The rectangle contained by rational straight lines commensurable in square only is irrational, and the side of the square equal to it is irrational. Let the latter be called *medial.*

Let the rectangle *AC* be contained by the rational straight lines *AB* and *BC* commensurable in square only.

I say that *AC* is irrational, and the side of the square equal to it is irrational, and let the latter be called medial.

Describe the square *AD* on *AB*. Then *AD* is rational.

And, since *AB* is incommensurable in length with *BC*, for by hypothesis they are commensurable in square only, while *AB* equals *BD*, therefore *DB* is also incommensurable in length with *BC*.

And *DB* is to *BC* as *AD* is to *AC*, therefore *DA* is incommensurable with *AC*.

But *DA* is rational, therefore *AC* is irrational, so that the side of the square *AC* is also irrational.

Let the latter be called *medial*.

Q.E.D.

The two lines *AB* and *BC* are rational lines. That means, according to Definition 3, that each is commensurable in square to some unmentioned, standard straight line. It’s convenient to assign that standard line a length of 1 and the standard square on it a square of area 1. Euclid did not, but doing so allows us to assign lengths to those lines. So if *a* is the length of *AB*, that means *a*^{2} is a rational number, that is, a ratio of two whole numbers. Likewise, if *b* is the length of *BC*, then *b*^{2} is a rational number.

Now *AB* and *BC* are also commensurable in square only. First of all, they’re commensurable in square, so the squares on them are in a numeric ratio, and that means *a*^{2}/*b*^{2} is a rational number, which we already know since both *a*^{2} and *b*^{2} are rational numbers themselves. But they’re also commensurable in square *only*. That means that *a*/*b* is an irrational number. In summary, *a*^{2} and *b*^{2} are rational, but *a*/*b* is irrational.

In the proof, it’s noted that the rectangle *AC* (which is a *a* by *b* rectangle), is irrational, that is, *ab* is an irrational number. The side of a square equal equal to that rectangle is also irrational, and it’s called *medial*. The length of that side is √(*ab*).

Another way of saying that is that the length √(*ab*) of a medial line is the geometric mean of two numbers *a* and *b* whose squares are rational and whose ratio is irrational.

Let *AB* be the standard line and *BC* be the diagonal of the standard square, then √(*ab*) = √(√2), the 4^{th} root of 2.

In fact, all medial lines have lengths 4^{th} roots of rational numbers whose square roots are irrational. If *a*^{2} and *b*^{2} are rational numbers but *a*/*b* is an irrational number, then √(*ab*) is a number whose fourth power, *a*^{2}*b*^{2}, is a rational number but whose square, *ab*, is an irrational number.

Thus, we could call a number whose square is an irrational number but whose fourth power is a rational number a *medial number*.

Only medial lines were defined in this proposition, but the adjective medial is applied to squares and rectangles starting with Proposition X.24. From the way it is used, we can tell that a *medial* square is one whose side is a medial line, and a medial rectangle is one equal to a square whose side is a medial line.

This proposition is used frequently in Book X starting with the next proposition.