---
book: 10
number: 22
id: "X.22"
kind: "theorem"
uses: ["[[book-10/proposition-21]]", "[[book-6/proposition-14]]", "[[book-6/proposition-22]]", "[[book-10/proposition-11]]", "[[book-10/proposition-13]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.22"
license: "CC-BY-SA-4.0"
---

# X.22

*The square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied*.

## Proof

Let *A* be medial and *CB* rational, and let a rectangular area *BD* equal to the square on *A* be applied to *BC*, producing *CD* as breadth; I say that *CD* is rational and incommensurable in length with *CB*.

For, since *A* is medial, the square on it is equal to a rectangular area contained by rational straight lines commensurable in square only. [[book-10/proposition-21|X. 21]]

Let the square on it be equal to *GF*.

But the square on it is also equal to *BD*; therefore *BD* is equal to *GF*.

But it is also equiangular with it; and in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional; [[book-6/proposition-14|VI. 14]] therefore, proportionally, as *BC* is to *EG*, so is *EF* to *CD*.

Therefore also, as the square on *BC* is to the square on *EG*, so is the square on *EF* to the square on *CD*. [[book-6/proposition-22|VI. 22]]

But the square on *CB* is commensurable with the square on *EG*, for each of these straight lines is rational; therefore the square on *EF* is also commensurable with the square on *CD*. [[book-10/proposition-11|X. 11]]

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

And, since *EF* is incommensurable in length with *EG*, for they are commensurable in square only, and, as *EF* is to *EG*, so is the square on *EF* to the rectangle *FE*, *EG*, [Lemma] therefore the square on *EF* is incommensurable with the rectangle *FE*, *EG*. [[book-10/proposition-11|X. 11]]

But the square on *CD* is commensurable with the square on *EF*, for the straight lines are rational in square; and the rectangle *DC*, *CB* is commensurable with the rectangle *FE*, *EG*, for they are equal to the square on *A*; therefore the square on *CD* is also incommensurable with the rectangle *DC*, *CB*. [[book-10/proposition-13|X. 13]]

But, as the square on *CD* is to the rectangle *DC*, *CB*, so is *DC* to *CB*; [Lemma] therefore *DC* is incommensurable in length with *CB*. [[book-10/proposition-11|X. 11]]

Therefore *CD* is rational and incommensurable in length with *CB*. Q. E. D.