Page 385 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· οὐδ᾿ ἄρα τὸ For, let the (square) on H be that (area) by which
ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς Θ λόγον ἔχει, ὃν τετράγωνος the (square) on BG is greater than the (square) on GC
ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ [Prop. 10.13 lem.]. Therefore, since as the (square) on
ΒΗ τῇ Θ μήκει. καὶ δύναται ἡ ΒΗ τῆς ΗΓ μεῖζον τῷ ἀπὸ BG (is) to the (square) on GC, so DE (is) to EF, thus,
τῆς Θ· ἡ ΗΒ ἄρα τῆς ΗΓ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου via conversion, as ED is to DF, so the (square) on BG
ἑαυτῇ μήκει. καί ἐστιν ἡ προσαρμόζουσα ἡ ΓΗ σύμμετρος (is) to the (square) on H [Prop. 5.19 corr.]. And ED does
τῇ ἐκκειμένῃ ῥητῇ τῇ Α μήκει· ἡ ἄρα ΒΓ ἀποτομή ἐστι not have to DF the ratio which (some) square number
πέμπτη. (has) to (some) square number. Thus, the (square) on
Εὕρηται ἄρα ἡ πέμπτη ἀποτομὴ ἡ ΒΓ· ὅπερ ἔδει δεῖξαι. BG 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
GB is greater than (the square on) GC by the (square)
þ length with the (previously) laid down rational (straight-
on (some straight-line) incommensurable in length with
(GB). And the attachment CG is commensurable in

†
line) A. Thus, BC is a fifth apotome [Def. 10.15].
Thus, the fifth apotome BC has been found. (Which
is) the very thing it was required to show.

† See footnote to Prop. 10.52.

Proposition 90
.
Εὑρεῖν τὴν ἕκτην ἀποτομήν. To find a sixth apotome.
᾿Εκκείσθω ῥητὴ ἡ Α καὶ τρεῖς ἀριθμοὶ οἱ Ε, ΒΓ, ΓΔ Let the rational (straight-line) A, and the three num-
λόγον μὴ ἔχοντες πρὸς ἀλλήλους, ὃν τετράγωνος ἀριθμὸς bers E, BC, and CD, not having to one another the ra-
πρὸς τετράγωνον ἀριθμόν· ἔτι δὲ καὶ ὁ ΓΒ πρὸς τὸν ΒΔ tio which (some) square number (has) to (some) square
λόγον μὴ ἐχετώ, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον number, be laid down. Furthermore, let CB also not have
ἀριθμόν· καὶ πεποιήσθω ὡς μὲν ὁ Ε πρὸς τὸν ΒΓ, οὕτως to BD the ratio which (some) square number (has) to
τὸ ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς ΖΗ, ὡς δὲ ὁ ΒΓ πρὸς τὸν (some) square number. And let it have been contrived
ΓΔ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ. that as E (is) to BC, so the (square) on A (is) to the
(square) on FG, and as BC (is) to CD, so the (square)
on FG (is) to the (square) on GH [Prop. 10.6 corr.].
Β ∆ Γ B D C
Α A
Ζ Θ Η F H G
Ε E

Κ K
᾿Επεὶ οὖν ἐστιν ὡς ὁ Ε πρὸς τὸν ΒΓ, οὕτως τὸ ἀπὸ τῆς Therefore, since as E is to BC, so the (square) on A
Α πρὸς τὸ ἀπὸ τῆς ΖΗ, σύμμετρον ἄρα τὸ ἀπὸ τῆς Α τῷ (is) to the (square) on FG, the (square) on A (is) thus
ἀπὸ τῆς ΖΗ. ῥητὸν δὲ τὸ ἀπὸ τῆς Α· ῥητὸν ἄρα καὶ τὸ ἀπὸ commensurable with the (square) on FG [Prop. 10.6].
τῆς ΖΗ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΖΗ. καὶ ἐπεὶ ὁ Ε πρὸς τὸν ΒΓ And the (square) on A (is) rational. Thus, the (square)
λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον on FG (is) also rational. Thus, FG is also a rational
ἀριθμόν, οὐδ᾿ ἄρα τὸ ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς ΖΗ λόγον (straight-line). And since E does not have to BC the ra-
ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· tio which (some) square number (has) to (some) square
ἀσύμμετρος ἄρα ἐστὶν ἡ Α τῆ ΖΗ μήκει. πάλιν, ἐπεί ἐστιν ὡς number, the (square) on A thus does not have to the
ὁ ΒΓ πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς (square) on FG the ratio which (some) square number
ΗΘ, σύμμετρον ἄρα τὸ ἀπὸ τῆς ΖΗ τῷ ἀπὸ τῆς ΗΘ. ῥητὸν (has) to (some) square number either. Thus, A is in-

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