Page 387 - Euclid's Elements of Geometry

ST EW iþ. aþ ELEMENTS BOOK 10

Proposition 91
.
᾿Εὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ ἀποτομῆς πρώτης, If an area is contained by a rational (straight-line) and
ἡ τὸ χωρίον δυναμένη ἀπορομή ἐστιν. a first apotome then the square-root of the area is an apo-
Περιεχέσθω γὰρ χωρίον τὸ ΑΒ ὑπὸ ῥητῆς τῆς ΑΓ καὶ tome.
ἀποτομῆς πρώτης τῆς ΑΔ· λέγω, ὅτι ἡ τὸ ΑΒ χωρίον δυ- For let the area AB have been contained by the ratio-
ναμένη ἀποτομή ἐστιν. nal (straight-line) AC and the first apotome AD. I say
that the square-root of area AB is an apotome.
Α ∆ Ε Ζ Η Ν A D E F G N
Λ Ο L P
Φ V
Σ Ξ S O
Υ U
Χ W
Γ Β Θ Ι Κ C B H I K

Ρ Μ R M
Τ T
᾿Επεὶ γὰρ ἀποτομή ἐστι πρώτη ἡ ΑΔ, ἔστω αὐτῇ προ- For since AD is a first apotome, let DG be its at-
σαρμόζουσα ἡ ΔΗ· αἱ ΑΗ, ΗΔ ἄρα ῥηταί εἰσι δυνάμει μόνον tachment. Thus, AG and DG are rational (straight-lines
σύμμετροι. καὶ ὅλη ἡ ΑΗ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ which are) commensurable in square only [Prop. 10.73].
τῇ ΑΓ, καὶ ἡ ΑΗ τῆς ΗΔ μεῖζον δύναται τῷ ἀπὸ συμμέτρου And the whole, AG, is commensurable (in length) with
ἑαυτῇ μήκει· ἐὰν ἄρα τῷ τετάρτῳ μέρει τοῦ ἀπὸ τῆς ΔΗ the (previously) laid down rational (straight-line) AC,
ἴσον παρὰ τὴν ΑΗ παραβληθῇ ἐλλεῖπον εἴδει τετραγώνῳ, and the square on AG is greater than (the square on)
εἰς σύμμετρα αὐτὴν διαιρεῖ. τετμήσθω ἡ ΔΗ δίχα κατὰ τὸ GD by the (square) on (some straight-line) commensu-
Ε, καὶ τῷ ἀπὸ τῆς ΕΗ ἴσον παρὰ τὴν ΑΗ παραβεβλήσθω rable in length with (AG) [Def. 10.11]. Thus, if (an area)
ἐλλεῖπον εἴδει τετραγώνῳ, καὶ ἔστω τὸ ὑπὸ τῶν ΑΖ, ΖΗ· equal to the fourth part of the (square) on DG is applied
σύμμετρος ἄρα ἐστὶν ἡ ΑΖ τῇ ΖΗ. καὶ διὰ τῶν Ε, Ζ, Η to AG, falling short by a square figure, then it divides
σημείων τῇ ΑΓ παράλληλοι ἤχθωσαν αἱ ΕΘ, ΖΙ, ΗΚ. (AG) into (parts which are) commensurable (in length)
Καὶ ἐπεὶ σύμμετρός ἐστιν ἡ ΑΖ τῇ ΖΗ μήκει, καὶ ἡ ΑΗ [Prop. 10.17]. Let DG have been cut in half at E. And
ἄρα ἑκατέρᾳ τῶν ΑΖ, ΖΗ σύμμετρός ἐστι μήκει. ἀλλὰ ἡ let (an area) equal to the (square) on EG have been ap-
ΑΗ σύμμετρός ἐστι τῇ ΑΓ· καὶ ἑκατέρα ἄρα τῶν ΑΖ, ΖΗ plied to AG, falling short by a square figure. And let it
σύμμετρός ἐστι τῇ ΑΓ μήκει. καί ἐστι ῥητὴ ἡ ΑΓ· ῥητὴ be the (rectangle contained) by AF and FG. AF is thus
ἄρα καὶ ἑκατέρα τῶν ΑΖ, ΖΗ· ὥστε καὶ ἑκάτερον τῶν ΑΙ, commensurable (in length) with FG. And let EH, FI,
ΖΚ ῥητόν ἐστιν. καὶ ἐπεὶ σύμμετρός ἐστιν ἡ ΔΕ τῇ ΕΗ and GK have been drawn through points E, F, and G
μήκει, καὶ ἡ ΔΗ ἄρα ἑκατέρᾳ τῶν ΔΕ, ΕΗ σύμμετρός ἐστι (respectively), parallel to AC.
μήκει. ῥητὴ δὲ ἡ ΔΗ καὶ ἀσύμμετρος τῇ ΑΓ μήκει· ῥητὴ And since AF is commensurable in length with FG,
ἄρα καὶ ἑκατέρα τῶν ΔΕ, ΕΗ καὶ ἀσύμμετρος τῇ ΑΓ μήκει· AG is thus also commensurable in length with each of
ἑκάτερον ἄρα τῶν ΔΘ, ΕΚ μέσον ἐστίν. AF and FG [Prop. 10.15]. But AG is commensurable
Κείσθω δὴ τῷ μὲν ΑΙ ἴσον τετράγωνον τὸ ΛΜ, τῷ δὲ (in length) with AC. Thus, each of AF and FG is also
ΖΚ ἴσον τετράγωνον ἀφῃρήσθω κοινὴν γωνίαν ἔχον αὐτῷ commensurable in length with AC [Prop. 10.12]. And
τὴν ὑπὸ ΛΟΜ τὸ ΝΞ· περὶ τὴν αὐτὴν ἄρα διάμετρόν ἐστι τὰ AC is a rational (straight-line). Thus, AF and FG (are)
ΛΜ, ΝΞ τετράγωνα. ἔστω αὐτῶν διάμετρος ἡ ΟΡ, καὶ κα- each also rational (straight-lines). Hence, AI and FK
ταγεγράφθω τὸ σχῆμα. ἐπεὶ οὖν ἴσον ἐστὶ τὸ ὑπὸ τῶν ΑΖ, are also each rational (areas) [Prop. 10.19]. And since
ΖΗ περιεχόμενον ὀρθογώνιον τῷ ἀπὸ τῆς ΕΗ τετραγώνῳ, DE is commensurable in length with EG, DG is thus
ἔστιν ἄρα ὡς ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως ἡ ΕΗ πρὸς τὴν ΖΗ. also commensurable in length with each of DE and EG
ἀλλ᾿ ὡς μὲν ἡ ΑΖ πρὸς τὴν ΕΗ, οὕτως τὸ ΑΙ πρὸς τὸ ΕΚ, [Prop. 10.15]. And DG (is) rational, and incommen-
ὡς δὲ ἡ ΕΗ πρὸς τὴν ΖΗ, οὕτως ἐστὶ τὸ ΕΚ πρὸς τὸ ΚΖ· surable in length with AC. DE and EG (are) thus
τῶν ἄρα ΑΙ, ΚΖ μέσον ἀνάλογόν ἐστι τὸ ΕΚ. ἔστι δὲ καὶ each rational, and incommensurable in length with AC
τῶν ΛΜ, ΝΞ μέσον ἀνάλογον τὸ ΜΝ, ὡς ἐν τοῖς ἔμπρο- [Prop. 10.13]. Thus, DH and EK are each medial (ar-
σθεν ἐδείχθη, καί ἐστι τὸ [μὲν] ΑΙ τῷ ΛΜ τετραγώνῳ ἴσον, eas) [Prop. 10.21].
τὸ δὲ ΚΖ τῷ ΝΞ· καὶ τὸ ΜΝ ἄρα τῷ ΕΚ ἴσον ἐστίν. ἀλλὰ So let the square LM, equal to AI, be laid down.
τὸ μὲν ΕΚ τῷ ΔΘ ἐστιν ἴσον, τὸ δὲ ΜΝ τῷ ΛΞ· τὸ ἄρα And let the square NO, equal to FK, have been sub-

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