A straight line commensurable with a minor straight line is minor.

Let *AB* be a minor straight line, and *CD* commensurable with *AB*.

I say that *CD* is also minor.

Make the same construction as before. Then, since *AE* and *EB* are incommensurable in square, therefore *CF* and *FD* are also incommensurable in square.

Now since *AE* is to *EB* as *CF* is to *FD*, therefore the square on *AE* is to the square on *EB* as the square on *CF* is to the square on *FD*.

Therefore, taken jointly, the sum of the squares on *AE* and *EB* is to the square on *EB* as the sum of the squares on *CF* and *FD* is to the square on *FD*.

But the square on *BE* is commensurable with the square on *DF*, therefore the sum of the squares on *AE* and *EB* is also commensurable with the sum of the squares on *CF* and *FD*.

But the sum of the squares on *AE* and *EB* is rational, therefore the sum of the squares on *CF* and *FD* is also rational.

Again, since the square on *AE* is to the rectangle *AE* by *EB* as the square on *CF* is to the rectangle *CF* by *FD*, while the square on *AE* is commensurable with the square on *CF*, therefore the rectangle *AE* by *EB* is also commensurable with the rectangle *CF* by *FD*.

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

Therefore *CF* and *FD* are straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial.

Therefore *CD* is minor.

Therefore, *a straight line commensurable with a minor straight line is minor.*

Q.E.D.