To a minor straight line only one straight line can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of squares on them rational but twice the rectangle contained by them medial.

Let *AB* be the minor straight line, and let *BC* be an annex to *AB*. Then *AC* and *CB* are straight lines incommensurable in square which make the sum of the squares on them rational, but twice the rectangle contained by them medial.

I say that no other straight line can be annexed to *AB* fulfilling the same conditions.

If possible, let *BD* be so annexed. Then *AD* and *DB* are both straight lines incommensurable in square which fulfill the aforesaid conditions.

Now, since the excess of the sum of the squares on *AD* and *DB* over the sum of the squares on *AC* and *CB* is also the excess of twice the rectangle *AD* by *DB* over twice the rectangle *AC* by *CB*, while the sum of the squares on *AD* and *DB* exceed the sum of the squares on *AC* and *CB* by a rational area, for both are rational, therefore twice the rectangle *AD* by *DB* also exceeds twice the rectangle *AC* by *CB* by a rational area, which is impossible, for both are medial.

Therefore, *to a minor straight line only one straight line can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of squares on them rational but twice the rectangle contained by them medial.*

Q.E.D.