If an area is contained by a rational straight line and a fifth apotome, then the side of the area is a straight line which produces with a rational area a medial whole.

Let the area *AB* be contained by the rational straight line *AC* and the fifth apotome *AD*.

I say that the side of the area *AB* is a straight line which produces with a rational area a medial whole.

Let *DG* be the annex to *AD*. Then *AG* and *GD* are rational straight lines commensurable in square only, the annex *GD* is commensurable in length with the rational straight line *AC* set out, and the square on the whole *AG* is greater than the square on the annex *DG* by the square on a straight line incommensurable with *AG*.

Therefore, if there is applied to *AG* a parallelogram equal to the fourth part of the square on *DG* and deficient by a square figure, then it divides it into in commensurable parts.

Bisect *DG* at the point *E*, apply to *AG* a parallelogram equal to the square on *EG* and deficient by a square figure, and let it be the rectangle *AF* by *FG*. Then *AF* is incommensurable in length with *FG*.

Now, since *AG* is incommensurable in length with *CA*, and both are rational, therefore *AK* is medial.

Again, since *DG* is rational and commensurable in length with *AC*, therefore *DK* is rational.

Now construct the square *LM* equal to *AI*, and subtract the square *NO*, equal to *FK* and about the same angle, the angle *LPM*. Then the squares *LM* and *NO* are about the same diameter. Let *PR* be their diameter, and draw the figure.

Similarly then we can prove that *LN* is the side of the area *AB*.

I say that *LN* is the straight line which produces with a rational area a medial whole.

Since *AK* was proved medial and equals the sum of the squares on *LP* and *PN*, therefore the sum of the squares on *LP* and *PN* is medial.

Again, since *DK* is rational and equals twice the rectangle *LP* by *PN*, therefore the latter is itself also rational.

And, since *AI* is incommensurable with *FK*, therefore the square on *LP* is also incommensurable with the square on *PN*. Therefore *LP* and *PN* are straight lines incommensurable in square which make the sum of the squares on them medial but twice the rectangle contained by them rational.

Therefore the remainder *LN* is the irrational straight line called that which produces with a rational area a medial whole, and it is the side of the area *AB*.

Therefore the side of the area *AB* is a straight line which produces with a rational area a medial whole.

Therefore, *if an area is contained by a rational straight line and a fifth apotome, then the side of the area is a straight line which produces with a rational area a medial whole.*

Q.E.D.