---
book: 10
number: 31
id: "X.31"
kind: "construction"
uses: ["[[book-10/proposition-29]]", "[[book-10/proposition-21]]", "[[book-10/proposition-11]]", "[[book-10/proposition-23]]", "[[book-10/proposition-14]]", "[[book-10/proposition-30]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.31"
license: "CC-BY-SA-4.0"
---

# X.31

*To find two medial straight lines commensurable in square only, containing a rational rectangle, 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

Let there be set out two rational straight lines *A*, *B* commensurable in square only and such that the square on *A*, being the greater, is greater than the square on *B* the less by the square on a straight line commensurable in length with *A*. [[book-10/proposition-29|X. 29]]

And let the square on *C* be equal to the rectangle *A*, *B*.

Now the rectangle *A*, *B* is medial; [[book-10/proposition-21|X. 21]] therefore the square on *C* is also medial; therefore *C* is also medial. [[book-10/proposition-21|X. 21]]

Let the rectangle *C*, *D* be equal to the square on *B*.

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

And since, as *A* is to *B*, so is the rectangle *A*, *B* to the square on *B*, while the square on *C* is equal to the rectangle *A*, *B*, and the rectangle *C*, *D* is equal to the square on *B*, therefore, as *A* is to *B*, so is the square on *C* to the rectangle *C*, *D*.

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

But *A* is commensurable with *B* in square only; therefore *C* is also commensurable with *D* 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

And since, as *A* is to *B*, so is *C* to *D*, and the square on *A* is greater than the square on *B* by the square on a straight line commensurable with *A*, therefore also the square on *C* is greater than the square on *D* by the square on a straight line commensurable with *C*. [[book-10/proposition-14|X. 14]]

Therefore two medial straight lines *C*, *D*, commensurable in square only and containing a rational rectangle, have been found, and the square on *C* is greater than the square on *D* by the square on a straight line commensurable in length with *C*.

Similarly also it can be proved that the square on *C* exceeds the square on *D* by the square on a straight line incommensurable with *C*, when the square on *A* is greater than the square on *B* by the square on a straight line incommensurable with *A*. [[book-10/proposition-30|X. 30]]