Page 303 - Euclid's Elements of Geometry

ST EW iþ. ¨mma ELEMENTS BOOK 10

Lemma
.
᾿Εὰν ὦσι δύο εὐθεῖαι, ἔστιν ὡς ἡ πρώτη πρὸς τὴν If there are two straight-lines then as the first is to the
δευτέραν, οὕτως τὸ ἀπὸ τῆς πρώτης πρὸς τὸ ὑπὸ τῶν δύο second, so the (square) on the first (is) to the (rectangle
εὐθειῶν. contained) by the two straight-lines.
Ζ Ε Η F E G

∆ D
῎Εστωσαν δύο εὐθεῖαι αἱ ΖΕ, ΕΗ. λέγω, ὅτι ἐστὶν ὡς ἡ Let FE and EG be two straight-lines. I say that as
ΖΕ πρὸς τὴν ΕΗ, οὕτως τὸ ἀπὸ τῆς ΖΕ πρὸς τὸ ὑπὸ τῶν FE is to EG, so the (square) on FE (is) to the (rectangle
ΖΕ, ΕΗ. contained) by FE and EG.
᾿Αναγεγράφθω γὰρ ἀπὸ τῆς ΖΕ τετράγωνον τὸ ΔΖ, καὶ For let the square DF have been described on FE.
συμπεπληρώσθω τὸ ΗΔ. ἐπεὶ οὖν ἐστιν ὡς ἡ ΖΕ πρὸς τὴν And let GD have been completed. Therefore, since as
ΕΗ, οὕτως τὸ ΖΔ πρὸς τὸ ΔΗ, καί ἐστι τὸ μὲν ΖΔ τὸ ἀπὸ FE is to EG, so FD (is) to DG [Prop. 6.1], and FD is
kbþ and EF is to the (square on) EF—that is to say, as GD
τῆς ΖΕ, τὸ δὲ ΔΗ τὸ ὑπὸ τῶν ΔΕ, ΕΗ, τουτέστι τὸ ὑπὸ the (square) on FE, and DG the (rectangle contained)
τῶν ΖΕ, ΕΗ, ἔστιν ἄρα ὡς ἡ ΖΕ πρὸς τὴν ΕΗ, οὕτως τὸ by DE and EG—that is to say, the (rectangle contained)
ἀπὸ τῆς ΖΕ πρὸς τὸ ὑπὸ τῶν ΖΕ, ΕΗ. ὁμοίως δὲ καὶ ὡς τὸ by FE and EG—thus as FE is to EG, so the (square)
ὑπὸ τῶν ΗΕ, ΕΖ πρὸς τὸ ἀπὸ τῆς ΕΖ, τουτέστιν ὡς τὸ ΗΔ on FE (is) to the (rectangle contained) by FE and EG.
πρὸς τὸ ΖΔ, οὕτως ἡ ΗΕ πρὸς τὴν ΕΖ· ὅπερ ἔδει δεῖξαι. And also, similarly, as the (rectangle contained) by GE

(is) to FD—so GE (is) to EF. (Which is) the very thing
it was required to show.
Proposition 22
.
Τὸ ἀπὸ μέσης παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ The square on a medial (straight-line), being ap-
ῥητὴν καὶ ἀσύμμετρον τῇ, παρ᾿ ἣν παράκειται, μήκει. plied to a rational (straight-line), produces as breadth a
(straight-line which is) rational, and incommensurable in
length with the (straight-line) to which it is applied.
Β B

Η G

Α Γ ∆ Ε Ζ A C D E F
῎Εστω μέση μὲν ἡ Α, ῥητὴ δὲ ἡ ΓΒ, καὶ τῷ ἀπὸ τῆς Let A be a medial (straight-line), and CB a rational
Α ἴσον παρὰ τὴν ΒΓ παραβεβλήσθω χωρίον ὀρθογώνιον τὸ (straight-line), and let the rectangular area BD, equal to
ΒΔ πλάτος ποιοῦν τὴν ΓΔ· λέγω, ὅτι ῥητή ἐστιν ἡ ΓΔ καὶ the (square) on A, have been applied to BC, producing
ἀσύμμετρος τῇ ΓΒ μήκει. CD as breadth. I say that CD is rational, and incommen-
᾿Επεὶ γὰρ μέση ἐστὶν ἡ Α, δύναται χωρίον περιεχόμενον surable in length with CB.
ὑπὸ ῥητῶν δυνάμει μόνον συμμέτρων. δυνάσθω τὸ ΗΖ. For since A is medial, the square on it is equal to a

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