To a second 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 medial rectangle.

Let *AB* be a second apotome of a medial straight line and *BC* 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 medial.

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 medial rectangle.

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

Set out a rational straight line *EF*. Apply *EG*, equal to the sum of the squares on *AC* and *CB*, to *EF* producing *EM* as breadth. Subtract *HG*, equal to twice the rectangle *AC* by *CB*, producing *HM* as breadth. Then the remainder *EL* equals the square on *AB*, so that *AB* is the side of *EL*.

Again, apply *EI*, equal to the sum of the squares on *AD* and *DB*, to *EF* producing *EN* as breadth.

But *EL* also equals the square on *AB*, therefore the remainder *HI* equals twice the rectangle *AD* by *DB*.

Now, since *AC* and *CB* are medial straight lines, therefore the squares on *AC* and *CB* are also medial. And they equal *EG*, therefore *EG* is also medial.

And it is applied to the rational straight line *EF*, producing *EM* as breadth, therefore *EM* is rational and incommensurable in length with *EF*.

Again, since the rectangle *AC* by *CB* is medial, twice the rectangle *AC* by *CB* is also medial. And it equals *HG*, therefore *HG* is also medial.

And it is applied to the rational straight line *EF*, producing *HM* as breadth, therefore *HM* is also rational and incommensurable in length with *EF*.

Since *AC* and *CB* are commensurable in square only, therefore *AC* is incommensurable in length with *CB*.

But *AC* is to *CB* as the square on *AC* is to the rectangle *AC* by *CB*, therefore the square on *AC* is incommensurable with the rectangle *AC* by *CB*.

But the sum of the squares on *AC* and *CB* is commensurable with the square on *AC*, while twice the rectangle *AC* by *CB* is commensurable with the rectangle *AC* by *CB*, therefore the sum of the squares on *AC* and *CB* is incommensurable with twice the rectangle *AC* by *CB*.

And *EG* equals the sum of the squares on *AC* and *CB*, while *GH* equals twice the rectangle *AC* by *CB*, therefore *EG* is incommensurable with *HG*.

But *EG* is to *HG* as *EM* is to *HM*, therefore *EM* is incommensurable in length with *MH*.

And both are rational, therefore *EM* and *MH* are rational straight lines commensurable in square only, therefore *EH* is an apotome, and *HM* an annex to it.

Similarly we can prove that *HN* is also an annex to it. Therefore to an apotome different straight lines are annexed which are commensurable with the wholes in square only, which is impossible.

Therefore, *to a second 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 medial rectangle.*

Q.E.D.