Page 382 - Euclid's Elements of Geometry

ST EW iþ.

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Β ∆ Γ B ELEMENTS BOOK 10
Α A
Ζ Θ Η F H G
Ε E

Κ K
᾿Εκκείσθω ῥητὴ ἡ Α, καὶ ἐκκείσθωσαν τρεῖς ἀριθμοὶ Let the rational (straight-line) A be laid down. And
οἱ Ε, ΒΓ, ΓΔ λόγον μὴ ἔχοντες πρὸς ἀλλήλους, ὃν let the three numbers, E, BC, and CD, not having to one
τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, ὁ δὲ ΓΒ another the ratio which (some) square number (has) to
πρὸς τὸν ΒΔ λόγον ἐχέτω, ὃν τετράγωνος ἀριθμὸς πρὸς (some) square number, be laid down. And let CB have
τετράγωνον ἀριθμόν, καὶ πεποιήσθω ὡς μὲν ὁ Ε πρὸς τὸν to BD the ratio which (some) square number (has) to
ΒΓ, οὕτως τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς (some) square number. And let it have been contrived
ΖΗ τετράγωνον, ὡς δὲ ὁ ΒΓ πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ that as E (is) to BC, so the square on A (is) to the
τῆς ΖΗ τετράγωνον πρὸς τὸ ἀπὸ τὴς ΗΘ. ἐπεὶ οὖν ἐστιν square on FG, and as BC (is) to CD, so the square on
ὡς ὁ Ε πρὸς τὸν ΒΓ, οὕτως τὸ ἀπὸ τῆς Α τετράγωνον FG (is) to the (square) on GH [Prop. 10.6 corr.]. There-
πρὸς τὸ ἀπὸ τῆς ΖΗ τετράγωνον, σύμμετρον ἄρα ἐστὶ fore, since as E is to BC, so the square on A (is) to the
τὸ ἀπὸ τῆς Α τετράγωνον τῷ ἀπὸ τῆς ΖΗ τετραγώνῳ. square on FG, the square on A is thus commensurable
ῥητὸν δὲ τὸ ἀπὸ τῆς Α τετράγωνον. ῥητὸν ἄρα καὶ τὸ ἀπὸ with the square on FG [Prop. 10.6]. And the square on
τῆς ΖΗ· ῥητὴ ἄρα ἐστὶν ἡ ΖΗ. καὶ ἐπεὶ ὁ Ε πρὸς τὸν ΒΓ A (is) rational. Thus, the (square) on FG (is) also ra-
λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον tional. Thus, FG is a rational (straight-line). And since
ἀριθμόν, οὐδ᾿ ἄρα τὸ ἀπὸ τῆς Α τετράγωνον πρὸς τὸ ἀπὸ E does not have to BC the ratio which (some) square
τῆς ΖΗ [τετράγωνον] λόγον ἔχει, ὅν τετράγωνος ἀριθμὸς number (has) to (some) square number, the square on A
πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ Α τῇ thus does not have to the [square] on FG the ratio which
ΖΗ μήκει. πάλιν, ἐπεί ἐστιν ὡς ὁ ΒΓ πρὸς τὸν ΓΔ, οὕτως τὸ (some) square number (has) to (some) square number
ἀπὸ τῆς ΖΗ τετράγωνον πρὸς τὸ ἀπὸ τῆς ΗΘ, σύμμετρον either. Thus, A is incommensurable in length with FG
ἄρα ἐστὶ τὸ ἀπὸ τῆς ΖΗ τῷ ἀπὸ τῆς ΗΘ. ῥητὸν δὲ τὸ ἀπὸ [Prop. 10.9]. Again, since as BC is to CD, so the square
τῆς ΖΗ· ῥητὸν ἄρα καὶ τὸ ἀπὸ τῆς ΗΘ· ῥητὴ ἄρα ἐστὶν ἡ ΗΘ. on FG is to the (square) on GH, the square on FG is thus
καὶ ἐπεὶ ὁ ΒΓ πρὸς τὸν ΓΔ λόγον οὐκ ἔχει, ὃν τετράγωνος commensurable with the (square) on GH [Prop. 10.6].
ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, οὐδ᾿ ἄρα τὸ ἀπὸ τῆς ΖΗ And the (square) on FG (is) rational. Thus, the (square)
πρὸς τὸ ἀπὸ τῆς ΗΘ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς on GH (is) also rational. Thus, GH is a rational (straight-
πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἐστὶν ἡ ΖΗ τῇ line). And since BC does not have to CD the ratio which
ΗΘ μήκει. καί εἰσιν ἀμφότεραι ῥηταί· αἱ ΖΗ, ΗΘ ἄρα ῥηταί (some) square number (has) to (some) square number,
εἰσι δυνάμει μόνον σύμμετροι· ἀποτομὴ ἄρα ἐστὶν ἡ ΖΘ. the (square) on FG thus does not have to the (square)
λέγω δή, ὅτι καὶ τρίτη. on GH the ratio which (some) square number (has) to
᾿Επεὶ γάρ ἐστιν ὡς μὲν ὁ Ε πρὸς τὸν ΒΓ, οὕτως τὸ ἀπὸ (some) square number either. Thus, FG is incommen-
τῆς Α τετράγωνον πρὸς τὸ ἀπὸ τῆς ΖΗ, ὡς δὲ ὁ ΒΓ πρὸς surable in length with GH [Prop. 10.9]. And both are
τὸν ΓΔ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ τῆς ΘΗ, δι᾿ ἴσου rational (straight-lines). FG and GH are thus rational
ἄρα ἐστὶν ὡς ὁ Ε πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ τῆς Α πρὸς (straight-lines which are) commensurable in square only.
τὸ ἀπὸ τῆς ΘΗ. ὁ δὲ Ε πρὸς τὸν ΓΔ λόγον οὐκ ἔχει, ὃν Thus, FH is an apotome [Prop. 10.73]. So, I say that (it
τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· οὐδ᾿ ἄρα τὸ is) also a third (apotome).
ἀπὸ τῆς Α πρὸς τὸ ἀπὸ τῆς ΗΘ λόγον ἔχει, ὃν τετράγωνος For since as E is to BC, so the square on A (is) to the
ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ἀσύμμετρος ἄρα ἡ Α τῇ (square) on FG, and as BC (is) to CD, so the (square)
ΗΘ μήκει. οὐδετέρα ἄρα τῶν ΖΗ, ΗΘ σύμμετρός ἐστι τῇ on FG (is) to the (square) on HG, thus, via equality, as
ἐκκειμένῃ ῥητῇ τῇ Α μήκει. ᾧ οὖν μεῖζόν ἐστι τὸ ἀπὸ τῆς E is to CD, so the (square) on A (is) to the (square) on
ΖΗ τοῦ ἀπὸ τῆς ΗΘ, ἔστω τὸ ἀπὸ τῆς Κ. ἐπεὶ οὖν ἐστιν HG [Prop. 5.22]. And E does not have to CD the ra-
ὡς ὁ ΒΓ πρὸς τὸν ΓΔ, οὕτως τὸ ἀπὸ τῆς ΖΗ πρὸς τὸ ἀπὸ tio which (some) square number (has) to (some) square
τῆς ΗΘ, ἀναστρέψαντι ἄρα ἐστὶν ὡς ὁ ΒΓ πρὸς τὸν ΒΔ, number. Thus, the (square) on A does not have to the
οὕτως τὸ ἀπὸ τῆς ΖΗ τετράγωνον πρὸς τὸ ἀπὸ τῆς Κ. ὁ δὲ (square) on GH the ratio which (some) square number
ΒΓ πρὸς τὸν ΒΔ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς (has) to (some) square number either. A (is) thus incom-
τετράγωνον ἀριθμόν. καὶ τὸ ἁπὸ τῆς ΖΗ ἄρα πρὸς τὸ ἀπὸ mensurable in length with GH [Prop. 10.9]. Thus, nei-
τῆς Κ λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ther of FG and GH is commensurable in length with the

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