Page 325 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ὑπὸ τῶν ΑΒ, ΒΓ· ὥστε καὶ τὸ ἁπὸ τῆς ΑΓ ἀσύμμετρόν ἐστι BC (is) rational, the sum of the (squares) on AB and
τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ. ῥητὸν δὲ τὸ δὶς ὑπὸ τῶν ΑΒ, ΒΓ· BC is thus incommensurable with twice the (rectangle
ἄλογον ἄρα τὸ ἀπὸ τῆς ΑΓ. ἄλογος ἄρα ἡ ΑΓ, καλείσθω δὲ contained) by AB and BC. Hence, the (square) on AC
ῥητὸν καὶ μέσον δυναμένη. ὅπερ ἔδει δεῖξαι. is also incommensurable with twice the (rectangle con-
tained) by AB and BC [Prop. 10.16]. And twice the
(rectangle contained) by AB and BC (is) rational. The
(square) on AC (is) thus irrational. Thus, AC (is) irra-
maþ rational plus a medial (area). (Which is) the very thing
tional [Def. 10.4]—let it be called the square-root of a
†
it was required to show.

q
q
† Thus, the square-root of a rational plus a medial (area) has a length expressible as [(1 + k ) + k]/[2 (1 + k )]+ [(1 + k ) − k]/[2 (1 + k )].
2
2 1/2
2
2 1/2
2 1/2
2
2 1/2
2
This and the corresponding irrational with a minus sign, whose length is expressible as q [(1 + k ) + k]/[2 (1 + k )]− q [(1 + k ) − k]/[2 (1 + k )]
√
4
2
2
2
2 2
(see Prop. 10.77), are the positive roots of the quartic x − (2/ 1 + k ) x + k /(1 + k ) = 0.
Proposition 41
.
᾿Εὰν δύο εὐθεῖαι δυνάμει ἀσύμμετροι συντεθῶσι ποιοῦς- If two straight-lines (which are) incommensurable in
αι τό τε συγκείμενον ἐκ τῶν ἀπ᾿ αὐτῶν τετραγώνων μέσον square, making the sum of the squares on them me-
καὶ τὸ ὑπ᾿ αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τῷ συγκειμένῳ dial, and the (rectangle contained) by them medial, and,
ἐκ τῶν ἀπ᾿ αὐτῶν τετραγώνων, ἡ ὅλη εὐθεῖα ἄλογός ἐστιν, moreover, incommensurable with the sum of the squares
καλείσθω δὲ δύο μέσα δυναμένη. on them, are added together then the whole straight-line
is irrational—let it be called the square-root of (the sum
of) two medial (areas).
Κ Θ K H
Α A
Η Ζ G F
Β B

Γ ∆ Ε C D E
Συγκείσθωσαν γὰρ δύο εὐθεῖαι δυνάμει ἀσύμμετροι αἱ For let the two straight-lines, AB and BC, incommen-
ΑΒ, ΒΓ ποιοῦσαι τὰ προκείμενα· λέγω, ὅτι ἡ ΑΓ ἄλογός surable in square, (and) fulfilling the prescribed (condi-
ἐστιν. tions), be laid down together [Prop. 10.35]. I say that
᾿Εκκείσθω ῥητὴ ἡ ΔΕ, καὶ παραβεβλήσθω παρὰ τὴν ΔΕ AC is irrational.
τοῖς μὲν ἀπὸ τῶν ΑΒ, ΒΓ ἴσον τὸ ΔΖ, τῷ δὲ δὶς ὑπὸ τῶν Let the rational (straight-line) DE be laid out, and let
ΑΒ, ΒΓ ἴσον τὸ ΗΘ· ὅλον ἄρα τὸ ΔΘ ἴσον ἐστὶ τῷ ἀπὸ (the rectangle) DF, equal to (the sum of) the (squares)
τῆς ΑΓ τετραγώνῳ. καὶ ἐπεὶ μέσον ἐστὶ τὸ συγκείμενον on AB and BC, and (the rectangle) GH, equal to twice
ἐκ τῶν ἀπὸ τῶν ΑΒ, ΒΓ, καί ἐστιν ἴσον τῷ ΔΖ, μέσον ἄρα the (rectangle contained) by AB and BC, have been ap-
ἐστὶ καὶ τὸ ΔΖ. καὶ παρὰ ῥητὴν τὴν ΔΕ παράκειται· ῥητὴ plied to DE. Thus, the whole of DH is equal to the
ἄρα ἐστὶν ἡ ΔΗ καὶ ἀσύμμετρος τῇ ΔΕ μήκει. διὰ τὰ αὐτὰ square on AC [Prop. 2.4]. And since the sum of the
δὴ καὶ ἡ ΗΚ ῥητή ἐστι καὶ ἀσύμμετρος τῇ ΗΖ, τουτέστι τῇ (squares) on AB and BC is medial, and is equal to DF,
ΔΕ, μήκει. καὶ ἐπεὶ ἀσύμμετρά ἐστι τὰ ἀπὸ τῶν ΑΒ, ΒΓ DF is thus also medial. And it is applied to the rational
τῷ δὶς ὑπὸ τῶν ΑΒ, ΒΓ, ἀσύμμετρόν ἐστι τὸ ΔΖ τῷ ΗΘ· (straight-line) DE. Thus, DG is rational, and incommen-

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