---
book: 10
number: 36
id: "X.36"
kind: "theorem"
uses: ["[[book-10/proposition-11]]", "[[book-10/proposition-6]]", "[[book-10/proposition-15]]", "[[book-10/proposition-13]]", "[[book-2/proposition-4]]", "[[book-10/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.36"
license: "CC-BY-SA-4.0"
---

# X.36

*If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be called* *binomial*.

## Proof

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

For, since *AB* is incommensurable in length with *BC*— for they are commensurable in square only— and, as *AB* is to *BC*, so is the rectangle *AB*, *BC* to the square on *BC*, therefore the rectangle *AB*, *BC* is incommensurable with the square on *BC*. [[book-10/proposition-11|X. 11]]

But twice the rectangle *AB*, *BC* is commensurable with the rectangle *AB*, *BC* [[book-10/proposition-6|X. 6]] , and the squares on *AB*, *BC* are commensurable with the square on *BC*—for *AB*, *BC* are rational straight lines commensurable in square only— [[book-10/proposition-15|X. 15]] therefore twice the rectangle *AB*, *BC* is incommensurable with the squares on *AB*, *BC*. [[book-10/proposition-13|X. 13]]

And, componendo, twice the rectangle *AB*, *BC* together with the squares on *AB*, *BC*, that is, the square on *AC* [[book-2/proposition-4|II. 4]] , is incommensurable with the sum of the squares on *AB*, *BC*. [[book-10/proposition-16|X. 16]]

But the sum of the squares on *AB*, *BC* is rational; therefore the square on *AC* is irrational, so that *AC* is also irrational. [[book-10/definitions#Definition 4|X. Def. 4]]

And let it be called *binomial*.