Page 383 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ἀριθμόν. σύμμετρός ἄρα ἐστὶν ἡ ΖΗ τῇ Κ μήκει, καὶ δύναται (previously) laid down rational (straight-line) A. There-
ἡ ΖΗ τῆς ΗΘ μεῖζον τῷ ἀπὸ συμμέτρου ἑαυτῇ. καὶ οὐδετέρα fore, let the (square) on K be that (area) by which the
τῶν ΖΗ, ΗΘ σύμμετρός ἐστι τῇ ἐκκειμένῃ ῥητῇ τῇ Α μήκει· (square) on FG is greater than the (square) on GH
ἡ ΖΘ ἄρα ἀποτομή ἐστι τρίτη. [Prop. 10.13 lem.]. Therefore, since as BC is to CD, so
Εὕρηται ἄρα ἡ τρίτη ἀποτομὴ ἡ ΖΘ· ὅπερ ἔδει δεῖξαι. the (square) on FG (is) to the (square) on GH, thus, via
conversion, as BC is to BD, so the square on FG (is) to
the square on K [Prop. 5.19 corr.]. And BC has to BD
the ratio which (some) square number (has) to (some)
square number. Thus, the (square) on FG also has to
the (square) on K the ratio which (some) square number
(has) to (some) square number. FG is thus commen-
surable in length with K [Prop. 10.9]. And the square
on FG is (thus) greater than (the square on) GH by
the (square) on (some straight-line) commensurable (in
phþ tional (straight-line) A. Thus, FH is a third apotome
length) with (FG). And neither of FG and GH is com-
mensurable in length with the (previously) laid down ra-

[Def. 10.13].
Thus, the third apotome FH has been found. (Which
is) very thing it was required to show.

† See footnote to Prop. 10.50.

.
Proposition 88
Εὑρεῖν τὴν τετάρτην ἀποτομήν. To find a fourth apotome.
Β Γ Η B C G
Α A

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

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