To an apotome only one rational straight line can be annexed which is commensurable with the whole in square only.

Let *AB* be an apotome, and *BC* an annex to it. Then *AC* and *CB* are rational straight lines commensurable in square only.

I say that no other rational straight line can be annexed to *AB* which is commensurable with the whole in square only.

If possible, let *BD* be so annexed. Then *AD* and *DB* are also rational straight lines commensurable in square only.

Now, since the excess of the sum of the squares on *AD* and *DB* over twice the rectangle *AD* by *DB* is also the excess of the sum of the squares on *AC* and *CB* over twice the rectangle *AC* by *CB*, for both exceed by the same, the square on *AB*, therefore, alternately, the excess of the sum of the squares on *AD* and *DB* over the sum of the squares on *AC* and *CB* is the excess of twice the rectangle *AD* by *DB* over twice the rectangle *AC* by *CB*.

But the sum of the squares on *AD* and *DB* exceeds 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, and a medial area does not exceeded a medial by a rational area.

Therefore no other rational straight line can be annexed to *AB* which is commensurable with the whole in square only.

Therefore only one rational straight line can be annexed to an apotome which is commensurable with the whole in square only.

Therefore, *to an apotome only one rational straight line can be annexed which is commensurable with the whole in square only.*

Q.E.D.