If two medial areas incommensurable with one another be added together, the remaining two irrational straight lines arise, namely either a second bimedial or a side of the sum of two medial areas.
若将两个不可公度的中项面相加，则所得之面的边要么是第二双中项线，要么是两个中项面之和的边。
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For let two medial areas AB, CD incommensurable with one another be added together; I say that the “side” of the area AD is either a second bimedial or a side of the sum of two medial areas. For AB is either greater or less than CD. First, if it so chance, let AB be greater than CD. Let the rational straight line EF be set out, and to EF let there be applied the rectangle EG equal to AB and producing EH as breadth, and the rectangle HI equal to CD and producing HK as breadth.
设两中项面AB、CD不可公度，且AB大于CD。作有理线段EF，在其上作矩形EG等于AB，宽为EH；再作矩形HI等于CD，宽为HK。
Now, since each of the areas AB, CD is medial, therefore each of the areas EG, HI is also medial. And they are applied to the rational straight line FE, producing EH, HK as breadth; therefore each of the straight lines EH, HK is rational and incommensurable in length with EF. [X. 22] And, since AB is incommensurable with CD, and AB is equal to EG, and CD to HI, therefore EG is also incommensurable with HI. But, as EG is to HI, so is EH to HK; [VI. 1] therefore EH is incommensurable in length with HK.
因AB、CD均为中项面，故EG、HI亦为中项面，且其宽EH、HK均为有理线段，但与EF长度不可公度。
[X. 11] Therefore EH, HK are rational straight lines commensurable in square only; therefore EK is binomial. [X. 36] But the square on EH is greater than the square on HK either by the square on a straight line commensurable with EH or by the square on a straight line incommensurable with it. First, let the square on it be greater by the square on a straight line commensurable in length with itself. Now neither of the straight lines EH, HK is commensurable in length with the rational straight line EF set out; therefore EK is a third binomial.
由于AB与CD不可公度，故EG与HI亦不可公度，从而EH与HK长度不可公度，因此EH、HK是仅平方可公度的有理线段，故EK为二项线。
[X. Deff. II. 3] But EF is rational; and, if an area be contained by a rational straight line and the third binomial, the “side” of the area is a second bimedial; [X. 56] therefore the “side” of EI, that is, of AD, is a second bimedial. Next, let the square on EH be greater than the square on HK by the square on a straight line incommensurable in length with EH. Now each of the straight lines EH, HK is incommensurable in length with EF; therefore EK is a sixth binomial. [X. Deff. II. 6] But, if an area be contained by a rational straight line and the sixth binomial, the “side” of the area is the side of the sum of two medial areas; [X. 59] so that the “side” of the area AD is also the side of the sum of two medial areas.
若EH上的正方形大于HK上的正方形一个与EH可公度的线段上的正方形，则EK为第三二项线，其面之边为第二双中项线；若不可公度，则EK为第六二项线，其面之边为两个中项面之和的边。