---
book: 10
number: 23
id: "X.23"
kind: "theorem"
uses: ["[[book-10/proposition-22]]", "[[book-6/proposition-1]]", "[[book-10/proposition-11]]", "[[book-10/proposition-13]]", "[[book-10/proposition-21]]", "[[book-10/proposition-18]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.23"
license: "CC-BY-SA-4.0"
---

# X.23

*A straight line commensurable with a medial straight line is medial*.

## Proof

Let *A* be medial, and let *B* be commensurable with *A*; I say that *B* is also medial.

For let a rational straight line *CD* be set out, and to *CD* let the rectangular area *CE* equal to the square on *A* be applied, producing *ED* as breadth; therefore *ED* is rational and incommensurable in length with *CD*. [[book-10/proposition-22|X. 22]]

And let the rectangular area *CF* equal to the square on *B* be applied to *CD*, producing *DF* as breadth.

Since then *A* is commensurable with *B*, the square on *A* is also commensurable with the square on *B*.

But *EC* is equal to the square on *A*, and *CF* is equal to the square on *B*; therefore *EC* is commensurable with *CF*.

And, as *EC* is to *CF*, so is *ED* to *DF*; [[book-6/proposition-1|VI. 1]] therefore *ED* is commensurable in length with *DF*. [[book-10/proposition-11|X. 11]]

But *ED* is rational and incommensurable in length with *DC*; therefore *DF* is also rational [[book-10/definitions#Definition 3|X. Def. 3]] and incommensurable in length with *DC*. [[book-10/proposition-13|X. 13]]

Therefore *CD*, *DF* are rational and commensurable in square only.

But the straight line the square on which is equal to the rectangle contained by rational straight lines commensurable in square only is medial; [[book-10/proposition-21|X. 21]] therefore the side of the square equal to the rectangle *CD*, *DF* is medial.

And *B* is the side of the square equal to the rectangle *CD*, *DF*; therefore *B* is medial.

Porism. From this it is manifest that an area commensurable with a medial area is medial.

[And in the same way as was explained in the case of rationals [[book-10/proposition-18|Lemma following X. 18]] it follows, as regards medials, that a straight line commensurable in length with a medial straight line is called *medial* and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also.

But, if any straight line be commensurable in square with a medial straight line, then, if it is also commensurable in length with it, the straight lines are called, in this case too, medial and commensurable in length and in square, but, if in square only, they are called medial straight lines commensurable in square only.]