第10卷命题 9 · 可公度线段平方比定理

But the squares on straight lines incommensurable in length have not to one another the ratio which a square number has to a square number; and squares which have not to one another the ratio which a square number has to a square number will not have their sides commensurable in length either. For let A, 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. 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] 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.] 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] 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.

设A、B长度可公度，则A与B的比等于某两数C与D的比。

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] 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.

由于相似图形面积比等于对应边比的二次比，且平方数之比等于边之比的二次比，故正方形A与B的面积比等于平方数C²与D²的比。

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.

反之，若正方形A与B的面积比等于平方数C²与D²的比，则A与B的比等于C与D的比，从而A与B长度可公度。

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] 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] 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.

若A与B长度不可公度，则正方形面积比不能等于平方数之比；反之亦然。推论：长度可公度的线段必平方可公度，但平方可公度的线段不一定长度可公度。