---
book: 10
number: 27
id: "X.27"
kind: "construction"
uses: ["[[book-6/proposition-13]]", "[[book-6/proposition-12]]", "[[book-6/proposition-17]]", "[[book-10/proposition-21]]", "[[book-10/proposition-11]]", "[[book-10/proposition-23]]", "[[book-5/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.27"
license: "CC-BY-SA-4.0"
---

# X.27

*To find medial straight lines commensurable in square only which contain a rational rectangle*.

## Proof

Let two rational straight lines *A*, *B* commensurable in square only be set out; let *C* be taken a mean proportional between *A*, *B*, [[book-6/proposition-13|VI. 13]] and let it be contrived that, as *A* is to *B*, so is *C* to *D*. [[book-6/proposition-12|VI. 12]]

Then, since *A*, *B* are rational and commensurable in square only, the rectangle *A*, *B*, that is, the square on *C* [[book-6/proposition-17|VI.17]], is medial. [[book-10/proposition-21|X. 21]]

Therefore *C* is medial. [[book-10/proposition-21|X. 21]]

And since, as *A* is to *B*, so is *C* to *D*, and *A*, *B* are commensurable in square only, therefore *C*, *D* are also commensurable in square only. [[book-10/proposition-11|X. 11]]

And *C* is medial; therefore *D* is also medial. [[book-10/proposition-23|X. 23]], addition

Therefore *C*, *D* are medial and commensurable in square only.

I say that they also contain a rational rectangle.

For since, as *A* is to *B*, so is *C* to *D*, therefore, alternately, as *A* is to *C*, so is *B* to *D*. [[book-5/proposition-16|V. 16]]

But, as *A* is to *C*, so is *C* to *B*; therefore also, as *C* is to *B*, so is *B* to *D*; therefore the rectangle *C*, *D* is equal to the square on *B*.

But the square on *B* is rational; therefore the rectangle *C*, *D* is also rational.

Therefore medial straight lines commensurable in square only have been found which contain a rational rectangle. Q. E. D.