Page 344 - Euclid's Elements of Geometry

ST EW iþ.

Α Η Ε Ζ ∆ Ρ Π A G E F D ELEMENTS BOOK 10
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Μ Ξ M O
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Β Θ Κ Λ Γ B H K L C

Σ Ο S P
Περιεχέσθω γὰρ χωρίον τὸ ΑΒΓΔ ὑπὸ ῥητῆς τῆς ΑΒ For let the area ABCD be contained by the rational
καὶ τῆς ἐκ δύο ὀνομάτων δυετέρας τῆς ΑΔ· λέγω, ὅτι ἡ τὸ (straight-line) AB and by the second binomial (straight-
ΑΓ χωρίον δυναμένη ἐκ δύο μέσων πρώτη ἐστίν. line) AD. I say that the square-root of area AC is a first
᾿Επεὶ γὰρ ἐκ δύο ὀνομάτων δευτέρα ἐστὶν ἡ ΑΔ, bimedial (straight-line).
διῃρήσθω εἰς τὰ ὀνόματα κατὰ τὸ Ε, ὥστε τὸ μεῖζον For since AD is a second binomial (straight-line), let it
ὄνομα εἶναι τὸ ΑΕ· αἱ ΑΕ, ΕΔ ἄρα ῥηταί εἰσι δυνάμει have been divided into its (component) terms at E, such
μόνον σύμμετροι, καὶ ἡ ΑΕ τῆς ΕΔ μεῖζον δύναται τῷ ἀπὸ that AE is the greater term. Thus, AE and ED are ratio-
συμμέτρου ἑαυτῇ, καὶ τὸ ἔλαττον ὄνομα ἡ ΕΔ σύμμετρόν nal (straight-lines which are) commensurable in square
ἐστι τῇ ΑΒ μήκει. τετμήσθω ἡ ΕΔ δίχα κατὰ τὸ Ζ, καὶ only, and the square on AE is greater than (the square
τῷ ἀπὸ τῆς ΕΖ ἴσον παρὰ τὴν ΑΕ παραβεβλήσθω ἐλλεῖπον on) ED by the (square) on (some straight-line) commen-
εἴδει τετραγώνῳ τὸ ὑπὸ τῶν ΑΗΕ· σύμμετρος ἄρα ἡ ΑΗ surable (in length) with (AE), and the lesser term ED
τῇ ΗΕ μήκει. καὶ διὰ τῶν Η, Ε, Ζ παράλληλοι ἤχθωσαν is commensurable in length with AB [Def. 10.6]. Let
ταῖς ΑΒ, ΓΔ αἱ ΗΘ, ΕΚ, ΖΛ, καὶ τῷ μὲν ΑΘ παραλλη- ED have been cut in half at F. And let the (rectan-
λογράμμῳ ἴσον τετράγωνον συνεστάτω τὸ ΣΝ, τῷ δὲ ΗΚ gle contained) by AGE, equal to the (square) on EF,
ἴσον τετράγωνον τὸ ΝΠ, καὶ κείσθω ὥστε ἐπ᾿ εὐθείας εἶναι have been applied to AE, falling short by a square fig-
τὴν ΜΝ τῇ ΝΞ· ἐπ᾿ εὐθείας ἄρα [ἐστὶ] καὶ ἡ ΡΝ τῆ ΝΟ. καὶ ure. AG (is) thus commensurable in length with GE
συμπεπληρώσθω τὸ ΣΠ τετράγωνον· φανερὸν δὴ ἐκ τοῦ [Prop. 10.17]. And let GH, EK, and FL have been
προδεδειγμένου, ὅτι τὸ ΜΡ μέσον ἀνάλογόν ἐστι τῶν ΣΝ, drawn through (points) G, E, and F (respectively), par-
ΝΠ, καὶ ἴσον τῷ ΕΛ, καὶ ὅτι τὸ ΑΓ χωρίον δύναται ἡ ΜΞ. allel to AB and CD. And let the square SN, equal to
δεικτέον δή, ὅτι ἡ ΜΞ ἐκ δύο μέσων ἐστὶ πρώτη. the parallelogram AH, have been constructed, and the
᾿Επεὶ ἀσύμμετρός ἐστιν ἡ ΑΕ τῇ ΕΔ μήκει, σύμμετρος square NQ, equal to GK. And let MN be laid down so
δὲ ἡ ΕΔ τῇ ΑΒ, ἀσύμμετρος ἄρα ἡ ΑΕ τῇ ΑΒ. καὶ ἐπεὶ as to be straight-on to NO. Thus, RN [is] also straight-on
σύμμετρός ἐστιν ἡ ΑΗ τῇ ΕΗ, σύμμετρός ἐστι καὶ ἡ ΑΕ to NP. And let the square SQ have been completed. So,
ἑκατέρᾳ τῶν ΑΗ, ΗΕ. ἀλλὰ ἡ ΑΕ ἀσύμμετρος τῇ ΑΒ μήκει· (it is) clear from what has been previously demonstrated
καὶ αἱ ΑΗ, ΗΕ ἄρα ἀσύμμετροί εἰσι τῇ ΑΒ. αἱ ΒΑ, ΑΗ, [Prop. 10.53 lem.] that MR is the mean proportional to
ΗΕ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ὥστε μέσον SN and NQ, and (is) equal to EL, and that MO is the
ἐστὶν ἑκάτερον τῶν ΑΘ, ΗΚ. ὥστε καὶ ἑκάτερον τῶν ΣΝ, square-root of the area AC. So, we must show that MO
ΝΠ μέσον ἐστίν. καὶ αἱ ΜΝ, ΝΞ ἄρα μέσαι εἰσίν. καὶ is a first bimedial (straight-line).
ἐπεὶ σύμμετρος ἡ ΑΗ τῇ ΗΕ μήκει, σύμμετρόν ἐστι καὶ Since AE is incommensurable in length with ED,
τὸ ΑΘ τῷ ΗΚ, τουτέστι τὸ ΣΝ τῷ ΝΠ, τουτέστι τὸ ἀπὸ and ED (is) commensurable (in length) with AB,
τῆς ΜΝ τῷ ἀπὸ τῆς ΝΞ [ὥστε δυνάμει εἰσὶ σύμμετροι αἱ AE (is) thus incommensurable (in length) with AB
ΜΝ, ΝΞ]. καὶ ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΑΕ τῇ ΕΔ μήκει, [Prop. 10.13]. And since AG is commensurable (in
ἀλλ᾿ ἡ μὲν ΑΕ σύμμετρός ἐστι τῇ ΑΗ, ἡ δὲ ΕΔ τῇ ΕΖ length) with EG, AE is also commensurable (in length)
σύμμετρος, ἀσύμμετρος ἄρα ἡ ΑΗ τῇ ΕΖ· ὥστε καὶ τὸ with each of AG and GE [Prop. 10.15]. But, AE is in-
ΑΘ τῷ ΕΛ ἀσύμμετρόν ἐστιν, τουτέστι τὸ ΣΝ τῷ ΜΡ, commensurable in length with AB. Thus, AG and GE
τουτέστιν ὁ ΟΝ τῇ ΝΡ, τουτέστιν ἡ ΜΝ τῇ ΝΞ ἀσύμμετρός are also (both) incommensurable (in length) with AB
ἐστι μήκει. ἐδείχθησαν δὲ αἱ ΜΝ, ΝΞ καὶ μέσαι οὖσαι καὶ [Prop. 10.13]. Thus, BA, AG, and (BA, and) GE are
δυνάμει σύμμετροι· αἱ ΜΝ, ΝΞ ἄρα μέσαι εἰσὶ δυνάμει μόνον (pairs of) rational (straight-lines which are) commensu-
σύμμετροι. λέγω δή, ὅτι καὶ ῥητὸν περιέχουσιν. ἐπεὶ γὰρ ἡ rable in square only. And, hence, each of AH and GK
ΔΕ ὑπόκειται ἑκατέρᾳ τῶν ΑΒ, ΕΖ σύμμετρος, σύμμετρος is a medial (area) [Prop. 10.21]. Hence, each of SN
ἄρα καὶ ἡ ΕΖ τῇ ΕΚ. καὶ ῥητὴ ἑκατέρα αὐτῶν· ῥητὸν ἄρα and NQ is also a medial (area). Thus, MN and NO
τὸ ΕΛ, τουτέστι τὸ ΜΡ· τὸ δὲ ΜΡ ἐστι τὸ ὑπὸ τῶν ΜΝΞ. are medial (straight-lines). And since AG (is) commen-
ἐὰν δὲ δύο μέσαι δυνάμει μόνον σύμμετροι συντεθῶσι ῥητὸν surable in length with GE, AH is also commensurable

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