---
book: 10
number: 20
id: "X.20"
kind: "theorem"
uses: ["[[book-6/proposition-1]]", "[[book-10/proposition-11]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.20"
license: "CC-BY-SA-4.0"
---

# X.20

*If a rational area be applied to a rational straight line, it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied*.

## Proof

For let the rational area *AC* be applied to *AB*, a straight line once more rational in any of the aforesaid ways, producing *BC* as breadth; I say that *BC* is rational and commensurable in length with *BA*. For on *AB* let the square *AD* be described; therefore *AD* is rational. [[book-10/definitions#Definition 4|X. Def. 4]]

But *AC* is also rational; therefore *DA* is commensurable with *AC*.

And, as *DA* is to *AC*, so is *DB* to *BC*. [[book-6/proposition-1|VI. 1]]

Therefore *DB* is also commensurable with *BC*; [[book-10/proposition-11|X. 11]] and *DB* is equal to *BA*; therefore *AB* is also commensurable with *BC*.

But *AB* is rational; therefore *BC* is also rational and commensurable in length with *AB*.

Therefore etc.