If an area be contained by a rational straight line and the third binomial, the “side” of the area is the irrational straight line called a second bimedial.
若一个面积由一条有理线段和第三条二项线围成，则该面积的“边”是被称为第二双中线的无理线段。
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For let the area ABCD be contained by the rational straight line AB and the third binomial AD divided into its terms at E, of which terms AE is the greater; I say that the “side” of the area AC is the irrational straight line called a second bimedial. For let the same construction be made as before. Now, since AD is a third binomial straight line, therefore AE, ED are rational straight lines commensurable in square only, the square on AE is greater than the square on ED by the square on a straight line commensurable with AE, and neither of the terms AE, ED is commensurable in length with AB.
设面积ABCD由有理线段AB和第三条二项线AD围成，AD被分为两段AE和ED，其中AE较大。
[X. Deff. II. 3] Then, in manner similar to the foregoing, we shall prove that MO is the “side” of the area AC, and MN, NO are medial straight lines commensurable in square only; so that MO is bimedial. It is next to be proved that it is also a second bimedial straight line.
由于AD是第三条二项线，故AE、ED是有理线段且仅平方可通约，AE上的正方形大于ED上的正方形一个与AE可通约的线段上的正方形，且AE、ED均不与AB长度可通约。
Since DE is incommensurable in length with AB, that is, with EK, and DE is commensurable with EF, therefore EF is incommensurable in length with EK. [X. 13] And they are rational; therefore FE, EK are rational straight lines commensurable in square only.
类似前述证明，可证MO是面积AC的“边”，且MN、NO是中线段且仅平方可通约，故MO是双中线。
Therefore EL, that is, MR, is medial. [X. 21] And it is contained by MN, NO; therefore the rectangle MN, NO is medial.
由于DE与AB（即EK）长度不可通约，而DE与EF可通约，故EF与EK长度不可通约；它们都是有理线段，故FE、EK仅平方可通约，因此EL（即MR）是中线段，且由MN、NO所成矩形是中线段。