A straight line commensurable with a major straight line is itself also major.
与一条中项线可公度的线段本身也是中项线。
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Let AB be major, and let CD be commensurable with AB; I say that CD is major. Let AB be divided at E; therefore AE, EB are straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial. [X. 39] Let the same construction be made as before. Then since, as AB is to CD, so is AE to CF, and EB to FD, therefore also, as AE is to CF, so is EB to FD.
设AB为中项线，CD与AB可公度。将AB分为AE和EB，则AE和EB是平方不可公度的线段，且它们的平方和是有理的，而它们所成矩形是中项的。
[V. 11] But AB is commensurable with CD; therefore AE, EB are also commensurable with CF, FD respectively. [X. 11] And since, as AE is to CF, so is EB to FD, alternately also, as AE is to EB, so is CF to FD; [V. 16] therefore also, componendo, as AB is to BE, so is CD to DF; [V. 18] therefore also, as the square on AB is to the square on BE, so is the square on CD to the square on DF. [VI. 20] Similarly we can prove that, as the square on AB is to the square on AE, so also is the square on CD to the square on CF.
作同样的构造。由于AB与CD之比等于AE与CF之比，也等于EB与FD之比，且AB与CD可公度，故AE与CF、EB与FD分别可公度。
Therefore also, as the square on AB is to the squares on AE, EB, so is the square on CD to the squares on CF, FD; therefore also, alternately, as the square on AB is to the square on CD, so are the squares on AE, EB to the squares on CF, FD. [V. 16] But the square on AB is commensurable with the square on CD; therefore the squares on AE, EB are also commensurable with the squares on CF, FD. And the squares on AE, EB together are rational; therefore the squares on CF, FD together are rational.
由比例关系可得，AB上的正方形与BE上的正方形之比等于CD上的正方形与DF上的正方形之比，类似地可证其他比例，从而AB上的正方形与CD上的正方形之比等于AE、EB上的正方形之和与CF、FD上的正方形之和之比。
Similarly also twice the rectangle AE, EB is commensurable with twice the rectangle CF, FD. And twice the rectangle AE, EB is medial; therefore twice the rectangle CF, FD is also medial. [X. 23, Por.] Therefore CF, FD are straight lines incommensurable in square which make, at the same time, the sum of the squares on them rational, but the rectangle contained by them medial; therefore the whole CD is the irrational straight line called major.
由于AB上的正方形与CD上的正方形可公度，故AE、EB上的正方形之和与CF、FD上的正方形之和可公度，且前者为有理，故后者也为有理。同理，二倍矩形AE、EB与二倍矩形CF、FD可公度，且前者为中项，故后者也为中项。因此CF、FD是平方不可公度的线段，且它们的平方和是有理的，而它们所成矩形是中项的，故CD是中项线。