If an area be contained by a rational straight line and the fifth binomial, the “side” of the area is the irrational straight line called the side of a rational plus a medial area.
若一个面积由一条有理线段和第五二项线围成，则该面积的“边”是无理线段，称为有理中项面之边。
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For let the area AC be contained by the rational straight line AB and the fifth binomial AD divided into its terms at E, so that AE is the greater term; I say that the “side” of the area AC is the irrational straight line called the side of a rational plus a medial area. For let the same construction be made as before shown; it is then manifest that MO is the “side” of the area AC. It is then to be proved that MO is the side of a rational plus a medial area.
设面积AC由有理线段AB和第五二项线AD围成，AD被分为两段AE和ED，其中AE是较大段。
For, since AG is incommensurable with GE, [X. 18] therefore AH is also commensurable with HE, [VI. 1, X. 11] that is, the square on MN with the square on NO; therefore MN, NO are incommensurable in square. And, since AD is a fifth binomial straight line, and ED the lesser segment, therefore ED is commensurable in length with AB.
作与前相同的构造，则MO是面积AC的“边”。需证MO是有理中项面之边。
[X. Deff. II. 5] But AE is incommensurable with ED; therefore AB is also incommensurable in length with AE. [X. 13] Therefore AK, that is, the sum of the squares on MN, NO, is medial. [X. 21] And, since DE is commensurable in length with AB, that is, with EK, while DE is commensurable with EF, therefore EF is also commensurable with EK.
由于AG与GE不可公度，故AH与HE可公度，即MN上的正方形与NO上的正方形可公度，因此MN与NO在平方上不可公度。
[X. 12] And EK is rational; therefore EL, that is, MR, that is, the rectangle MN, NO, is also rational. [X. 19] Therefore MN, NO are straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational.
因AD是第五二项线，ED与AB长度可公度，而AE与ED不可公度，故AB与AE长度不可公度，从而MN、NO上的正方形之和为中项面；又因DE与EF可公度，故矩形MN·NO为有理面。因此MN、NO是平方不可公度且平方和为中项面、乘积为有理面的线段。