To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line.
给定一条线段A，求作两条线段，一条仅长度上与A不可公度，另一条在平方上也与A不可公度。
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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. Let two numbers B, C be set out which have not to one another the ratio which a square number has to a square number, that is, which are not similar plane numbers; and let it be contrived that, as B is to C, so is the square on A to the square on D —for we have learnt how to do this— [X. 6, Por.] therefore the square on A is commensurable with the square on D.
取两个数B和C，它们之比不是平方数之比，即不是相似平面数。
[X. 6] And, since B has not to C the ratio which a square number has to a square number, therefore neither has the square on A to the square on D the ratio which a square number has to a square number; therefore A is incommensurable in length with D.
作比B:C等于A上的正方形与D上的正方形之比，则A上的正方形与D上的正方形可公度。
[X. 9] Let E be taken a mean proportional between A, D; therefore, as A is to D, so is the square on A to the square on E.
由于B与C之比不是平方数之比，故A上的正方形与D上的正方形之比也不是平方数之比，因此A与D长度上不可公度。
[V. Def. 9] But A is incommensurable in length with D; therefore the square on A is also incommensurable with the square on E; [X. 11] therefore A is incommensurable in square with E.
取E为A与D的比例中项，则A与D之比等于A上的正方形与E上的正方形之比；因A与D长度不可公度，故A上的正方形与E上的正方形不可公度，所以A与E平方上不可公度。