Page 381 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

᾿Εκκείσθω ῥητὴ ἡ Α καὶ τῇ Α σύμμετρος μήκει ἡ ΗΓ. Let the rational (straight-line) A, and GC (which is)
ῥητὴ ἄρα ἐστὶν ἡ ΗΓ. καὶ ἐκκείσθωσαν δύο τετράγωνοι commensurable in length with A, be laid down. Thus,
ἀριθμοὶ οἱ ΔΕ, ΕΖ, ὧν ἡ ὑπεροχὴ ὁ ΔΖ μὴ ἔστω τετράγωνος. GC is a rational (straight-line). And let the two square
καὶ πεποιήσθω ὡς ὁ ΖΔ πρὸς τὸν ΔΕ, οὕτως τὸ ἀπὸ numbers DE and EF be laid down, and let their differ-
τῆς ΓΗ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΗΒ τετράγωνον. ence DF be not square [Prop. 10.28 lem. I]. And let it
σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΓΗ τετράγωνον τῷ ἀπὸ have been contrived that as FD (is) to DE, so the square
τῆς ΗΒ τετραγώνῳ. ῥητὸν δὲ τὸ ἀπὸ τῆς ΓΗ. ῥητὸν ἄρα on CG (is) to the square on GB [Prop. 10.6 corr.]. Thus,
[ἐστὶ] καὶ τὸ ἀπὸ τῆς ΗΒ· ῥητὴ ἄρα ἐστὶν ἡ ΒΗ. καὶ ἐπεὶ τὸ the square on CG is commensurable with the square on
ἀπὸ τῆς ΗΓ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΗΒ λόγον οὐκ GB [Prop. 10.6]. And the (square) on CG (is) rational.
ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, Thus, the (square) on GB [is] also rational. Thus, BG is a
ἀσύμμετρός ἐστιν ἡ ΓΗ τῇ ΗΒ μήκει. καί εἰσιν ἀμφότεραι rational (straight-line). And since the square on GC does
ῥηταί· αἱ ΓΗ, ΗΒ ἄρα ρηταί εἰσι δυνάμει μόνον σύμμετροι· not have to the (square) on GB the ratio which (some)
ἡ ΒΓ ἄρα ἀποτομή ἐστιν. λέγω δή, ὅτι καὶ δευτέρα. square number (has) to (some) square number, CG is
incommensurable in length with GB [Prop. 10.9]. And
they are both rational (straight-lines). Thus, CG and GB
are rational (straight-lines which are) commensurable in
square only. Thus, BC is an apotome [Prop. 10.73]. So,
I say that it is also a second (apotome).
Β Γ Η B C G
Α A
Ε Ζ ∆ E F D
Θ H
῟Ωι γὰρ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΒΗ τοῦ ἀπὸ τῆς ΗΓ, For let the (square) on H be that (area) by which
ἔστω τὸ ἀπὸ τῆς Θ. ἐπεὶ οὖν ἐστιν ὡς τὸ ἀπὸ τῆς ΒΗ the (square) on BG is greater than the (square) on GC
πρὸς τὸ ἀπὸ τῆς ΗΓ, οὕτως ὁ ΕΔ ἀριθμὸς πρὸς τὸν ΔΖ [Prop. 10.13 lem.]. Therefore, since as the (square) on
ἀριθμόν, ἀναστρέψαντι ἄρα ἐστὶν ὡς τὸ ἀπὸ τῆς ΒΗ πρὸς BG is to the (square) on GC, so the number ED (is)
τὸ ἀπὸ τῆς Θ, οὕτως ὁ ΔΕ πρὸς τὸν ΕΖ. καί ἐστιν ἑκάτερος to the number DF, thus, also, via conversion, as the
τῶν ΔΕ, ΕΖ τετράγωνος· τὸ ἄρα ἀπὸ τῆς ΒΗ πρὸς τὸ ἀπὸ (square) on BG is to the (square) on H, so DE (is)
τῆς Θ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον to EF [Prop. 5.19 corr.]. And DE and EF are each
ἀριθμόν· σύμμετρος ἄρα ἐστὶν ἡ ΒΗ τῇ Θ μήκει. καὶ δύναται square (numbers). Thus, the (square) on BG has to the
ἡ ΒΗ τῆς ΗΓ μεῖζον τῷ ἀπὸ τῆς Θ· ἡ ΒΗ ἄρα τῆς ΗΓ (square) on H the ratio which (some) square number
μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει. καί ἐστιν ἡ (has) to (some) square number. Thus, BG is commen-
προσαρμόζουσα ἡ ΓΗ τῇ ἐκκειμένῃ ῥητῇ σύμμετρος τῇ Α. surable in length with H [Prop. 10.9]. And the square on
ἡ ΒΓ ἄρα ἀποτομή ἐστι δευτέτα. BG is greater than (the square on) GC by the (square)
Εὕρηται ἄρα δευτέρα ἀποτομὴ ἡ ΒΓ· ὅπερ ἔδει δεῖξαι. on H. Thus, the square on BG is greater than (the square
on) GC by the (square) on (some straight-line) commen-
pzþ down rational (straight-line) A. Thus, BC is a second
surable in length with (BG). And the attachment CG
is commensurable (in length) with the (prevously) laid

†
apotome [Def. 10.12].
Thus, the second apotome BC has been found.
(Which is) the very thing it was required to show.

† See footnote to Prop. 10.49.

Proposition 87
.
Εὑρεῖν τὴν τρίτην ἀποτομήν. To find a third apotome.

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