---
book: 10
number: 58
id: "X.58"
kind: "theorem"
uses: ["[[book-10/proposition-18]]", "[[book-6/proposition-1]]", "[[book-10/proposition-11]]", "[[book-10/proposition-13]]", "[[book-10/proposition-21]]", "[[book-10/proposition-12]]", "[[book-10/proposition-19]]", "[[book-10/proposition-40]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.58"
license: "CC-BY-SA-4.0"
---

# X.58

If an area be contained by a rational straight line and the fifth binomial, the side of the area is the irrational straight line called the side of a rational plus a medial area.

## Proof

For let the area *AC* be contained by the rational straight line *AB* and the fifth binomial *AD* divided into its terms at *E*, so that *AE* is the greater term; I say that the side of the area *AC* is the irrational straight line called the side of a rational plus a medial area.

For let the same construction be made as before shown; it is then manifest that *MO* is the side of the area *AC.*

It is then to be proved that *MO* is the side of a rational plus a medial area.

For, since *AG* is incommensurable with *GE*, [[book-10/proposition-18|X. 18]] therefore *AH* is also commensurable with *HE*, [[book-6/proposition-1|VI. 1]], [[book-10/proposition-11|X. 11]] that is, the square on *MN* with the square on *NO*; therefore *MN*, *NO* are incommensurable in square.

And, since *AD* is a fifth binomial straight line, and *ED* the lesser segment, therefore *ED* is commensurable in length with *AB*. [[book-10/definitions#Definition 5 (part 2)|X. Deff. II. 5]]

But *AE* is incommensurable with *ED*; therefore *AB* is also incommensurable in length with *AE.* [[book-10/proposition-13|X. 13]]

Therefore *AK*, that is, the sum of the squares on *MN*, *NO*, is medial. [[book-10/proposition-21|X. 21]]

And, since *DE* is commensurable in length with *AB*, that is, with *EK*, while *DE* is commensurable with *EF*, therefore *EF* is also commensurable with *EK.* [[book-10/proposition-12|X. 12]]

And *EK* is rational; therefore *EL*, that is, *MR*, that is, the rectangle *MN*, *NO*, is also rational. [[book-10/proposition-19|X. 19]]

Therefore *MN*, *NO* are straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational.

Therefore *MO* is the side of a rational plus a medial area [[book-10/proposition-40|X. 40]] and is the side of the area *AC.* Q. E. D.