Book 10
9
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.
9
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.
9
But it has not; therefore A is not commensurable in length with B.
9
Therefore etc.
9
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.
9
[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] 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]
9
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.
9
For, if they have, they will be similar plane numbers: which is contrary to the hypothesis.
9
Therefore numbers which are not similar plane numbers have not to one another the ratio which a square number has to a square number.]
[PROPOSITION 10.
10
To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line.
10
Let A be the assigned straight line; thus it is required to find two straight lines incommensurable, the one in length only, and the other in square also, with A.
