Page 384 - Euclid's Elements of Geometry

ST EW iþ.

ἀριθμόν· οὐδ᾿ ἄρα τὸ ἀπὸ τῆς ΗΒ πρὸς τὸ ἀπὸ τῆς Θ λόγον is) also a fourth (apotome).] ELEMENTS BOOK 10
ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· Now, let the (square) on H be that (area) by which
ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ Θ μήκει. καὶ δύναται ἡ the (square) on BG is greater than the (square) on GC
ΒΗ τῆς ΗΓ μεῖζον τῷ ἀπὸ τῆς Θ· ἡ ἄρα ΒΗ τῆς ΗΓ μεῖζον [Prop. 10.13 lem.]. Therefore, since as DE is to EF,
δύναται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ. καὶ ἐστιν ὅλη ἡ ΒΗ so the (square) on BG (is) to the (square) on GC, thus,
σύμμετρος τῇ ἐκκειμένῃ ῥητῇ μήκει τῇ Α. ἡ ἄρα ΒΓ ἀπο- also, via conversion, as ED is to DF, so the (square) on
τομή ἐστι τετάρτη. GB (is) to the (square) on H [Prop. 5.19 corr.]. And ED
Εὕρηται ἄρα ἡ τετάρτη ἀποτομή· ὅπερ ἔδει δεῖξαι. does not have to DF the ratio which (some) square num-
ber (has) to (some) square number. Thus, the (square) on
GB does not have to the (square) on H the ratio which
(some) square number (has) to (some) square number
either. Thus, BG is incommensurable in length with H
[Prop. 10.9]. And the square on BG is greater than (the
square on) GC by the (square) on H. Thus, the square on
BG is greater than (the square) on GC by the (square) on
pjþ with the the (previously) laid down rational (straight-
(some straight-line) incommensurable (in length) with
(BG). And the whole, BG, is commensurable in length
†
line) A. Thus, BC is a fourth apotome [Def. 10.14].
Thus, a fourth apotome has been found. (Which is)
the very thing it was required to show.
† See footnote to Prop. 10.51.

Proposition 89
.
Εὐρεῖν τὴν πέμπτην ἀποτομήν. To find a fifth apotome.
Β Γ Η B C G
Α A

∆ Ζ Ε D F E
Θ H
᾿Εκκείσθω ῥητὴ ἡ Α, καὶ τῇ Α μήκει σύμμετρος ἔστω ἡ Let the rational (straight-line) A be laid down, and let
ΓΗ· ῥητὴ ἄρα [ἐστὶν] ἡ ΓΗ. καὶ ἐκκείσθωσαν δύο ἀριθμοὶ οἱ CG be commensurable in length with A. Thus, CG [is]
ΔΖ, ΖΕ, ὥστε τὸν ΔΕ πρὸς ἑκάτερον τῶν ΔΖ, ΖΕ λόγον a rational (straight-line). And let the two numbers DF
πάλιν μὴ ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον and FE be laid down such that DE again does not have
ἀριθμόν· καὶ πεποιήσθω ὡς ὁ ΖΕ πρὸς τὸν ΕΔ, οὕτως τὸ to each of DF and FE the ratio which (some) square
ἀπὸ τῆς ΓΗ πρὸς τὸ ἀπὸ τῆς ΗΒ. ῥητὸν ἄρα καὶ τὸ ἀπὸ τῆς number (has) to (some) square number. And let it have
ΗΒ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΒΗ. καὶ ἐπεί ἐστιν ὡς ὁ ΔΕ πρὸς been contrived that as FE (is) to ED, so the (square) on
τὸν ΕΖ, οὕτως τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ, ὁ δὲ CG (is) to the (square) on GB. Thus, the (square) on GB
ΔΕ πρὸς τὸν ΕΖ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς (is) also rational [Prop. 10.6]. Thus, BG is also rational.
πρὸς τετράγωνον ἀριθμόν, οὐδ᾿ ἄρα τὸ ἀπὸ τῆς ΒΗ πρὸς And since as DE is to EF, so the (square) on BG (is) to
τὸ ἀπὸ τῆς ΗΓ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς the (square) on GC. And DE does not have to EF the ra-
τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ ΗΓ tio which (some) square number (has) to (some) square
μήκει. καί εἰσιν ἀμφότεραι ῥηταί· αἱ ΒΗ, ΗΓ ἄρα ῥηταί εἰσι number. The (square) on BG thus does not have to the
δυνάμει μόνον σύμμετροι· ἡ ΒΓ ἄρα ἀποτομή ἐστιν. λέγω (square) on GC the ratio which (some) square number
δή, ὅτι καὶ πέμπτη. (has) to (some) square number either. Thus, BG is in-
῟Ωι γὰρ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΒΗ τοῦ ἀπὸ τῆς ΗΓ, commensurable in length with GC [Prop. 10.9]. And
ἔστω τὸ ἀπὸ τῆς Θ. ἐπεὶ οὖν ἐστιν ὡς τὸ ἀπὸ τὴς ΒΗ πρὸς they are both rational (straight-lines). BG and GC are
τὸ ἀπὸ τῆς ΗΓ, οὕτως ὁ ΔΕ πρὸς τὸν ΕΖ, ἀναστρέψαντι thus rational (straight-lines which are) commensurable
ἄρα ἐστὶν ὡς ὁ ΕΔ πρὸς τὸν ΔΖ, οὕτως τὸ ἀπὸ τῆς ΒΗ πρὸς in square only. Thus, BC is an apotome [Prop. 10.73].
τὸ ἀπὸ τῆς Θ, ὁ δὲ ΕΔ πρὸς τὸν ΔΖ λόγον οὐκ ἔχει, ὃν So, I say that (it is) also a fifth (apotome).

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