If two medial straight lines commensurable in square only and containing a medial rectangle be added together, the whole is irrational; and let it be called a second bimedial straight line.
若两条仅平方可通约且所成矩形为中项线的中项线相加，则整体为无理线，称为第二双中项线。
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For let two medial straight lines AB, BC commensurable in square only and containing a medial rectangle be added together; I say that AC is irrational. For let a rational straight line DE be set out, and let the parallelogram DF equal to the square on AC be applied to DE, producing DG as breadth. [I. 44 ] Then, since the square on AC is equal to the squares on AB, BC and twice the rectangle AB, BC, [II. 4 ] let EH, equal to the squares on AB, BC, be applied to DE; therefore the remainder HF is equal to twice the rectangle AB, BC. And, since each of the straight lines AB, BC is medial, therefore the squares on AB, BC are also medial.
设两条中项线AB、BC仅平方可通约且所成矩形为中项线，相加得AC。
But, by hypothesis, twice the rectangle AB, BC is also medial. And EH is equal to the squares on AB, BC, while FH is equal to twice the rectangle AB, BC; therefore each of the rectangle EH, HF is medial. And they are applied to the rational straight line DE; therefore each of the straight lines DH, HG is rational and incommensurable in length with DE. [X. 22 ] Since then AB is incommensurable in length with BC, and, as AB is to BC, so is the square on AB to the rectangle AB, BC, therefore the square on AB is incommensurable with the rectangle AB, BC.
取有理线DE，作矩形DF等于AC上的正方形，得宽DG。因AC上的正方形等于AB、BC上的正方形之和加上二倍矩形AB、BC，故作EH等于两正方形和，则HF等于二倍矩形。
[X. 11 ] But the sum of the squares on AB, BC is commensurable with the square on AB, [X. 15 ] and twice the rectangle AB, BC is commensurable with the rectangle AB, BC. [X. 6 ] Therefore the sum of the squares on AB, BC is incommensurable with twice the rectangle AB, BC. [X. 13 ] But EH is equal to the squares on AB, BC, and HF is equal to twice the rectangle AB, BC. Therefore EH is incommensurable with HF, so that DH is also incommensurable in length with HG.
由于AB、BC均为中项线，两正方形和及二倍矩形均为中项线，且贴于有理线DE，故DH、HG均为有理线且与DE长度不可通约。
[VI. 1 , X. 11 ] Therefore DH, HG are rational straight lines commensurable in square only; so that DG is irrational. [X. 36 ] But DE is rational; and the rectangle contained by an irrational and a rational straight line is irrational; [cf. X. 20 ] therefore the area DF is irrational, and the side of the square equal to it is irrational. [X. Def. 4 ] But AC is the side of the square equal to DF; therefore AC is irrational.
因AB与BC长度不可通约，推得两正方形和与二倍矩形不可通约，故EH与HF不可通约，从而DH与HG仅平方可通约，DG为无理线。矩形DF为无理，其等积正方形边AC为无理线。