Page 386 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

δὲ τὸ ἀπὸ τῆς ΖΗ· ῥητὸν ἄρα καὶ τὸ ἀπὸ τῆς ΗΘ· ῥητὴ ἄρα commensurable in length with FG [Prop. 10.9]. Again,
καὶ ἡ ΗΘ. καὶ ἐπεὶ ὁ ΒΓ πρὸς τὸν ΓΔ λόγον οὐκ ἔχει, ὃν since as BC is to CD, so the (square) on FG (is) to the
τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδ᾿ ἄρα τὸ (square) on GH, the (square) on FG (is) thus commen-
ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ λόγον ἔχει, ὃν τετράγωνος surable with the (square) on GH [Prop. 10.6]. And the
ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν (square) on FG (is) rational. Thus, the (square) on GH
ἡ ΖΗ τῇ ΗΘ μήκει. καί εἰσιν ἀμφότεραι ῥηταί· αἱ ΖΗ, ΗΘ (is) also rational. Thus, GH (is) also rational. And since
ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἡ ἄρα ΖΘ ἀποτομή BC does not have to CD the ratio which (some) square
ἐστιν. λέγω δή, ὅτι καὶ ἕκτη. number (has) to (some) square number, the (square) on
᾿Επεὶ γάρ ἐστιν ὡς μὲν ὁ Ε πρὸς τὸν ΒΓ, οὕτως τὸ ἀπὸ FG thus does not have to the (square) on GH the ra-
τῆς Α πρὸς τὸ ἀπὸ τῆς ΖΗ, ὡς δὲ ὁ ΒΓ πρὸς τὸν ΓΔ, οὕτως tio which (some) square (number) has to (some) square
τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ, δι᾿ ἴσου ἄρα ἐστὶν ὡς ὁ (number) either. Thus, FG is incommensurable in length
Ε πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς ΗΘ. ὁ with GH [Prop. 10.9]. And both are rational (straight-
δὲ Ε πρὸς τὸν ΓΔ λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς lines). Thus, FG and GH are rational (straight-lines
πρὸς τετράγωνον ἀριθμόν· οὐδ᾿ ἄρα τὸ ἀπὸ τῆς Α πρὸς which are) commensurable in square only. Thus, FH is
τὸ ἀπὸ τῆς ΗΘ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς an apotome [Prop. 10.73]. So, I say that (it is) also a
τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ Α τῇ ΗΘ sixth (apotome).
μήκει· οὐδετέρα ἄρα τῶν ΖΗ, ΗΘ σύμμετρός ἐστι τῇ Α ῥητῇ For since as E is to BC, so the (square) on A (is)
μήκει. ᾧ οὖν μεῖζόν ἐστι τὸ ἀπὸ τῆς ΖΗ τοῦ ἀπὸ τῆς ΗΘ, to the (square) on FG, and as BC (is) to CD, so the
ἔστω τὸ ἀπὸ τῆς Κ. ἐπεὶ οὖν ἐστιν ὡς ὁ ΒΓ πρὸς τὸν ΓΔ, (square) on FG (is) to the (square) on GH, thus, via
οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΗΘ, ἀναστρέψαντι equality, as E is to CD, so the (square) on A (is) to
ἄρα ἐστὶν ὡς ὁ ΓΒ πρὸς τὸν ΒΔ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς the (square) on GH [Prop. 5.22]. And E does not have
τὸ ἀπὸ τῆς Κ. ὁ δὲ ΓΒ πρὸς τὸν ΒΔ λόγον οὐκ ἔχει, ὃν to CD the ratio which (some) square number (has) to
τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· οὐδ᾿ ἄρα τὸ (some) square number. Thus, the (square) on A does not
ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς Κ λόγον ἔχει, ὃν τετράγωνος have to the (square) GH the ratio which (some) square
ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ number (has) to (some) square number either. A is thus
ΖΗ τῇ Κ μήκει. καὶ δύναται ἡ ΖΗ τῆς ΗΘ μεῖζον τῷ ἀπὸ incommensurable in length with GH [Prop. 10.9]. Thus,
τῆς Κ· ἡ ΖΗ ἄρα τῆς ΗΘ μεῖζον δύναται τῷ ἀπὸ ἀσυμμέτρου neither of FG and GH is commensurable in length with
ἑαυτῇ μήκει. καὶ οὐδετέρα τῶν ΖΗ, ΗΘ σύμμετρός ἐστι τῇ the rational (straight-line) A. Therefore, let the (square)
ἐκκειμένῃ ῥητῇ μήκει τῇ Α. ἡ ἄρα ΖΘ ἀποτομή ἐστιν ἕκτη. on K be that (area) by which the (square) on FG is
Εὕρηται ἄρα ἡ ἕκτη ἀποτομὴ ἡ ΖΘ· ὅπερ ἔδει δεῖξαι. greater than the (square) on GH [Prop. 10.13 lem.].
Therefore, since as BC is to CD, so the (square) on FG
(is) to the (square) on GH, thus, via conversion, as CB is
to BD, so the (square) on FG (is) to the (square) on K
[Prop. 5.19 corr.]. And CB does not have to BD the ra-
tio which (some) square number (has) to (some) square
number. Thus, the (square) on FG does not have to the
(square) on K the ratio which (some) square number
(has) to (some) square number either. FG is thus in-
commensurable in length with K [Prop. 10.9]. And the
square on FG is greater than (the square on) GH by the
(square) on K. Thus, the square on FG is greater than
(the square on) GH by the (square) on (some straight-
line) incommensurable in length with (FG). And neither
of FG and GH is commensurable in length with the (pre-
viously) laid down rational (straight-line) A. Thus, FH
is a sixth apotome [Def. 10.16].
Thus, the sixth apotome FH has been found. (Which
is) the very thing it was required to show.
† See footnote to Prop. 10.53.

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