---
book: 10
number: 19
id: "X.19"
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_1.19"
license: "CC-BY-SA-4.0"
---

# X.19

*The rectangle contained by rational straight lines commensurable in length is rational*.

## Proof

For let the rectangle *AC* be contained by the rational straight lines *AB*, *BC* commensurable in length; I say that *AC* is rational.

For on *AB* let the square *AD* be described; therefore *AD* is rational. [[book-10/definitions#Definition 4|X. Def. 4]]

And, since *AB* is commensurable in length with *BC*, while *AB* is equal to *BD*, therefore *BD* is commensurable in length with *BC*.

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

Therefore *DA* is commensurable with *AC*. [[book-10/proposition-11|X. 11]]

But *DA* is rational; therefore *AC* is also rational. [[book-10/definitions#Definition 4|X. Def. 4]]

Therefore etc.