---
book: 10
number: 39
id: "X.39"
kind: "theorem"
uses: ["[[book-10/proposition-33]]", "[[book-10/proposition-6]]", "[[book-10/proposition-23]]", "[[book-10/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.39"
license: "CC-BY-SA-4.0"
---

# X.39

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

## Proof

For let two straight lines *AB*, *BC* incommensurable in square, and fulfilling the given conditions [[book-10/proposition-33|X. 33]] , be added together; I say that *AC* is irrational.

For, since the rectangle *AB*, *BC* is medial, twice the rectangle *AB*, *BC* is also medial. [[book-10/proposition-6|X. 6 and 23, Por.]]

But the sum of the squares on *AB*, *BC* is rational; therefore twice the rectangle *AB*, *BC* is incommensurable with the sum of the squares on *AB*, *BC*, so that the squares on *AB*, *BC* together with twice the rectangle *AB*, *BC* that is, the square on *AC*, is also incommensurable with the sum of the squares on *AB*, *BC*; [[book-10/proposition-16|X. 16]] 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 *major*. Q. E. D.