---
book: 10
number: 40
id: "X.40"
kind: "theorem"
uses: ["[[book-10/proposition-34]]", "[[book-10/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.40"
license: "CC-BY-SA-4.0"
---

# X.40

*If two straight lines incommensurable in square which make the sum of the squares on them medial*, *but the rectangle contained by them rational*, *be added together, the whole straight line is irrational; and let it be called the* *side of a rational plus a medial area*.

## Proof

For let two straight lines *AB*, *BC* incommensurable in square, and fulfilling the given conditions [[book-10/proposition-34|X. 34]] , be added together; I say that *AC* is irrational.

For, since the sum of the squares on *AB*, *BC* is medial, while twice the rectangle *AB*, *BC* is rational, therefore the sum of the squares on *AB*, *BC* is incommensurable with twice the rectangle *AB*, *BC*; so that the square on *AC* is also incommensurable with twice the rectangle *AB*, *BC*. [[book-10/proposition-16|X. 16]]

But twice the rectangle *AB*, *BC* is rational; therefore the square on *AC* is irrational.

Therefore *AC* is irrational. [[book-10/definitions#Definition 4|X. Def. 4]]

And let it be called the *side of a rational plus a medial area*. Q. E. D.