To a first apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle.

Let *AB* be a first apotome of a medial straight line, and let *KC* be an annex to *AB*. Then *AC* and *CB* are medial straight lines commensurable in square only such that the rectangle *AC* by *CB* which they contain is rational.

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

If possible, let *DB* also be so annexed. Then *AD* and *DB* are medial straight lines commensurable in square only such that the rectangle *AD* by *DB* which they contain is rational.

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 they 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 also the excess of twice the rectangle *AD* by *DB* over twice the rectangle *AC* by *CB*.

But twice the rectangle *AD* by *DB* exceeds twice the rectangle *AC* by *CB* by a rational area, for both are rational.

Therefore the sum of the squares on *AD* and *DB* also exceeds the sum of the squares on *AC* and *CB* by a rational area, which is impossible, for both are medial, and a medial area does not exceed a medial by a rational area.

Therefore, *to a first apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle.*

Q.E.D.