---
book: 10
number: 56
id: "X.56"
kind: "theorem"
uses: ["[[book-10/proposition-13]]", "[[book-10/proposition-21]]", "[[book-10/proposition-38]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.56"
license: "CC-BY-SA-4.0"
---

# X.56

If an area be contained by a rational straight line and the third binomial, the side of the area is the irrational straight line called a second bimedial.

## Proof

For let the area *ABCD* be contained by the rational straight line *AB* and the third binomial *AD* divided into its terms at *E*, of which terms *AE* is the greater; I say that the side of the area *AC* is the irrational straight line called a second bimedial.

For let the same construction be made as before.

Now, since *AD* is a third binomial straight line, therefore *AE*, *ED* are rational straight lines commensurable in square only, the square on *AE* is greater than the square on *ED* by the square on a straight line commensurable with *AE*, and neither of the terms *AE*, *ED* is commensurable in length with *AB*. [[book-10/definitions#Definition 3 (part 2)|X. Deff. II. 3]]

Then, in manner similar to the foregoing, we shall prove that *MO* is the side of the area *AC*, and *MN*, *NO* are medial straight lines commensurable in square only; so that *MO* is bimedial.

It is next to be proved that it is also a second bimedial straight line.

Since *DE* is incommensurable in length with *AB*, that is, with *EK*, and *DE* is commensurable with *EF*, therefore *EF* is incommensurable in length with *EK*. [[book-10/proposition-13|X. 13]]

And they are rational; therefore *FE*, *EK* are rational straight lines commensurable in square only.

Therefore *EL*, that is, *MR*, is medial. [[book-10/proposition-21|X. 21]]

And it is contained by *MN*, *NO*; therefore the rectangle *MN*, *NO* is medial.

Therefore *MO* is a second bimedial straight line. [[book-10/proposition-38|X. 38]] Q. E. D.