Page 418 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ἀπὸ τῆς ΚΖ. καί ἐστιν ὡς τὸ ἀπὸ τῆς ΘΚ πρὸς τὸ ἀπὸ τῆς following, so all of the leading (magnitudes) are to all of
ΚΖ, οὕτως ἡ ΘΚ πρὸς τὴν ΚΕ, ἐπεὶ αἱ τρεῖς αἱ ΘΚ, ΚΖ, the following [Prop. 5.12]. And as FK (is) to KE, so CD
ΚΕ ἀνάλογόν εἰσιν. σύμμετρος ἄρα ἡ ΘΚ τῇ ΚΕ μήκει. is to DB [Prop. 5.11]. And, thus, as HK (is) to KF, so
ὥστε καὶ ἡ ΘΕ τῇ ΕΚ σύμμετρός ἐστι μήκει. καὶ ἐπεὶ τὸ CD is to DB [Prop. 5.11]. And the (square) on CD (is)
ἀπὸ τῆς Α ἴσον ἐστὶ τῷ ὑπὸ τῶν ΕΘ, ΒΔ, ῥητὸν δέ ἐστι commensurable with the (square) on DB [Prop. 10.36].
τὸ ἀπὸ τῆς Α, ῥητὸν ἄρα ἐστὶ καὶ τὸ ὑπὸ τῶν ΕΘ, ΒΔ. καὶ The (square) on HK is thus also commensurable with
παρὰ ῥητὴν τὴν ΒΔ παράκειται· ῥητὴ ἄρα ἐστὶν ἡ ΕΘ καὶ the (square) on KF [Props. 6.22, 10.11]. And as the
σύμμετρος τῇ ΒΔ μήκει· ὥστε καὶ ἡ σύμμετρος αὐτῇ ἡ ΕΚ (square) on HK is to the (square) on KF, so HK (is) to
ῥητή ἐστι καὶ σύμμετρος τῇ ΒΔ μήκει. ἐπεὶ οὖν ἐστιν ὡς ἡ KE, since the three (straight-lines) HK, KF, and KE
ΓΔ πρὸς ΔΒ, οὕτως ἡ ΖΚ πρὸς ΚΕ, αἱ δὲ ΓΔ, ΔΒ δυνάμει are proportional [Def. 5.9]. HK is thus commensurable
μόνον εἰσὶ σύμμετροι, καὶ αἱ ΖΚ, ΚΕ δυνάμει μόνον εἰσὶ in length with KE [Prop. 10.11]. Hence, HE is also com-
σύμμετροι. ῥητὴ δέ ἐστιν ἡ ΚΕ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΖΚ. αἱ mensurable in length with EK [Prop. 10.15]. And since
ΖΚ, ΚΕ ἄρα ῥηταὶ δυνάμει μόνον εἰσὶ σύμμετροι· ἀποτομὴ the (square) on A is equal to the (rectangle contained) by
ἄρα ἐστὶν ἡ ΕΖ. EH and BD, and the (square) on A is rational, the (rect-
῎Ητοι δὲ ἡ ΓΔ τῆς ΔΒ μεῖζον δύναται τῷ ἀπὸ συμμέτρου angle contained) by EH and BD is thus also rational.
ἑαυτῇ ἢ τῷ ἀπὸ ἀσυμμέτρου. And it is applied to the rational (straight-line) BD. Thus,
Εἰ μὲν οὖν ἡ ΓΔ τῆς ΔΒ μεῖζον δύναται τῷ ἀπὸ EH is rational, and commensurable in length with BD
συμμέτρου [ἑαυτῇ], καὶ ἡ ΖΚ τῆς ΚΕ μεῖζον δυνήσεται τῷ [Prop. 10.20]. And, hence, the (straight-line) commensu-
ἀπὸ συμμέτρου ἑαυτῇ. καὶ εἰ μὲν σύμμετρός ἐστιν ἡ ΓΔ τῇ rable (in length) with it, EK, is also rational [Def. 10.3],
ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΖΚ· εἰ δὲ ἡ ΒΔ, καὶ ἡ ΚΕ· εἰ and commensurable in length with BD [Prop. 10.12].
δὲ οὐδετέρα τῶν ΓΔ, ΔΒ, καὶ οὐδετέρα τῶν ΖΚ, ΚΕ. Therefore, since as CD is to DB, so FK (is) to KE, and
Εἰ δὲ ἡ ΓΔ τῆς ΔΒ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου CD and DB are (straight-lines which are) commensu-
ἑαυτῇ, καὶ ἡ ΖΚ τῆς ΚΕ μεῖζον δυνήσεται τῷ ἀπὸ rable in square only, FK and KE are also commensu-
ἀσυμμέτρου ἑαυτῇ. καὶ εἰ μὲν ἡ ΓΔ σύμμετρός ἐστι τῇ rable in square only [Prop. 10.11]. And KE is rational.
ἐκκειμένῃ ῥητῇ μήκει, καὶ ἡ ΖΚ· εἰ δὲ ἡ ΒΔ, καὶ ἡ ΚΕ· εἰ Thus, FK is also rational. FK and KE are thus rational
δὲ οὐδετέρα τῶν ΓΔ, ΔΒ, καὶ οὐδετέρα τῶν ΖΚ, ΚΕ· ὥστε (straight-lines which are) commensurable in square only.
ἀποτομή ἐστιν ἡ ΖΕ, ἧς τὰ ὀνόματα τὰ ΖΚ, ΚΕ σύμμετρά Thus, EF is an apotome [Prop. 10.73].
ἐστι τοῖς τῆς ἐκ δύο ὀνομάτων ὀνόμασι τοῖς ΓΔ, ΔΒ καὶ ἐν And the square on CD is greater than (the square on)
τῷ αὐτῷ λόγῳ, καὶ τὴν αὐτῆν τάξιν ἔχει τῇ ΒΓ· ὅπερ ἔδει DB either by the (square) on (some straight-line) com-
δεῖξαι. mensurable, or by the (square) on (some straight-line)
incommensurable, (in length) with (CD).
Therefore, if the square on CD is greater than (the
square on) DB by the (square) on (some straight-line)
commensurable (in length) with [CD] then the square
on FK will also be greater than (the square on) KE by
the (square) on (some straight-line) commensurable (in
length) with (FK) [Prop. 10.14]. And if CD is com-
mensurable in length with a (previously) laid down ra-
tional (straight-line), (so) also (is) FK [Props. 10.11,
10.12]. And if BD (is commensurable), (so) also (is)
KE [Prop. 10.12]. And if neither of CD or DB (is com-
mensurable), neither also (are) either of FK or KE.
And if the square on CD is greater than (the square
on) DB by the (square) on (some straight-line) incom-
mensurable (in length) with (CD) then the square on
FK will also be greater than (the square on) KE by
the (square) on (some straight-line) incommensurable
(in length) with (FK) [Prop. 10.14]. And if CD is com-
mensurable in length with a (previously) laid down ra-
tional (straight-line), (so) also (is) FK [Props. 10.11,
10.12]. And if BD (is commensurable), (so) also (is) KE

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