---
book: 10
number: 38
id: "X.38"
kind: "theorem"
uses: ["[[book-1/proposition-44]]", "[[book-2/proposition-4]]", "[[book-10/proposition-22]]", "[[book-10/proposition-11]]", "[[book-10/proposition-15]]", "[[book-10/proposition-6]]", "[[book-10/proposition-13]]", "[[book-6/proposition-1]]", "[[book-10/proposition-36]]", "[[book-10/proposition-20]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.38"
license: "CC-BY-SA-4.0"
---

# X.38

*If two medial straight lines commensurable in square only and containing a medial rectangle be added together, the whole is irrational; and let it be called a* *second bimedial* *straight line*.

## Proof

For let two medial straight lines *AB*, *BC* commensurable in square only and containing a medial rectangle be added together; I say that *AC* is irrational.

For let a rational straight line *DE* be set out, and let the parallelogram *DF* equal to the square on *AC* be applied to *DE*, producing *DG* as breadth. [[book-1/proposition-44|I. 44]]

Then, since the square on *AC* is equal to the squares on *AB*, *BC* and twice the rectangle *AB*, *BC*, [[book-2/proposition-4|II. 4]] let *EH*, equal to the squares on *AB*, *BC*, be applied to *DE*; therefore the remainder *HF* is equal to twice the rectangle *AB*, *BC*.

And, since each of the straight lines *AB*, *BC* is medial, therefore the squares on *AB*, *BC* are also medial.

But, by hypothesis, twice the rectangle *AB*, *BC* is also medial.

And *EH* is equal to the squares on *AB*, *BC*, while *FH* is equal to twice the rectangle *AB*, *BC*; therefore each of the rectangle *EH*, *HF* is medial.

And they are applied to the rational straight line *DE*; therefore each of the straight lines *DH*, *HG* is rational and incommensurable in length with *DE*. [[book-10/proposition-22|X. 22]]

Since then *AB* is incommensurable in length with *BC*, and, as *AB* is to *BC*, so is the square on *AB* to the rectangle *AB*, *BC*, therefore the square on *AB* is incommensurable with the rectangle *AB*, *BC*. [[book-10/proposition-11|X. 11]]

But the sum of the squares on *AB*, *BC* is commensurable with the square on *AB*, [[book-10/proposition-15|X. 15]] and twice the rectangle *AB*, *BC* is commensurable with the rectangle *AB*, *BC*. [[book-10/proposition-6|X. 6]]

Therefore the sum of the squares on *AB*, *BC* is incommensurable with twice the rectangle *AB*, *BC*. [[book-10/proposition-13|X. 13]]

But *EH* is equal to the squares on *AB*, *BC*, and *HF* is equal to twice the rectangle *AB*, *BC*.

Therefore *EH* is incommensurable with *HF*, so that *DH* is also incommensurable in length with *HG*. [[book-6/proposition-1|VI. 1]] , [[book-10/proposition-11|X. 11]]

Therefore *DH*, *HG* are rational straight lines commensurable in square only; so that *DG* is irrational. [[book-10/proposition-36|X. 36]]

But *DE* is rational; and the rectangle contained by an irrational and a rational straight line is irrational; cf. [[book-10/proposition-20|X. 20]] therefore the area *DF* is irrational, and the side of the square equal to it is irrational. [[book-10/definitions#Definition 4|X. Def. 4]]

But *AC* is the side of the square equal to *DF*; therefore *AC* is irrational.

And let it be called a *second bimedial* straight line. Q. E. D.