Page 358 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

σύμμετρος ἔστω ἡ ΓΔ· λέγω, ὅτι ἡ ΓΔ ἐκ δύο ὀνομάτων Let AB be a binomial (straight-line), and let CD be
ἐστὶ καὶ τῇ τάξει ἡ αὐτὴ τῇ ΑΒ. commensurable in length with AB. I say that CD is a bi-
nomial (straight-line), and (is) the same in order as AB.
Α Ε Β A E B

Γ Ζ ∆ C F D

᾿Επεὶ γὰρ ἐκ δύο ὀνομάτων ἐστὶν ἡ ΑΒ, διῃρήσθω εἰς τὰ For since AB is a binomial (straight-line), let it have
ὀνόματα κατὰ τὸ Ε, καὶ ἔστω μεῖζον ὄνομα τὸ ΑΕ· αἱ ΑΕ, been divided into its (component) terms at E, and let
ΕΒ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι. γεγονέτω ὡς AE be the greater term. AE and EB are thus rational
ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΑΕ πρὸς τὴν ΓΖ· καὶ λοιπὴ (straight-lines which are) commensurable in square only
ἄρα ἡ ΕΒ πρὸς λοιπὴν τὴν ΖΔ ἐστιν, ὡς ἡ ΑΒ πρὸς τὴν [Prop. 10.36]. Let it have been contrived that as AB (is)
ΓΔ. σύμμετρος δὲ ἡ ΑΒ τῇ ΓΔ μήκει· σύμμετρος ἄρα ἐστὶ to CD, so AE (is) to CF [Prop. 6.12]. Thus, the remain-
καὶ ἡ μὲν ΑΕ τῇ ΓΖ, ἡ δὲ ΕΒ τῇ ΖΔ. καί εἰσι ῥηταὶ αἱ ΑΕ, der EB is also to the remainder FD, as AB (is) to CD
ΕΒ· ῥηταὶ ἄρα εἰσὶ καὶ αἱ ΓΖ, ΖΔ. καὶ ἐστιν ὡς ἡ ΑΕ πρὸς [Props. 6.16, 5.19 corr.]. And AB (is) commensurable
ΓΖ, ἡ ΕΒ πρὸς ΖΔ. ἐναλλὰξ ἄρα ἐστὶν ὡς ἡ ΑΕ πρὸς ΕΒ, ἡ in length with CD. Thus, AE is also commensurable
ΓΖ πρὸς ΖΔ. αἱ δὲ ΑΕ, ΕΒ δυνάμει μόνον [εἰσὶ] σύμμετροι· (in length) with CF, and EB with FD [Prop. 10.11].
καὶ αἱ ΓΖ, ΖΔ ἄρα δυνάμει μόνον εἰσὶ σύμμετροι. καί εἰσι And AE and EB are rational. Thus, CF and FD are
ῥηταί· ἐκ δύο ἄρα ὀνομάτων ἐστὶν ἡ ΓΔ. λέγω δή, ὅτι τῇ also rational. And as AE is to CF, (so) EB (is) to FD
τάξει ἐστὶν ἡ αὐτὴ τῇ ΑΒ. [Prop. 5.11]. Thus, alternately, as AE is to EB, (so)
῾Η γὰρ ΑΕ τῆς ΕΒ μεῖζον δύναται ἤτοι τῷ ἀπὸ CF (is) to FD [Prop. 5.16]. And AE and EB [are]
συμμέτρου ἑαυτῇ ἢ τῷ ἀπὸ ἀσυμμέτρου. εἰ μὲν οὖν ἡ commensurable in square only. Thus, CF and FD are
ΑΕ τῆς ΕΒ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ also commensurable in square only [Prop. 10.11]. And
ἡ ΓΖ τῆς ΖΔ μεῖζον δυνήσεται τῷ ἀπὸ συμμέτρου ἑαυτῇ. they are rational. CD is thus a binomial (straight-line)
καὶ εἰ μὲν σύμμετρός ἐστιν ἡ ΑΕ τῇ ἐκκειμένῃ ῥητῇ, καὶ [Prop. 10.36]. So, I say that it is the same in order as
ἡ ΓΖ σύμμετρος αὐτῇ ἔσται, καὶ διὰ τοῦτο ἑκατέρα τῶν AB.
ΑΒ, ΓΔ ἐκ δύο ὀνομάτων ἐστὶ πρώτη, τουτέστι τῇ τάξει For the square on AE is greater than (the square on)
ἡ αὐτή. εἰ δὲ ἡ ΕΒ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ, EB by the (square) on (some straight-line) either com-
καὶ ἡ ΖΔ σύμμετρός ἐστιν αὐτῇ, καὶ διὰ τοῦτο πάλιν τῇ mensurable or incommensurable (in length) with (AE).
τάξει ἡ αὐτὴ ἔσται τῇ ΑΒ· ἑκατέρα γὰρ αὐτῶν ἔσται ἐκ δύο Therefore, if the square on AE is greater than (the square
ὀνομάτων δευτέρα. εἰ δὲ οὐδετέρα τῶν ΑΕ, ΕΒ σύμμετρός on) EB by the (square) on (some straight-line) com-
ἐστι τῇ ἐκκειμένῃ ῥητῇ, οὐδετέρα τῶν ΓΖ, ΖΔ σύμμετρος mensurable (in length) with (AE) then the square on
αὐτῇ ἔσται, καί ἐστιν ἑκατέρα τρίτη. εἰ δὲ ἡ ΑΕ τῆς ΕΒ CF will also be greater than (the square on) FD by
μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ, καὶ ἡ ΓΖ τὴς ΖΔ the (square) on (some straight-line) commensurable (in
μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ. καὶ εἰ μὲν ἡ ΑΕ length) with (CF) [Prop. 10.14]. And if AE is com-
σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῃ, καὶ ἡ ΓΖ σύμμετρός mensurable (in length) with (some previously) laid down
ἐστιν αὐτῇ, καὶ ἐστιν ἑκατέρα τετάρτη. εἰ δὲ ἡ ΕΒ, καὶ rational (straight-line) then CF will also be commensu-
ἡ ΖΔ, καὶ ἔσται ἑκατέρα πέμπτη. εἰ δὲ οὐδετέρα τῶν ΑΕ, rable (in length) with it [Prop. 10.12]. And, on account
ΕΒ, καὶ τῶν ΓΖ, ΖΔ οὐδετέρα σύμμετρός ἐστι τῇ ἐκκειμένῃ of this, AB and CD are each first binomial (straight-
ῥητῇ, καὶ ἔσται ἑκατέρα ἕκτη. lines) [Def. 10.5]—that is to say, the same in order. And if
῞Ωστε ἡ τῇ ἐκ δύο ὀνομάτων μήκει σύμμετρος ἐκ δύο EB is commensurable (in length) with the (previously)
ὀνομάτων ἐστὶ καὶ τῇ τάξει ἡ αὐτή· ὅπερ ἔδει δεῖξαι. laid down rational (straight-line) then FD is also com-
mensurable (in length) with it [Prop. 10.12], and, again,
on account of this, (CD) will be the same in order as
AB. For each of them will be second binomial (straight-
lines) [Def. 10.6]. And if neither of AE and EB is com-
mensurable (in length) with the (previously) laid down
rational (straight-line) then neither of CF and FD will
be commensurable (in length) with it [Prop. 10.13], and
each (of AB and CD) is a third (binomial straight-line)

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