Page 506 - Euclid's Elements of Geometry

ST EW igþ. aþ ELEMENTS BOOK 13

Proposition 1
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᾿Εὰν εὐθεῖα γραμμὴ ἄκρον καὶ μέσον λόγον τμηθῇ, If a straight-line is cut in extreme and mean ratio
τὸ μεῖζον τμῆμα προσλαβὸν τὴν ἡμίσειαν τῆς ὅλης πεν- then the square on the greater piece, added to half of
ταπλάσιον δύναται τοῦ ἀπὸ τῆς ἡμισείας τετραγώνου. the whole, is five times the square on the half.
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Εὐθεῖα γὰρ γραμμὴ ἡ ΑΒ ἄκρον καὶ μέσον λόγον For let the straight-line AB have been cut in extreme
τετμήσθω κατὰ τὸ Γ σημεῖον, καὶ ἔστω μεῖζον τμῆμα τὸ and mean ratio at point C, and let AC be the greater
ΑΓ, καὶ ἐκβεβλήσθω ἐπ᾿ εὐθείας τῇ ΓΑ εὐθεῖα ἡ ΑΔ, καὶ piece. And let the straight-line AD have been produced
κείσθω τῆς ΑΒ ἡμίσεια ἡ ΑΔ· λέγω, ὅτι πενταπλάσιόν ἐστι in a straight-line with CA. And let AD be made (equal
τὸ ἀπὸ τῆς ΓΔ τοῦ ἀπὸ τῆς ΔΑ. to) half of AB. I say that the (square) on CD is five times
᾿Αναγεγράφθωσαν γὰρ ἀπὸ τῶν ΑΒ, ΔΓ τετράγωνα the (square) on DA.
τὰ ΑΕ, ΔΖ, καὶ καταγεγράφθω ἐν τῷ ΔΖ τὸ σχῆμα, καὶ For let the squares AE and DF have been described
διήχθω ἡ ΖΓ ἐπὶ τὸ Η. καὶ ἐπεὶ ἡ ΑΒ ἄκρον καὶ μέσον on AB and DC (respectively). And let the figure in DF
λόγον τέτμηται κατὰ τὸ Γ, τὸ ἄρα ὑπὸ τῶν ΑΒΓ ἴσον ἐστὶ have been drawn. And let FC have been drawn across to
τῷ ἀπὸ τῆς ΑΓ. καί ἐστι τὸ μὲν ὑπὸ τῶν ΑΒΓ τὸ ΓΕ, τὸ G. And since AB has been cut in extreme and mean ratio
δὲ ἀπὸ τῆς ΑΓ τὸ ΖΘ· ἴσον ἄρα τὸ ΓΕ τῷ ΖΘ. καὶ ἐπεὶ at C, the (rectangle contained) by ABC is thus equal to
διπλῆ ἐστιν ἡ ΒΑ τῆς ΑΔ, ἴση δὲ ἡ μὲν ΒΑ τῇ ΚΑ, ἡ δὲ the (square) on AC [Def. 6.3, Prop. 6.17]. And CE is
ΑΔ τῇ ΑΘ, διπλῆ ἄρα καὶ ἡ ΚΑ τῆς ΑΘ. ὡς δὲ ἡ ΚΑ πρὸς the (rectangle contained) by ABC, and FH the (square)
τὴν ΑΘ, οὕτως τὸ ΓΚ πρὸς τὸ ΓΘ· διπλάσιον ἄρα τὸ ΓΚ on AC. Thus, CE (is) equal to FH. And since BA is
τοῦ ΓΘ. εἰσὶ δὲ καὶ τὰ ΛΘ, ΘΓ διπλάσια τοῦ ΓΘ. ἴσον ἄρα double AD, and BA (is) equal to KA, and AD to AH,
τὸ ΚΓ τοῖς ΛΘ, ΘΓ. ἐδείχθη δὲ καὶ τὸ ΓΕ τῷ ΘΖ ἴσον· KA (is) thus also double AH. And as KA (is) to AH, so
ὅλον ἄρα τὸ ΑΕ τετράγωνον ἴσον ἐστὶ τῷ ΜΝΞ γνώμονι. CK (is) to CH [Prop. 6.1]. Thus, CK (is) double CH.
καὶ ἐπεὶ διπλῆ ἐστιν ἡ ΒΑ τῆς ΑΔ, τετραπλάσιόν ἐστι τὸ And LH plus HC is also double CH [Prop. 1.43]. Thus,
ἀπὸ τῆς ΒΑ τοῦ ἀπὸ τῆς ΑΔ, τουτέστι τὸ ΑΕ τοῦ ΔΘ. KC (is) equal to LH plus HC. And CE was also shown
ἴσον δὲ τὸ ΑΕ τῷ ΜΝΞ γνώμονι· καὶ ὁ ΜΝΞ ἄρα γνώμων (to be) equal to HF. Thus, the whole square AE is equal
τετραπλάσιός ἐστι τοῦ ΑΟ· ὅλον ἄρα τὸ ΔΖ πενταπλάσιόν to the gnomon MNO. And since BA is double AD, the
ἐστι τοῦ ΑΟ. καί ἐστι τὸ μὲν ΔΖ τὸ ἀπὸ τῆς ΔΓ, τὸ δὲ ΑΟ (square) on BA is four times the (square) on AD—that
τὸ ἀπὸ τῆς ΔΑ· τὸ ἄρα ἀπὸ τῆς ΓΔ πενταπλάσιόν ἐστι τοῦ is to say, AE (is four times) DH. And AE (is) equal to
ἀπὸ τῆς ΔΑ. gnomon MNO. And, thus, gnomon MNO is also four
᾿Εὰν ἄρα εὐθεῖα ἄκρον καὶ μέσον λόγον τμηθῇ, τὸ μεῖζον times AP. Thus, the whole of DF is five times AP. And
τμῆμα προσλαβὸν τὴν ἡμίσειαν τῆς ὅλης πενταπλάσιον DF is the (square) on DC, and AP the (square) on DA.
δύναται τοῦ ἀπὸ τῆς ἡμισείας τετραγώνου· ὅπερ ἔδει δεῖξαι. Thus, the (square) on CD is five times the (square) on
DA.
Thus, if a straight-line is cut in extreme and mean ra-
tio then the square on the greater piece, added to half of

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