Page 349 - Euclid's Elements of Geometry

ST EW iþ.
njþ

the square root of a rational plus a medial area (see Prop. 10.40), since ρ is rational. ELEMENTS BOOK 10
Proposition 59
.
᾿Εὰν χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο If an area is contained by a rational (straight-line) and
ὀνομάτων ἕκτης, ἡ τὸ χωρίον δυναμένη ἄλογός ἐστιν ἡ a sixth binomial (straight-line) then the square-root of
καλουμένη δύο μέσα δυναμένη. the area is the irrational (straight-line which is) called
the square-root of (the sum of) two medial (areas). †
Α Η Ε Ζ ∆ Ρ Π A G E F D R Q

Μ Ξ M O
Ν N
Β Θ Κ Λ Γ B H K L C

Σ Ο S P
Χωρίον γὰρ τὸ ΑΒΓΔ περιεχέσθω ὑπὸ ῥητῆς τῆς ΑΒ For let the area ABCD be contained by the rational
καὶ τῆς ἐκ δύο ὀνομάτων ἕκτης τῆς ΑΔ διῃρημένης εἰς τὰ (straight-line) AB and the sixth binomial (straight-line)
ὀνόματα κατὰ τὸ Ε, ὥστε τὸ μεῖζον ὄνομα εἶναι τὸ ΑΕ· AD, which has been divided into its (component) terms
λέγω, ὅτι ἡ τὸ ΑΓ δυναμένη ἡ δύο μέσα δυναμένη ἐστίν. at E, such that AE is the greater term. So, I say that the
Κατεσκευάσθω [γὰρ] τὰ αὐτὰ τοῖς προδεδειγμένοις. square-root of AC is the square-root of (the sum of) two
φανερὸν δή, ὅτι [ἡ] τὸ ΑΓ δυναμένη ἐστὶν ἡ ΜΞ, καὶ medial (areas).
ὅτι ἀσύμμετρός ἐστιν ἡ ΜΝ τῇ ΝΞ δυνάμει. καὶ ἐπεὶ [For] let the same construction be made as that shown
ἀσύμμετρός ἐστιν ἡ ΕΑ τῇ ΑΒ μήκει, αἱ ΕΑ, ΑΒ ἄρα previously. So, (it is) clear that MO is the square-root of
ῥηταί εἰσι δυνάμει μόνον σύμμετροι· μέσον ἄρα ἐστὶ τὸ ΑΚ, AC, and that MN is incommensurable in square with
τουτέστι τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΜΝ, ΝΞ. πάλιν, NO. And since EA is incommensurable in length with
ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΕΔ τῇ ΑΒ μήκει, ἀσύμμετρος ἄρα AB [Def. 10.10], EA and AB are thus rational (straight-
ἐστὶ καὶ ἡ ΖΕ τῇ ΕΚ· αἱ ΖΕ, ΕΚ ἄρα ῥηταί εἰσι δυνάμει lines which are) commensurable in square only. Thus,
μόνον σύμμετροι· μέσον ἄρα ἐστὶ τὸ ΕΛ, τουτέστι τὸ ΜΡ, AK—that is to say, the sum of the (squares) on MN
τουτέστι τὸ ὑπὸ τῶν ΜΝΞ. καὶ ἐπεὶ ἀσύμμετρος ἡ ΑΕ τῇ and NO—is medial [Prop. 10.21]. Again, since ED
ΕΖ, καὶ τὸ ΑΚ τῷ ΕΛ ἀσύμμετρόν ἐστιν. ἀλλὰ τὸ μὲν is incommensurable in length with AB [Def. 10.10],
ΑΚ ἐστι τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΜΝ, ΝΞ, τὸ FE is thus also incommensurable (in length) with EK
δὲ ΕΛ ἐστι τὸ ὑπὸ τῶν ΜΝΞ· ἀσύμμετρον ἄρα ἐστὶ τὸ [Prop. 10.13]. Thus, FE and EK are rational (straight-
συγκείμενον ἐκ τῶν ἀπὸ τῶν ΜΝΞ τῷ ὑπὸ τῶν ΜΝΞ. καί lines which are) commensurable in square only. Thus,
ἐστι μέσον ἑκάτερον αὐτῶν, καὶ αἱ ΜΝ, ΝΞ δυνάμει εἰσὶν EL—that is to say, MR—that is to say, the (rectangle
ἀσύμμετροι. contained) by MNO—is medial [Prop. 10.21]. And since
῾Η ΜΞ ἄρα δύο μέσα δυναμένη ἐστὶ καὶ δύναται τὸ ΑΓ· AE is incommensurable (in length) with EF, AK is also
ὅπερ ἔδει δεῖξαι. incommensurable with EL [Props. 6.1, 10.11]. But, AK
is the sum of the (squares) on MN and NO, and EL is
the (rectangle contained) by MNO. Thus, the sum of the
(squares) on MNO is incommensurable with the (rect-
angle contained) by MNO. And each of them is medial.
And MN and NO are incommensurable in square.
Thus, MO is the square-root of (the sum of) two me-
dial (areas) [Prop. 10.41]. And (it is) the square-root of
AC. (Which is) the very thing it was required to show.

† If the rational straight-line has unit length then this proposition states that the square-root of a sixth binomial straight-line is the square root of
√ √
′
the sum of two medial areas: i.e., a sixth binomial straight-line has a length k + k whose square-root can be written
«
„ q
q
′
′
′′
′′
k 1/4 [1 + k /(1 + k ′′ 2 1/2 ]/2 + [1 − k /(1 + k ′′ 2 1/2 ]/2 , where k ′′ 2 = (k − k )/k . This is the length of the square-root of the sum of
)
)
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