If a rational area be applied to a rational straight line, it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied.
若将一有理面积贴于一条有理线段，则所得宽度为一条与所贴线段长度可公度的有理线段。
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For let the rational area AC be applied to AB, a straight line once more rational in any of the aforesaid ways, producing BC as breadth; I say that BC is rational and commensurable in length with BA. For on AB let the square AD be described; therefore AD is rational.
设有理面积AC贴于有理线段AB，得宽度BC。
[X. Def. 4] But AC is also rational; therefore DA is commensurable with AC.
在AB上作正方形AD，则AD为有理面积。
And, as DA is to AC, so is DB to BC. [VI. 1] Therefore DB is also commensurable with BC; [X. 11] and DB is equal to BA; therefore AB is also commensurable with BC.
因AC亦为有理，故DA与AC可公度；由VI.1，DA比AC等于DB比BC，故DB与BC可公度。
But AB is rational; therefore BC is also rational and commensurable in length with AB.
DB等于BA，故AB与BC可公度；AB为有理，故BC为有理且与AB长度可公度。