A straight line commensurable with a major straight line is itself also major.

Let *AB* be major, and let *CD* be commensurable with *AB*.

I say that *CD* is major.

Divide *AB* at *E*. Then *AE* and *EB* are straight lines incommensurable in square which make the sum of the squares on them rational but the rectangle contained by them medial.

Make the same construction as before.

Since *AB* is to *CD* as *AE* is to *CF*, and *EB* is to *FD*, therefore *AE* is to *CF* as *EB* is to *FD*.

But *AB* is commensurable with *CD*, therefore *AE* and *EB* are commensurable with *CF* and *FD* respectively.

Since *AE* is to *CF* as *EB* is to *FD*, alternately, also *AE* is to *EB* as *CF* is to *FD*, therefore, taken jointly, *AB* is to *BE* as *CD* is to *DF*.

Therefore the square on *AB* is to the square on *BE* as the square on *CD* is to the square on *DF*.

Similarly we can prove that the square on *AB* is to the square on *AE* as the square on *CD* is to the square on *CF*. Therefore the square on *AB* is to the squares on *AE* and *EB* as the square on *CD* is to the squares on *CF* and *FD*, therefore, alternately, the square on *AB* is to the square on *CD*, so are the squares on *AE* and *EB* to the squares on *CF* and *FD*.

But the square on *AB* is commensurable with the square on *CD*, therefore the squares on *AE* and *EB* are also commensurable with the squares on *CF* and *FD*.

And the squares on *AE* and *EB* together are rational, therefore the squares on *CF* and *FD* together are rational.

Similarly also twice the rectangle *AE* by *EB* is commensurable with twice the rectangle *CF* by *FD*.

And twice the rectangle *AE* by *EB* is medial, therefore twice the rectangle *CF* by *FD* is also medial.

Therefore *CF* and *FD* are straight lines incommensurable in square which make, at the same time, the sum of the squares on them rational, but the rectangle contained by them medial, therefore the whole *CD* is the irrational straight line called major.

Therefore a straight line commensurable with the major straight line is major.

Therefore, *a straight line commensurable with a major straight line is itself also major.*

Q.E.D.