A straight line commensurable with that which produces with a rational area a medial whole is a straight line which produces with a rational area a medial whole.
与能产生有理面中项面的线段可公度的线段，也是能产生有理面中项面的线段。
本页以“有理面中项面线段的共度性”整体图解辅助阅读；点、线、角、圆索引已按命题文字和证明步骤校订，可与证明和问答联动。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let AB be a straight line which produces with a rational area a medial whole, and CD commensurable with AB; I say that CD is also a straight line which produces with a rational area a medial whole.
设AB为能产生有理面中项面的线段，CD与AB可公度。
For let BE be the annex to AB; therefore AE, EB are straight lines incommensurable in square which make the sum of the squares on AE, EB medial, but the rectangle contained by them rational.
取BE为AB的附加线段，则AE、EB是平方不可公度的线段，且它们的平方和是中项面，而它们所成矩形是有理面。
[X. 77] Let the same construction be made.
作同样的构造，可类似证明CF、FD与AE、EB成相同比例，且AE、EB的平方和与CF、FD的平方和可公度，矩形AE、EB与矩形CF、FD可公度。
Then we can prove, in manner similar to the foregoing, that CF, FD are in the same ratio as AE, EB, the sum of the squares on AE, EB is commensurable with the sum of the squares on CF, FD, and the rectangle AE, EB with the rectangle CF, FD; so that CF, FD are also straight lines incommensurable in square which make the sum of the squares on CF, FD medial, but the rectangle contained by them rational.
因此CF、FD也是平方不可公度的线段，且它们的平方和是中项面，而它们所成矩形是有理面。