The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number also have their sides commensurable in length. But the squares on straight lines incommensurable in length do not have to one another the ratio which a square number has to a square number; and squares which do not have to one another the ratio which a square number has to a square number also do not have their sides commensurable in length either.

Let *A* and *B* be commensurable in length.

I say that the square on *A* has to the square on *B* the ratio which a square number has to a square number.

Since *A* is commensurable in length with *B*, therefore *A* has to *B* the ratio which a number has to a number. Let it have to it the ratio which *C* has to *D*.

Since then *A* is to *B* as *C* is 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, 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, therefore the square on *A* is to the square on *B* as the square on *C* is 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*.

Since the square on *A* is to the square on *B* as the square on *C* is 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 *A* is to *B* as *C* is 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*.

Next, let *A* be incommensurable in length with *B*.

I say that the square on *A* does not have to the square on *B* the ratio which a square number has to a square number.

Above

If the square on *A* does have to the square on *B* the ratio which a square number has to a square number, then *A* is commensurable with *B*.

But it is not, therefore the square on *A* does not have to the square on *B* the ratio which a square number has to a square number.

Finally, 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*.

Above

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

But it does not, therefore *A* is not commensurable in length with *B*.

Therefore, *the squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number also have their sides commensurable in length. But the squares on straight lines incommensurable in length do not have to one another the ratio which a square number has to a square number; and squares which do not have to one another the ratio which a square number has to a square number also do not have their sides commensurable in length either.*

Q.E.D.

And it is clear 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 also commensurable in length.*

VIII.26 and converse

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, and that, if two numbers have to one another the ratio which a square number has to a square number, then they are similar plane numbers.*

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

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For, if they have, then they are similar plane numbers, which is contrary to the hypothesis. Therefore numbers which are not similar plane numbers do not have to one another the ratio which a square number has to a square number.

Propositions X.5 through X.8 stated that lines are commensurable if and only if they they had a numeric ratio, that is a ratio of a number to another. This proposition states that lines are commensurable if and only if the squares on them have a ratio of a square number to another square number.

For example, the diagonal of a square and the side of the square are not commensurable since the squares on them are in the ratio 2:1, and 2:1 is not the ratio of a square number to a square number, see the guide to proposition VIII.8.

As in the proof of X.5, again there is the assumption that numbers are magnitudes. For example, toward the beginning of the proof there is the following argument: Since the ratio of magnitudes *A* : *B* equals the ratio of numbers *C* : *D*, and the ratio of the square on *A* to the square on *B* (a ratio of two plane figures) is duplicate of the ratio *A* : *B*, and the ratio of the square on *C* to the square on *D* (a ratio of two numbers) is duplicate of the ratio *C* : *D*, therefore the ratio of the square on *A* to the square on *B* equals the ratio of the square on *C* to the square on *D*.

Such a conclusion is warranted if numbers are magnitudes, but if they are, then several of the propositions in Book V fail since they rely on the existence of fourth proportionals.

In the second stage of the proof of the proposition, the converse is argued: Since the squares are proportional, therefore *A* : *B* = *C* : *D*. A separate proposition should be supplied with a proof to justify that step.

The proposition is used repeatedly in Book X starting with the next. It is also used in Book XIII in propositions XIII.6 and XIII.11.