---
book: 10
number: 30
id: "X.30"
kind: "construction"
uses: ["[[book-10/proposition-28]]", "[[book-10/proposition-6]]", "[[book-5/proposition-19]]", "[[book-3/proposition-31]]", "[[book-1/proposition-47]]", "[[book-10/proposition-9]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.30"
license: "CC-BY-SA-4.0"
---

# X.30

*To find two rational straight lines commensurable in square only and such that the square on the greater is greater is greater than the square on the less by the square on a straight line incommensurable in length with the greater*.

## Proof

Let there be set out a rational straight line *AB*, and two square numbers *CE*, *ED* such that their sum *CD* is not square; [[book-10/proposition-28|Lemma 2]] let there be described on *AB* the semicircle *AFB*, let it be contrived that, as *DC* is to *CE*, so is the square on *BA* to the square on *AF*, [[book-10/proposition-6|X. 6, Por.]] and let *FB* be joined.

Then, in a similar manner to the preceding, we can prove that *BA*, *AF* are rational straight lines commensurable in square only.

And since, as *DC* is to *CE*, so is the square on *BA* to the square on *AF*, therefore, convertendo, as *CD* is to *DE*, so is the square on *AB* to the square on *BF*. [[book-5/proposition-19|V. 19, Por.]], [[book-3/proposition-31|III. 31]], [[book-1/proposition-47|I. 47]]

But *CD* has not to *DE* the ratio which a square number has to a square number; therefore neither has the square on *AB* to the square on *BF* the ratio which a square number has to a square number; therefore *AB* is incommensurable in length with *BF*. [[book-10/proposition-9|X. 9]]

And the square on *AB* is greater than the square on *AF* by the square on *FB* incommensurable with *AB*.

Therefore *AB*, *AF* are rational straight lines commensurable in square only, and the square on *AB* is greater than the square on *AF* by the square on *FB* incommensurable in length with *AB*. Q. E. D.