For, since A is commensurable in length with B, therefore A has to B the ratio which a number has to a number. [[X. 5](elem.10.5)]

Let it have to it the ratio which C has to D.

Since then, as A is to B, so is C to D, while the ratio of the square on A to the square on B is duplicate of the ratio of A to B, for similar figures are in the duplicate ratio of their corresponding sides; [[VI. 20, Por.](elem.6.20.p.1)] and the ratio of the square on C to the square on D is duplicate of the ratio of C to D, for between two square numbers there is one mean proportional number, and the square number has to the square number the ratio duplicate of that which the side has to the side; [[VIII. 11](elem.8.11)] therefore also, as the square on A is to the square on B, so is the square on C to the square on D.

Next, as the square on A is to the square on B, so let the square on C be to the square on D; I say that A is commensurable in length with B.

For since, as the square on A is to the square on B, so is the square on C to the square on D, while the ratio of the square on A to the square on B is duplicate of the ratio of A to B, and the ratio of the square on C to the square on D is duplicate of the ratio of C to D, therefore also, as A is to B, so is C to D.

Therefore A has to B the ratio which the number C has to the number D; therefore A is commensurable in length with B. [[X. 6](elem.10.6)]

Next, let A be incommensurable in length with B; I say that the square on A has not to the square on B the ratio which a square number has to a square number.

For, if the square on A has to the square on B the ratio which a square number has to a square number, A will be commensurable with B.

But it is not; therefore the square on A has not to the square on B the ratio which a square number has to a square number.

Again, let the square on A not have to the square on B the ratio which a square number has to a square number; I say that A is incommensurable in length with B.

For, if A is commensurable with B, the square on A will have to the square on B the ratio which a square number has to a square number.

But it has not; therefore A is not commensurable in length with B.

Therefore etc.

PORISM.
And it is manifest from what has been proved that straight lines commensurable in length are always commensurable in square also, but those commensurable in square are not always commensurable in length also.

LEMMA.
[It has been proved in the arithmetical books that similar plane numbers have to one another the ratio which a square number has to a square number, [[VIII. 26](elem.8.26)] and that, if two numbers have to one another the ratio which a square number has to a square number, they are similar plane numbers. [Converse of [VIII. 26](elem.8.26)]

And it is manifest from these propositions that numbers which are not similar plane numbers, that is, those which have not their sides proportional, have not to one another the ratio which a square number has to a square number.

For, if they have, they will be similar plane numbers: which is contrary to the hypothesis.

Therefore numbers which are not similar plane numbers have not to one another the ratio which a square number has to a square number.]