Page 348 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

τῆς ἐκ δύο ὀνομάτων πέμπτης τῆς ΑΔ διῃρημένης εἰς τὰ For let the area AC be contained by the rational
ὀνόματα κατὰ τὸ Ε, ὥστε τὸ μεῖζον ὄνομα εἶναι τὸ ΑΕ· (straight-line) AB and the fifth binomial (straight-line)
λέγω [δή], ὅτι ἡ τὸ ΑΓ χωρίον δυναμένη ἄλογός ἐστιν ἡ AD, which has been divided into its (component) terms
καλουμένη ῥητὸν καὶ μέσον δυναμένη. at E, such that AE is the greater term. [So] I say that
the square-root of area AC is the irrational (straight-line
which is) called the square-root of a rational plus a me-
dial (area).
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
Κατεσκευάσθω γὰρ τὰ αὐτὰ τοῖς πρότερον δεδειγμένοις· For let the same construction be made as that shown
φανερὸν δή, ὅτι ἡ τὸ ΑΓ χωρίον δυναμένη ἐστὶν ἡ ΜΞ. previously. So, (it is) clear that MO is the square-root of
δεικτέον δή, ὅτι ἡ ΜΞ ἐστιν ἡ ῥητὸν καὶ μέσον δυναμένη. area AC. So, we must show that MO is the square-root
᾿Επεὶ γὰρ ἀσύμμετρός ἐστιν ἡ ΑΗ τῇ ΗΕ, ἀσύμμετρον of a rational plus a medial (area).
ἄρα ἐστὶ καὶ τὸ ΑΘ τῷ ΘΕ, τουτέστι τὸ ἀπὸ τῆς ΜΝ τῷ ἀπὸ For since AG is incommensurable (in length) with
τῆς ΝΞ· αἱ ΜΝ, ΝΞ ἄρα δυνάμει εἰσὶν ἀσύμμετροι. καὶ ἐπεὶ GE [Prop. 10.18], AH is thus also incommensurable
ἡ ΑΔ ἐκ δύο ὀνομάτων ἐστὶ πέμπτη, καί [ἐστιν] ἔλασσον with HE—that is to say, the (square) on MN with the
αὐτῆς τμῆμα τὸ ΕΔ, σύμμετρος ἄρα ἡ ΕΔ τῇ ΑΒ μήκει. (square) on NO [Props. 6.1, 10.11]. Thus, MN and
ἀλλὰ ἡ ΑΕ τῇ ΕΔ ἐστιν ἀσύμμετρος· καὶ ἡ ΑΒ ἄρα τῇ NO are incommensurable in square. And since AD is
ΑΕ ἐστιν ἀσύμμετρος μήκει [αἱ ΒΑ, ΑΕ ῥηταί εἰσι δυνάμει a fifth binomial (straight-line), and ED [is] its lesser seg-
μόνον σύμμετροι]· μέσον ἄρα ἐστὶ τὸ ΑΚ, τουτέστι τὸ ment, ED (is) thus commensurable in length with AB
συγκείμενον ἐκ τῶν ἀπὸ τῶν ΜΝ, ΝΞ. καὶ ἐπεὶ σύμμετρός [Def. 10.9]. But, AE is incommensurable (in length)
ἐστιν ἡ ΔΕ τῇ ΑΒ μήκει, τουτέστι τῇ ΕΚ, ἀλλὰ ἡ ΔΕ τῇ with ED. Thus, AB is also incommensurable in length
ΕΖ σύμμετρός ἐστιν, καὶ ἡ ΕΖ ἄρα τῇ ΕΚ σύμμετρός ἐστιν. with AE [BA and AE are rational (straight-lines which
καὶ ῥητὴ ἡ ΕΚ· ῥητὸν ἄρα καὶ τὸ ΕΛ, τουτέστι τὸ ΜΡ, are) commensurable in square only] [Prop. 10.13]. Thus,
τουτέστι τὸ ὑπὸ ΜΝΞ· αἱ ΜΝ, ΝΞ ἄρα δυνάμει ἀσύμμετροί AK—that is to say, the sum of the (squares) on MN
εἰσι ποιοῦσαι τὸ μὲν συγκείμενον ἐκ τῶν ἀπ᾿ αὐτῶν τε- and NO—is medial [Prop. 10.21]. And since DE is
τραγώνων μέσον, τὸ δ᾿ ὑπ᾿ αὐτῶν ῥητόν. commensurable in length with AB—that is to say, with
῾Η ΜΞ ἄρα ῥητὸν καὶ μέσον δυναμένη ἐστὶ καὶ δύναται EK—but, DE is commensurable (in length) with EF,
τὸ ΑΓ χωρίον· ὅπερ ἔδει δεῖξαι. EF is thus also commensurable (in length) with EK
[Prop. 10.12]. And EK (is) rational. Thus, EL—that
is to say, MR—that is to say, the (rectangle contained)
by MNO—(is) also rational [Prop. 10.19]. MN and NO
are thus (straight-lines which are) incommensurable in
square, making the sum of the squares on them medial,
and the (rectangle contained) by them rational.
Thus, MO is the square-root of a rational plus a me-
dial (area) [Prop. 10.40]. And (it is) the square-root of
area 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 fifth binomial straight-line is the square root of
√
′
a rational plus a medial area: i.e., a fifth binomial straight-line has a length k ( 1 + k + 1) whose square-root can be written
q q p
′
)
ρ [(1 + k ′′ 2 1/2 + k ]/[2 (1 + k ′′ 2 )] + ρ [(1 + k ′′ 2 1/2 − k ]/[2 (1 + k ′′ 2 )], where ρ = k (1 + k ′′ 2 ) and k ′′ 2 = k . This is the length of
′′
)
′′
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