Page 380 - Euclid's Elements of Geometry

ST EW iþ.

Β Γ Η B ELEMENTS BOOK 10
G
C
Α A
Ε Ζ ∆ E F D
Θ H
᾿Εκκείσθω ῥητὴ ἡ Α, καὶ τῇ Α μήκει σύμμετρος ἔστω Let the rational (straight-line) A be laid down. And
ἡ ΒΗ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΒΗ. καὶ ἐκκείσθωσαν δύο let BG be commensurable in length with A. BG is thus
τετράγωνοι ἀριθμοὶ οἱ ΔΕ, ΕΖ, ὧν ἡ ὑπεροχὴ ὁ ΖΔ μὴ also a rational (straight-line). And let two square num-
ἔστω τετράγωνος· οὐδ᾿ ἄρα ὁ ΕΔ πρὸς τὸν ΔΖ λόγον ἔχει, bers DE and EF be laid down, and let their difference
ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν. καὶ πε- FD be not square [Prop. 10.28 lem. I]. Thus, ED does
ποιήσθω ὡς ὁ ΕΔ πρὸς τὸν ΔΖ, οὕτως τὸ ἀπὸ τῆς ΒΗ not have to DF the ratio which (some) square number
τετράγωνον πρὸς τὸ ἀπὸ τὴς ΗΓ τετράγωνον· σύμμετρον (has) to (some) square number. And let it have been
ἄρα ἐστὶ τὸ ἀπὸ τῆς ΒΗ τῷ ἀπὸ τῆς ΗΓ. ῥητὸν δὲ τὸ ἀπὸ contrived that as ED (is) to DF, so the square on BG
τῆς ΒΗ· ῥητὸν ἄρα καὶ τὸ ἀπὸ τῆς ΗΓ· ῥητὴ ἄρα ἐστὶ καὶ (is) to the square on GC [Prop. 10.6. corr.]. Thus, the
ἡ ΗΓ. καὶ ἐπεὶ ὁ ΕΔ πρὸς τὸν ΔΖ λόγον οὐκ ἔχει, ὃν (square) on BG is commensurable with the (square) on
τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδ᾿ ἄρα τὸ GC [Prop. 10.6]. And the (square) on BG (is) ratio-
ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ λόγον ἔχει, ὃν τετράγωνος nal. Thus, the (square) on GC (is) also rational. Thus,
ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν GC is also rational. And since ED does not have to DF
ἡ ΒΗ τῇ ΗΓ μήκει. καί εἰσιν ἀμφότεραι ῥηταί· αἱ ΒΗ, ΗΓ the ratio which (some) square number (has) to (some)
ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἡ ἄρα ΒΓ ἀποτομή square number, the (square) on BG thus does not have to
ἐστιν. λέγω δή, ὅτι καὶ πρώτη. the (square) on GC the ratio which (some) square num-
῟Ωι γὰρ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΒΗ τοῦ ἀπὸ τῆς ΗΓ, ber (has) to (some) square number either. Thus, BG is
ἔστω τὸ ἀπὸ τῆς Θ. καὶ ἐπεί ἐστιν ὡς ὁ ΕΔ πρὸς τὸν incommensurable in length with GC [Prop. 10.9]. And
ΖΔ, οὕτως τὸ ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ τῆς ΗΓ, καὶ ἀνα- they are both rational (straight-lines). Thus, BG and GC
στρέψαντι ἄρα ἐστὶν ὡς ὁ ΔΕ πρὸς τὸν ΕΖ, οὕτως τὸ ἀπὸ are rational (straight-lines which are) commensurable in
τῆς ΗΒ πρὸς τὸ ἀπὸ τῆς Θ. ὁ δὲ ΔΕ πρὸς τὸν ΕΖ λόγον square only. Thus, BC is an apotome [Prop. 10.73]. So,
ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· I say that (it is) also a first (apotome).
ἑκάτερος γὰρ τετράγωνός ἐστιν· καὶ τὸ ἀπὸ τῆς ΗΒ ἄρα Let the (square) on H be that (area) by which the
πρὸς τὸ ἀπὸ τῆς Θ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς (square) on BG is greater than the (square) on GC
πρὸς τετράγωνον ἀριθμόν· σύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ [Prop. 10.13 lem.]. And since as ED is to FD, so the
Θ μήκει. καὶ δύναται ἡ ΒΗ τῆς ΗΓ μεῖζον τῷ ἀπὸ τῆς Θ· (square) on BG (is) to the (square) on GC, thus, via con-
ἡ ΒΗ ἄρα τῆς ΗΓ μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ version, as DE is to EF, so the (square) on GB (is) to
μήκει. καί ἐστιν ἡ ὅλη ἡ ΒΗ σύμμετρος τῇ ἐκκειμένῃ ῥητῇ the (square) on H [Prop. 5.19 corr.]. And DE has to EF
μήκει τῇ Α. ἡ ΒΓ ἄρα ἀποτομή ἐστι πρώτη. the ratio which (some) square-number (has) to (some)
Εὕρηται ἄρα ἡ πρώτη ἀποτομὴ ἡ ΒΓ· ὅπερ ἔδει εὑρεῖν. square-number. For each is a square (number). Thus, the
(square) on GB also has to the (square) on H the ra-
tio which (some) square number (has) to (some) square
number. Thus, BG is commensurable 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)
pþ with the (previously) laid down rational (straight-line) A.
on (some straight-line) commensurable in length with
(BG). And the whole, BG, is commensurable in length

Thus, BC is a first apotome [Def. 10.11].
Thus, the first apotome BC has been found. (Which
is) the very thing it was required to find.

† See footnote to Prop. 10.48.
Proposition 86
.
Εὑρεῖν τὴν δευτέραν ἀποτομήν. To find a second apotome.

380