Page 393 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

῎Εστω γὰρ τῇ ΑΔ προσαρμόζουσα ἡ ΔΗ· αἱ ἄρα line). For let DG be an attachment to AD. Thus, AG
ΑΗ, ΗΔ ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ ἡ ΑΗ and DG are rational (straight-lines which are) commen-
σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ τῇ ΑΓ μήκει, ἡ δὲ ὅλη surable in square only [Prop. 10.73], and AG is com-
ἡ ΑΗ τῆς προσαρμοζούσης τῆς ΔΗ μεῖζον δύναται τῷ mensurable in length with the (previously) laid down ra-
ἀπὸ ἀσυμμέτρου ἑαυτῇ μήκει. ἐπεὶ οὖν ἡ ΑΗ τῆς ΗΔ tional (straight-line) AC, and the square on the whole,
μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ μήκει, ἐὰν ἄρα AG, is greater than (the square on) the attachment, DG,
τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΔΗ ἴσον παρὰ τὴν ΑΗ πα- by the square on (some straight-line) incommensurable
ραβληθῇ ἐλλεῖπον εἴδει τετραγώνῳ, εἰς ἀσύμμετρα αὐτὴν in length with (AG) [Def. 10.14]. Therefore, since the
διελεῖ. τετμήσθω οὖν ἡ ΔΗ δίχα κατὰ τὸ Ε, καὶ τῷ ἀπὸ square on AG is greater than (the square on) GD by
τῆς ΕΗ ἴσον παρὰ τὴν ΑΗ παραβεβλήσθω ἐλλεῖπον εἴδει the (square) on (some straight-line) incommensurable in
τετραγώνῳ, καὶ ἔστω τὸ ὑπὸ τῶν ΑΖ, ΖΗ· ἀσύμμετρος length with (AG), thus if (some area), equal to the fourth
ἄρα ἐστὶ μήκει ἡ ΑΖ τῇ ΖΗ. ἤχθωσαν οὖν διὰ τῶν Ε, Ζ, Η part of the (square) on DG, is applied to AG, falling short
παράλληλοι ταῖς ΑΓ, ΒΔ αἱ ΕΘ, ΖΙ, ΗΚ. ἐπεὶ οὖν ῥητή ἐστιν by a square figure, then it divides (AG) into (parts which
ἡ ΑΗ καὶ σύμμετρος τῇ ΑΓ μήκει, ῥητὸν ἄρα ἐστὶν ὅλον τὸ are) incommensurable (in length) [Prop. 10.18]. There-
ΑΚ. πάλιν, ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΔΗ τῇ ΑΓ μήκει, καί fore, let DG have been cut in half at E, and let (some
εἰσιν ἀμφότεραι ῥηταί, μέσον ἄρα ἐστὶ τὸ ΔΚ. πάλιν, ἐπεὶ area), equal to the (square) on EG, have been applied
ἀσύμμετρός ἐστὶν ἡ ΑΖ τῇ ΖΗ μήκει, ἀσύμμετρον ἄρα καὶ to AG, falling short by a square figure, and let it be the
τὸ ΑΙ τῷ ΖΚ. (rectangle contained) by AF and FG. Thus, AF is in-
Συνεστάτω οὖν τῷ μὲν ΑΙ ἴσον τετράγωνον τὸ ΛΜ, τῷ commensurable in length with FG. Therefore, let EH,
δὲ ΖΚ ἴσον ἀφῃρήσθω περὶ τὴν αὐτὴν γωνίαν τὴν ὑπὸ τῶν FI, and GK have been drawn through E, F, and G (re-
ΛΟΜ τὸ ΝΞ. περὶ τὴν αὐτὴν ἄρα διάμετόν ἐστι τὰ ΛΜ, ΝΞ spectively), parallel to AC and BD. Therefore, since AG
τετράγωνα. ἔστω αὐτῶν διάμετος ἡ ΟΡ, καὶ καταγεγράφθω is rational, and commensurable in length with AC, the
τὸ σχῆμα. ἐπεὶ οὖν τὸ ὑπὸ τῶν ΑΖ, ΖΗ ἴσον ἐστὶ τῷ ἀπὸ whole (area) AK is thus rational [Prop. 10.19]. Again,
τῆς ΕΗ, ἀνάλογον ἄρα ἐστὶν ὡς ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως since DG is incommensurable in length with AC, and
ἡ ΕΗ πρὸς τὴν ΖΗ. ἀλλ᾿ ὡς μὲν ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως both are rational (straight-lines), DK is thus a medial
ἐστὶ τὸ ΑΙ πρὸς τὸ ΕΚ, ὡς δὲ ἡ ΕΗ πρὸς τὴν ΖΗ, οὕτως (area) [Prop. 10.21]. Again, since AF is incommensu-
ἐστὶ τὸ ΕΚ πρὸς τὸ ΖΚ· τῶν ἄρα ΑΙ, ΖΚ μέσον ἀνάλογόν rable in length with FG, AI (is) thus also incommensu-
ἐστι τὸ ΕΚ. ἔστι δὲ καὶ τῶν ΛΜ, ΝΞ τετραγώνων μέσον rable with FK [Props. 6.1, 10.11].
ἀνάλογον τὸ ΜΝ, καί ἐστιν ἴσον τὸ μὲν ΑΙ τῷ ΛΜ, τὸ δὲ Therefore, let the square LM, equal to AI, have been
ΖΚ τῷ ΝΞ· καὶ τὸ ΕΚ ἄρα ἴσον ἐστὶ τῷ ΜΝ. ἀλλὰ τῷ μὲν constructed. And let NO, equal to FK, (and) about the
ΕΚ ἴσον ἐστὶ τὸ ΔΘ, τῷ δὲ ΜΝ ἴσον ἐστὶ τὸ ΛΞ· ὅλον same angle, LPM, have been subtracted (from LM).
ἄρα τὸ ΔΚ ἴσον ἐστὶ τῷ ΥΦΧ γνώμονι καὶ τῷ ΝΞ. ἐπεὶ Thus, the squares LM and NO are about the same diag-
οὐν ὅλον τὸ ΑΚ ἴσον ἐστὶ τοῖς ΛΜ, ΝΞ τετραγώνοις, ὧν onal [Prop. 6.26]. Let PR be their (common) diagonal,
τὸ ΔΚ ἴσον ἐστὶ τῷ ΥΦΧ γνώμονι καὶ τῷ ΝΞ τετραγώνῳ, and let the (rest of the) figure have been drawn. There-
λοιπὸν ἄρα τὸ ΑΒ ἴσον ἐστὶ τῷ ΣΤ, τουτέστι τῷ ἀπὸ τῆς fore, since the (rectangle contained) by AF and FG is
ΛΝ τετραγώνῳ· ἡ ΛΝ ἄρα δύναται τὸ ΑΒ χωρίον. λέγω, equal to the (square) on EG, thus, proportionally, as AF
ὅτι ἡ ΛΝ ἄλογός ἐστιν ἡ καλουμένη ἐλάσσων. is to EG, so EG (is) to FG [Prop. 6.17]. But, as AF (is)
᾿Επεὶ γὰρ ῥητόν ἐστι τὸ ΑΚ καί ἐστιν ἴσον τοῖς ἀπὸ τῶν to EG, so AI is to EK, and as EG (is) to FG, so EK is
ΛΟ, ΟΝ τετράγωνοις, τὸ ἄρα συγκείμενον ἐκ τῶν ἀπὸ τῶν to FK [Prop. 6.1]. Thus, EK is the mean proportional to
ΛΟ, ΟΝ ῥητόν ἐστιν. πάλιν, ἐπεὶ τὸ ΔΚ μέσον ἐστίν, καί AI and FK [Prop. 5.11]. And MN is also the mean pro-
ἐστιν ἴσον τὸ ΔΚ τῷ δὶς ὑπὸ τῶν ΛΟ, ΟΝ, τὸ ἄρα δὶς ὑπὸ portional to the squares LM and NO [Prop. 10.13 lem.],
τῶν ΛΟ, ΟΝ μέσον ἐστίν. καὶ ἐπεὶ ἀσύμμετρον ἐδείχθη τὸ and AI is equal to LM, and FK to NO. EK is thus
ΑΙ τῷ ΖΚ, ἀσύμμετρον ἄρα καὶ τὸ ἀπὸ τῆς ΛΟ τετράγωνον also equal to MN. But, DH is equal to EK, and LO is
τῷ ἀπὸ τῆς ΟΝ τετραγώνῳ. αἱ ΛΟ, ΟΝ ἄρα δυνάμει εἰσὶν equal to MN [Prop. 1.43]. Thus, the whole of DK is
ἀσύμμετροι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ᾿ αὐτῶν equal to the gnomon UV W and NO. Therefore, since
τετραγώνων ῥητόν, τὸ δὲ δὶς ὑπ᾿ αὐτῶν μέσον. ἡ ΛΝ ἄρα the whole of AK is equal to the (sum of the) squares LM
ἄλογός ἐστιν ἡ καλουμένη ἐλάσσων· καὶ δύναται τὸ ΑΒ and NO, of which DK is equal to the gnomon UV W
χωρίον. and the square NO, the remainder AB is thus equal to
῾Η ἄρα τὸ ΑΒ χωρίον δυναμένη ἐλάσσων ἐστίν· ὅπερ ST —that is to say, to the square on LN. Thus, LN is the
ἔδει δεῖξαι. square-root of area AB. I say that LN is the irrational
(straight-line which is) called minor.

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