If four straight lines be proportional, and the square on the first be greater than the square on the second by the square on a straight line commensurable with the first, the square on the third will also be greater than the square on the fourth by the square on a straight line commensurable with the third.
若四条线段成比例，且第一条线段上的正方形比第二条线段上的正方形大一个与第一条线段可公度的线段上的正方形，则第三条线段上的正方形也比第四条线段上的正方形大一个与第三条线段可公度的线段上的正方形。若第一条线段上的正方形比第二条线段上的正方形大一个与第一条线段不可公度的线段上的正方形，则第三条线段上的正方形也比第四条线段上的正方形大一个与第三条线段不可公度的线段上的正方形。
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And, if the square on the first be greater than the square on the second by the square on a straight line incommensurable with the first, the square on the third will also be greater than the square on the fourth by the square on a straight line in- commensurable with the third. Let A, B, C, D be four straight lines in proportion, so that, as A is to B, so is C to D; and let the square on A be greater than the square on B by the square on E, and let the square on C be greater than the square on D by the square on F; I say that, if A is commensurable with E, C is also commensurable with F, and, if A is incommensurable with E, C is also incommensurable with F.
设A、B、C、D成比例，即A比B等于C比D。设A上的正方形比B上的正方形大E上的正方形，C上的正方形比D上的正方形大F上的正方形。
For since, as A is to B, so is C to D, therefore also, as the square on A is to the square on B, so is the square on C to the square on D. [VI. 22] But the squares on E, B are equal to the square on A, and the squares on D, F are equal to the square on C.
由比例关系，A上的正方形比B上的正方形等于C上的正方形比D上的正方形。又A上的正方形等于E、B上的正方形之和，C上的正方形等于D、F上的正方形之和。
Therefore, as the squares on E, B are to the square on B, so are the squares on D, F to the square on D; therefore, separando, as the square on E is to the square on B, so is the square on F to the square on D; [V. 17] therefore also, as E is to B, so is F to D; [VI. 22] therefore, inversely, as B is to E, so is D to F. But, as A is to B, so also is C to D; therefore, ex aequali, as A is to E, so is C to F.
因此，E、B上的正方形之和比B上的正方形等于D、F上的正方形之和比D上的正方形。分离后得E上的正方形比B上的正方形等于F上的正方形比D上的正方形。
[V. 22] Therefore, if A is commensurable with E, C is also commensurable with F, and, if A is incommensurable with E, C is also incommensurable with F.
从而E比B等于F比D，反比得B比E等于D比F。结合A比B等于C比D，由等比得A比E等于C比F。故若A与E可公度，则C与F可公度；若A与E不可公度，则C与F不可公度。