Page 301 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ὀρθογώνιον περιεχέσθω τὸ ΑΓ· λέγω, ὅτι ῥητόν ἐστι τὸ rational straight-lines AB and BC (which are) commen-
ΑΓ. surable in length. I say that AC is rational.
∆ D

Γ C

Α Β A B
᾿Αναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔ· For let the square AD have been described on AB.
kþ
ῥητὸν ἄρα ἐστὶ τὸ ΑΔ. καὶ ἐπεὶ σύμμετρός ἐστιν ἡ ΑΒ τῇ AD is thus rational [Def. 10.4]. And since AB is com-
ΒΓ μήκει, ἴση δέ ἐστιν ἡ ΑΒ τῇ ΒΔ, σύμμετρος ἄρα ἐστὶν mensurable in length with BC, and AB is equal to BD,
ἡ ΒΔ τῇ ΒΓ μήκει. καί ἐστιν ὡς ἡ ΒΔ πρὸς τὴν ΒΓ, οὕτως BD is thus commensurable in length with BC. And as
τὸ ΔΑ πρὸς τὸ ΑΓ. σύμμετρον ἄρα ἐστὶ τὸ ΔΑ τῷ ΑΓ. BD is to BC, so DA (is) to AC [Prop. 6.1]. Thus, DA
ῥητὸν δὲ τὸ ΔΑ· ῥητὸν ἄρα ἐστὶ καὶ τὸ ΑΓ. is commensurable with AC [Prop. 10.11]. And DA (is)
Τὸ ἄρα ὑπὸ ῥητῶν μήκει συμμέτρων, καὶ τὰ ἑξῆς. rational. Thus, AC is also rational [Def. 10.4]. Thus,
the . . . by rational straight-lines . . . commensurable, and
so on . . . .

.
Proposition 20
᾿Εὰν ῥητὸν παρὰ ῥητὴν παραβληθῇ, πλάτος ποιεῖ ῥητὴν If a rational (area) is applied to a rational (straight-
καὶ σύμμετρον τῇ, παρ᾿ ἣν παράκειται, μήκει. line) then it produces as breadth a (straight-line which is)
rational, and commensurable in length with the (straight-
line) to which it is applied.
∆ D

Β Α B A

Γ C
῾Ρητὸν γὰρ τὸ ΑΓ παρὰ ῥητὴν τὴν ΑΒ παραβεβλήσθω For let the rational (area) AC have been applied to the
πλάτος ποιοῦν τὴν ΒΓ· λέγω, ὅτι ῥητή ἐστιν ἡ ΒΓ καὶ rational (straight-line) AB, producing the (straight-line)
σύμμετρος τῇ ΒΑ μήκει. BC as breadth. I say that BC is rational, and commen-
᾿Αναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔ· surable in length with BA.
ῥητὸν ἄρα ἐστὶ τὸ ΑΔ. ῥητὸν δὲ καὶ τὸ ΑΓ· σύμμετρον ἄρα For let the square AD have been described on AB.

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