Page 389 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

Χωρίον γὰρ τὸ ΑΒ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΓ καὶ For let the area AB have been contained by the ratio-
ἀποτομῆς δευτέρας τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ χωρίον nal (straight-line) AC and the second apotome AD. I say
δυναμένη μέσης ἀποτομή ἐστι πρώτη. that the square-root of area AB is the first apotome of a
῎Εστω γὰρ τῇ ΑΔ προσαρμόζουσα ἡ ΔΗ· αἱ ἄρα medial (straight-line).
ΑΗ, ΗΔ ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ ἡ προ- For let DG be an attachment to AD. Thus, AG and
σαρμόζουσα ἡ ΔΗ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ τῇ GD are rational (straight-lines which are) commensu-
ΑΓ, ἡ δὲ ὅλη ἡ ΑΗ τῆς προσαρμοζούσης τῆς ΗΔ μεῖζον rable in square only [Prop. 10.73], and the attachment
δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει. ἐπεὶ οὖν ἡ ΑΗ DG is commensurable (in length) with the (previously)
τῆς ΗΔ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, ἐὰν ἄρα laid down rational (straight-line) AC, and the square on
τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΗΔ ἴσον παρὰ τὴν ΑΗ πα- the whole, AG, is greater than (the square on) the at-
ραβληθῇ ἐλλεῖπον εἴδει τετραγώνῳ, εἰς σύμμετρα αὐτὴν tachment, GD, by the (square) on (some straight-line)
διαιρεῖ. τετμήσθω οὖν ἡ ΔΗ δίχα κατὰ τὸ Ε· καὶ τῷ ἀπὸ commensurable in length with (AG) [Def. 10.12]. There-
τῆς ΕΗ ἴσον παρὰ τὴν ΑΗ παραβεβλήσθω ἐλλεῖπον εἴδει fore, since the square on AG is greater than (the square
τετραγώνῳ, καὶ ἔστω τὸ ὑπὸ τῶν ΑΖ, ΖΗ· σύμμετρος ἄρα on) GD by the (square) on (some straight-line) commen-
ἐστὶν ἡ ΑΖ τῇ ΖΗ μήκει. καὶ ἡ ΑΗ ἄρα ἑκατέρᾳ τῶν ΑΖ, surable (in length) with (AG), thus if (an area) equal
ΖΗ σύμμετρός ἐστι μήκει. ῥητὴ δὲ ἡ ΑΗ καὶ ἀσύμμετρος to the fourth part of the (square) on GD is applied
τῇ ΑΓ μήκει· καὶ ἑκατέρα ἄρα τῶν ΑΖ, ΖΗ ῥητή ἐστι καὶ to AG, falling short by a square figure, then it divides
ἀσύμμετρος τῇ ΑΓ μήκει· ἑκάτερον ἄρα τῶν ΑΙ, ΖΚ μέσον (AG) into (parts which are) commensurable (in length)
ἐστίν. πάλιν, ἐπεὶ σύμμετρός ἐστιν ἡ ΔΕ τῇ ΕΗ, καὶ ἡ ΔΗ [Prop. 10.17]. Therefore, let DG have been cut in half at
ἄρα ἑκατέρᾳ τῶν ΔΕ, ΕΗ σύμμετρός ἐστιν. ἀλλ᾿ ἡ ΔΗ E. And let (an area) equal to the (square) on EG have
σύμμετρός ἐστι τῇ ΑΓ μήκει [ῥητὴ ἄρα καὶ ἑκατέρα τῶν been applied to AG, falling short by a square figure. And
ΔΕ, ΕΗ καὶ σύμμετρος τῇ ΑΓ μήκει]. ἑκάτερον ἄρα τῶν let it be the (rectangle contained) by AF and FG. Thus,
ΔΘ, ΕΚ ῥητόν ἐστιν. AF is commensurable in length with FG. AG is thus
Συνεστάτω οὖν τῷ μὲν ΑΙ ἴσον τετράγωνον τὸ ΛΜ, τῷ also commensurable in length with each of AF and FG
δὲ ΖΚ ἴσον ἀφῃρήσθω τὸ ΝΞ περὶ τὴν αὐτὴν γωνίαν ὂν τῷ [Prop. 10.15]. And AG (is) a rational (straight-line), and
ΛΜ τὴν ὑπὸ τῶν ΛΟΜ· περὶ τὴν αὐτὴν ἄρα ἐστὶ διάμετρον incommensurable in length with AC. AF and FG are
τὰ ΛΜ, ΝΞ τετράγωνα. ἔστω αὐτῶν διάμετρος ἡ ΟΡ, καὶ thus also each rational (straight-lines), and incommen-
καταγεγράφθω τὸ σχῆμα. ἐπεὶ οὖν τὰ ΑΙ, ΖΚ μέσα ἐστὶ surable in length with AC [Prop. 10.13]. Thus, AI and
καί ἐστιν ἴσα τοῖς ἀπὸ τῶν ΛΟ, ΟΝ, καὶ τὰ ἀπὸ τῶν ΛΟ, FK are each medial (areas) [Prop. 10.21]. Again, since
ΟΝ [ἄρα] μέσα ἐστίν· καὶ αἱ ΛΟ, ΟΝ ἄρα μέσαι εἰσὶ δυνάμει DE is commensurable (in length) with EG, thus DG is
μόνον σύμμετροι. καὶ ἐπεὶ τὸ ὑπὸ τῶν ΑΖ, ΖΗ ἴσον ἐστὶ τῷ also commensurable (in length) with each of DE and EG
ἀπὸ τῆς ΕΗ, ἔστιν ἄρα ὡς ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως ἡ ΕΗ [Prop. 10.15]. But, DG is commensurable in length with
πρὸς τὴν ΖΗ· ἀλλ᾿ ὡς μὲν ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως τὸ ΑΙ AC [thus, DE and EG are also each rational, and com-
πρὸς τὸ ΕΚ· ὡς δὲ ἡ ΕΗ πρὸς τὴν ΖΗ, οὕτως [ἐστὶ] τὸ ΕΚ mensurable in length with AC]. Thus, DH and EK are
πρὸς τὸ ΖΚ· τῶν ἄρα ΑΙ, ΖΚ μέσον ἀνάλογόν ἐστι τὸ ΕΚ. each rational (areas) [Prop. 10.19].
ἔστι δὲ καὶ τῶν ΛΜ, ΝΞ τετραγώνων μέσον ἀνάλογον τὸ Therefore, let the square LM, equal to AI, have
ΜΝ· καί ἐστιν ἴσον τὸ μὲν ΑΙ τῷ ΛΜ, τὸ δὲ ΖΚ τῷ ΝΞ· καὶ been constructed. And let NO, equal to FK, which is
τὸ ΜΝ ἄρα ἴσον ἐστὶ τῷ ΕΚ. ἀλλὰ τῷ μὲν ΕΚ ἴσον [ἐστὶ] about the same angle LPM as LM, have been subtracted
τὸ ΔΘ, τῷ δὲ ΜΝ ἴσον τὸ ΛΞ· ὅλον ἄρα τὸ ΔΚ ἴσον ἐστὶ (from LM). Thus, the squares LM and NO are about
τῷ ΥΦΧ γνώμονι καὶ τῷ ΝΞ. ἐπεὶ οὖν ὅλον τὸ ΑΚ ἴσον the same diagonal [Prop. 6.26]. Let PR be their (com-
ἐστὶ τοῖς ΛΜ, ΝΞ, ὧν τὸ ΔΚ ἴσον ἐστὶ τῷ ΥΦΧ γνώμονι mon) diagonal, and let the (rest of the) figure have been
καὶ τῷ ΝΞ, λοιπὸν ἄρα τὸ ΑΒ ἴσον ἐστὶ τῷ ΤΣ. τὸ δὲ ΤΣ drawn. Therefore, since AI and FK are medial (areas),
ἐστι τὸ ἀπὸ τῆς ΛΝ· τὸ ἀπὸ τῆς ΛΝ ἄρα ἴσον ἐστὶ τῷ ΑΒ and are equal to the (squares) on LP and PN (respec-
χωρίῳ· ἡ ΛΝ ἄρα δύναται τὸ ΑΒ χωρίον. λέγω [δή], ὅτι ἡ tively), [thus] the (squares) on LP and PN are also me-
ΛΝ μέσης ἀποτομή ἐστι πρώτη. dial. Thus, LP and PN are also medial (straight-lines
†
᾿Επεὶ γὰρ ῥητόν ἐστι τὸ ΕΚ καί ἐστιν ἴσον τῷ ΛΞ, ῥητὸν which are) commensurable in square only. And since
ἄρα ἐστὶ τὸ ΛΞ, τουτέστι τὸ ὑπὸ τῶν ΛΟ, ΟΝ. μέσον δὲ the (rectangle contained) by AF and FG is equal to
ἐδείχθη τὸ ΝΞ· ἀσύμμετρον ἄρα ἐστὶ τὸ ΛΞ τῷ ΝΞ· ὡς the (square) on EG, thus as AF is to EG, so EG (is)
δὲ τὸ ΛΞ πρὸς τὸ ΝΞ, οὕτως ἐστὶν ἡ ΛΟ πρὸς ΟΝ· αἱ to FG [Prop. 10.17]. But, as AF (is) to EG, so AI
ΛΟ, ΟΝ ἄρα ἀσύμμετροί εἰσι μήκει. αἱ ἄρα ΛΟ, ΟΝ μέσαι (is) to EK. And as EG (is) to FG, so EK [is] to FK
εἰσὶ δυνάμει μόνον σύμμετροι ῥητὸν περιέχουσαι· ἡ ΛΝ ἄρα [Prop. 6.1]. Thus, EK is the mean proportional to AI

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