---
book: 10
number: 29
id: "X.29"
kind: "construction"
uses: ["[[book-10/proposition-28]]", "[[book-10/proposition-6]]", "[[book-10/proposition-9]]", "[[book-5/proposition-19]]", "[[book-3/proposition-31]]", "[[book-1/proposition-47]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.29"
license: "CC-BY-SA-4.0"
---

# X.29

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

## Proof

For let there be set out any rational straight line *AB*, and two square numbers *CD*, *DE* such that their difference *CE* is not square; [[book-10/proposition-28|Lemma 1]] let there be described on *AB* the semicircle *AFB*, and 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.]]

Let *FB* be joined.

Since, as the square on *BA* is to the square on *AF*, so is *DC* to *CE*, therefore the square on *BA* has to the square on *AF* the ratio which the number *DC* has to the number *CE*; therefore the square on *BA* is commensurable with the square on *AF*. [[book-10/proposition-6|X. 6]]

But the square on *AB* is rational; [[book-10/definitions#Definition 4|X. Def. 4]] therefore the square on *AF* is also rational; [*id.*] therefore *AF* is also rational.

And, since *DC* has not to *CE* the ratio which a square number has to a square number, neither has the square on *BA* to the square on *AF* the ratio which a square number has to a square number; therefore *AB* is incommensurable in length with *AF*. [[book-10/proposition-9|X. 9]]

Therefore *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 to *DE* the ratio which a square number has to a square number: therefore also the square on *AB* has to the square on *BF* the ratio which a square number has to a square number; therefore *AB* is commensurable in length with *BF*. [[book-10/proposition-9|X. 9]]

And the square on *AB* is equal to the squares on *AF*, *FB*; therefore the square on *AB* is greater than the square on *AF* by the square on *BF* commensurable with *AB*.

Therefore there have been found two rational straight lines *BA*, *AF* commensurable in square only and such that the square on the greater *AB* is greater than the square on the less *AF* by the square on *BF* commensurable in length with *AB*.