, καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ, τὸ δὲ ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν ΓΖ, ΖΔ· καὶ αἱ ΓΖ, ΖΔ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τετραγώνων μέσον καὶ τὸ ὑπʼ αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τὸ συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τῷ ὑπʼ αὐτῶν.
For let BE be the annex to AB, and let the same construction be made; therefore AE, EB are incommensurable in square, producing the sum of the squares on them medial, and the rectangle contained by them medial, and further the sum of the squares on them incommensurable with the rectangle contained by them. καί εἰσιν, ὡς ἐδείχθη, αἱ ΑΕ, ΕΒ σύμμετροι ταῖς ΓΖ, ΖΔ, καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ, τὸ δὲ ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν ΓΖ, ΖΔ· And, as was proved, AE, EB are commensurable with GZ, ZD, and the sum of the squares on AE, EB with the sum of the squares on GZ, ZD, and the rectangle contained by AE, EB with the rectangle contained by GZ, ZD; καὶ αἱ ΓΖ, ΖΔ ἄρα δυνάμει εἰσὶν ἀσύμμετροι ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τετραγώνων μέσον καὶ τὸ ὑπʼ αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τὸ συγκείμενον ἐκ τῶν ἀπʼ αὐτῶν τῷ ὑπʼ αὐτῶν. therefore GZ, ZD are also incommensurable in square, producing both the sum of the squares on them medial, and the rectangle contained by them medial, and further the sum of the squares on them incommensurable with the rectangle contained by them. ἡ ΓΔ ἄρα μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν· ὅπερ ἔδει δεῖξαι.
ἡ ΓΔ ἄρα μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν· ὅπερ ἔδει δεῖξαι.
Therefore GD is that which produces with a medial area a medial whole; which was to be proved.
§10.prop3.108ἀπὸ ῥητοῦ μέσου ἀφαιρουμένου ἡ τὸ λοιπὸν χωρίον δυναμένη μία δύο ἀλόγων γίνεται ἤτοι ἀποτομὴ ἢ ἐλάσσων. ἀπὸ γὰρ ῥητοῦ τοῦ ΒΓ μέσον ἀφῃρήσθω τὸ ΒΔ·
If from a rational area a medial area be subtracted, the straight line producing the remaining area is one of two irrational straight lines, either an apotome or a minor. ἀπὸ γὰρ ῥητοῦ τοῦ ΒΓ μέσον ἀφῃρήσθω τὸ ΒΔ· λέγω, ὅτι ἡ τὸ λοιπὸν δυναμένη τὸ ΕΓ μία δύο ἀλόγων γίνεται ἤτοι ἀποτομὴ ἢ ἐλάσσων.
λέγω, ὅτι ἡ τὸ λοιπὸν δυναμένη τὸ ΕΓ μία δύο ἀλόγων γίνεται ἤτοι ἀποτομὴ ἢ ἐλάσσων. Ἐκκείσθω γὰρ ῥητὴ ἡ ΖΗ, καὶ τῷ μὲν ΒΓ ἴσον παρὰ τὴν ΖΗ παραβεβλήσθω ὀρθογώνιον παραλληλόγραμμον τὸ ΗΘ, τῷ δὲ ΔΒ ἴσον ἀφῃρήσθω τὸ ΗΚ·
For from the rational area BC let the medial area BD be subtracted; I say that the straight line producing the remaining area EG is one of two irrational straight lines, either an apotome or a minor. Ἐκκείσθω γὰρ ῥητὴ ἡ ΖΗ, καὶ τῷ μὲν ΒΓ ἴσον παρὰ τὴν ΖΗ παραβεβλήσθω ὀρθογώνιον παραλληλόγραμμον τὸ ΗΘ, τῷ δὲ ΔΒ ἴσον ἀφῃρήσθω τὸ ΗΚ· λοιπὸν ἄρα τὸ ΕΓ ἴσον ἐστὶ τῷ ΛΘ.
λοιπὸν ἄρα τὸ ΕΓ ἴσον ἐστὶ τῷ ΛΘ. ἐπεὶ οὖν ῥητὸν μέν ἐστι τὸ ΒΓ, μέσον δὲ τὸ ΒΔ, ἴσον δὲ τὸ μὲν ΒΓ τῷ ΗΘ, τὸ δὲ ΒΔ τῷ ΗΚ, ῥητὸν μὲν ἄρα ἐστὶ τὸ ΗΘ, μέσον δὲ τὸ ΗΚ. καὶ παρὰ ῥητὴν τὴν ΖΗ παράκειται· ῥητὴ μὲν ἄρα ἡ ΖΘ καὶ σύμμετρος τῇ ΖΗ μήκει, ῥητὴ δὲ ἡ ΖΚ καὶ ἀσύμμετρος τῇ ΖΗ μήκει· ἀσύμμετρος ἄρα ἐστὶν ἡ ΖΘ τῇ ΖΚ μήκει. αἱ ΖΘ, ΖΚ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι·
For let the rational straight line ZH be set out, and to ZH let there be applied the rectangular parallelogram HΘ equal to BC, and let HK equal to DB be subtracted; therefore the remainder EG is equal to LΘ. ἐπει οὖν ῥητὸν μέν ἐστι τὸ ΒΓ, μέσον δὲ τὸ ΒΔ, ἴσον δὲ τὸ μὲν ΒΓ τῷ ΗΘ, τὸ δὲ ΒΔ τῷ ΗΚ, ῥητὸν μὲν ἄρα ἐστὶ τὸ ΗΘ, μέσον δὲ τὸ ΗΚ. Since then BC is rational, and BD is medial, and BC is equal to HΘ, and BD to HK, therefore HΘ is rational, and HK is medial. καὶ παρὰ ῥητην τὴν ΖΗ παράκειται· ῥητὴ μὲν ἄρα ἡ ΖΘ καὶ σύμμετρος τῇ ΖΗ μήκει, ῥητὴ δὲ ἡ ΖΚ καὶ ἀσύμμετρος τῇ ΖΗ μήκει· ἀσύμμετρος ἄρα ἐστὶν ἡ ΖΘ τῇ ΖΚ μήκει. And they are applied to the rational straight line ZH; therefore ZΘ is rational and commensurable in length with ZH, and ZK is rational and incommensurable in length with ZH; therefore ZΘ is incommensurable in length with ZK. αἱ ΖΘ, ΖΚ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΚΘ, προσαρμόζουσα δὲ αὐτῇ ἡ ΚΖ.
ἀποτομὴ ἄρα ἐστὶν ἡ ΚΘ, προσαρμόζουσα δὲ αὐτῇ ἡ ΚΖ. ἤτοι δὴ ἡ ΘΖ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἢ οὔ.
Therefore ZΘ, ZK are rational straight lines commensurable in square only; therefore KΘ is an apotome, and KZ is the annex to it. ἤτοι δὴ ἡ ΘΖ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἢ οὔ. Either then ΘZ is greater in square than ZK by the square on a straight line commensurable with itself, or not.
δυνάσθω πρότερον τῷ ἀπὸ συμμέτρου. καί ἐστιν ὅλη ἡ ΘΖ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ· ἀποτομὴ ἄρα πρώτη ἐστὶν ἡ ΚΘ. τὸ δʼ ὑπὸ ῥητῆς καὶ ἀποτομῆς πρώτης περιεχόμενον ἡ δυναμένη ἀποτομή ἐστιν. ἡ ἄρα τὸ ΛΘ, τουτέστι τὸ ΕΓ, δυναμένη ἀποτομή ἐστιν.
δυνάσθω πρότερον τῷ ἀπὸ συμμέτρου. καί ἐστιν ὅλη ἡ ΘΖ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ· ἀποτομὴ ἄρα πρώτη ἐστὶν ἡ ΚΘ. Let it first be greater in square by the square on a straight line commensurable with itself; and the whole ΘZ is commensurable in length with the rational straight line ZH set out; therefore KΘ is a first apotome. τὸ δʼ ὑπὸ ῥητῆς καὶ ἀποτομῆς πρώτης περιεχόμενον ἡ δυναμένη ἀποτομή ἐστιν. ἡ ἄρα τὸ ΛΘ, τουτέστι τὸ ΕΓ, δυναμένη ἀποτομή ἐστιν. But the straight line producing the area contained by a rational straight line and a first apotome is an apotome. Therefore the straight line producing LΘ, that is, EG, is an apotome.
εἰ δὲ ἡ ΘΖ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, καί ἐστιν ὅλη ἡ ΖΘ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ, ἀποτομὴ τετάρτη ἐστὶν ἡ ΚΘ. τὸ δʼ ὑπὸ ῥητῆς καὶ ἀποτομῆς τετάρτης περιεχόμενον ἡ δυναμένη ἐλάσσων ἐστίν· ὅπερ ἔδει δεῖξαι.
εἰ δὲ ἡ ΘΖ τῆς ΖΚ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, καί ἐστιν ὅλη ἡ ΖΘ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ ΖΗ, ἀποτομὴ τετάρτη ἐστὶν ἡ ΚΘ. But if ΘZ is greater in square than ZK by the square on a straight line incommensurable with itself, and the whole ZΘ is commensurable in length with the rational straight line ZH set out, KΘ is a fourth apotome. τὸ δʼ ὑπὸ ῥητῆς καὶ ἀποτομῆς τετάρτης περιεχόμενον ἡ δυναμένη ἐλάσσων ἐστίν· ὅπερ ἔδει δεῖξαι. But the straight line producing the area contained by a rational straight line and a fourth apotome is a minor; which was to be proved.

Notes

  1. §10.prop3.107ποιοῦσαι τό τε συγκείμενον — The participle ποιοῦσαι modifies αἱ ΑΕ, ΕΒ, governing three parallel double-accusative constructions (object + complement): (1) τό τε συγκείμενον... μέσον (making the sum medial), (2) καὶ τὸ ὑπʼ αὐτῶν μέσον (and the rectangle medial), and (3) καὶ ἔτι ἀσύμμετρον τὸ συγκείμενον... τῷ ὑπʼ αὐτῶν (and further making the sum incommensurable with the rectangle). In the third structure, the adjective ἀσύμμετρον serves as a predicative complement to τὸ συγκείμενον and takes the dative τῷ ὑπʼ αὐτῶν.
  2. §10.prop3.108δυναμένη — The present participle δυναμένη (from δύναμαι) in Euclid's geometry means 'being equal in square to' or 'the side of the square equal to [an area]'. Here it is used substantively to mean 'the straight line producing (or equal in square to) the remaining area'.
  3. §10.prop3.108μεῖζον δύναται — The expression μεῖζον δύναται means 'is greater in square', and is accompanied by the dative of difference (here τῷ ἀπὸ συμμέτρου / τῷ ἀπὸ ἀσυμμέτρου, 'by the square on a straight line commensurable/incommensurable'). This expresses the algebraic relationship x^2 - y^2 = z^2, where the square on x exceeds the square on y by the square on z which is commensurable or incommensurable in length with x.

Cite this passage

Euclid, Elements §10.prop3.107-10.prop3.108. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:10.prop3.107-10.prop3.108

Please note the AI-draft status of the translation and the date accessed.

Translation, notes and summary are AI-generated drafts, revised through reader feedback.