and to CD let the rectangular area CE equal to the square on A be applied, producing ED as breadth; therefore ED is rational and incommensurable in length with CD. [[X. 22](elem.10.22)]
And let the rectangular area CF equal to the square on B be applied to CD, producing DF as breadth.

Since then A is commensurable with B, the square on A is also commensurable with the square on B.

But EC is equal to the square on A, and CF is equal to the square on B; therefore EC is commensurable with CF.

And, as EC is to CF, so is ED to DF; [[VI. 1](elem.6.1)] therefore ED is commensurable in length with DF. [[X. 11](elem.10.11)]

But ED is rational and incommensurable in length with DC; therefore DF is also rational [[X. Def. 3](elem.10.def.3)] and incommensurable in length with DC. [[X. 13](elem.10.13)]

Therefore CD, DF are rational and commensurable in square only.

But the straight line the square on which is equal to the rectangle contained by rational straight lines commensurable in square only is medial; [[X. 21](elem.10.21)] therefore the side of the square equal to the rectangle CD, DF is medial.

And B is the side of the square equal to the rectangle CD, DF; therefore B is medial.

PORISM.
From this it is manifest that an area commensurable with a medial area is medial.

[And in the same way as was explained in the case of rationals [[Lemma following X. 18](elem.10.18.l.1)] it follows, as regards medials, that a straight line commensurable in length with a medial straight line is called medial and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also.

But, if any straight line be commensurable in square with a medial straight line, then, if it is also commensurable in length with it, the straight lines are called, in this case too, medial and commensurable in length and in square, but, if in square only, they are called medial straight lines commensurable in square only.]