---
book: 10
number: 37
id: "X.37"
kind: "theorem"
uses: ["[[book-10/proposition-36]]", "[[book-2/proposition-4]]", "[[book-10/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.37"
license: "CC-BY-SA-4.0"
---

# X.37

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

## Proof

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

For, since *AB* is incommensurable in length with *BC*, therefore the squares on *AB*, *BC* are also incommensurable with twice the rectangle *AB*, *BC*; cf. [[book-10/proposition-36|X. 36, ll. 9-20]] and, componendo, the squares on *AB*, *BC* together with twice the rectangle *AB*, *BC*, that is, the square on *AC* [[book-2/proposition-4|II. 4]], is incommensurable with the rectangle *AB*, *BC*. [[book-10/proposition-16|X. 16]]

But the rectangle *AB*, *BC* is rational, for, by hypothesis, *AB*, *BC* are straight lines containing a rational rectangle; therefore the square on *AC* is irrational; therefore *AC* is irrational. [[book-10/definitions#Definition 4|X. Def. 4]]

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