If two medial straight lines commensurable in square only and containing a rational rectangle are added together, the whole is irrational; let it be called the *first bimedial* straight line.

Let two medial straight lines *AB* and *BC* commensurable in square only and containing a rational rectangle be added together.

I say that the whole *AC* is irrational.

Since *AB* is incommensurable in length with *BC*, therefore the sum of the squares on *AB* and *BC* is also incommensurable with twice the rectangle *AB* by *BC*, and, taken jointly, the sum of the squares on *AB* and *BC* together with twice the rectangle *AB* by *BC*, that is, the square on *AC*, is incommensurable with the rectangle *AB* by *BC*.

But the rectangle *AB* by *BC* is rational, for, by hypothesis, *AB* and *BC* are straight lines containing a rational rectangle, therefore the square on *AC* is irrational. Therefore *AC* is irrational. And let it be called a *first bimedial* straight line.

Q.E.D.