If an area be contained by a rational straight line and the fourth binomial, the “side” of the area is the irrational straight line called major.
若一个面积由一条有理线段和第四条二项线围成，则该面积的“边”是无理线段，称为“大线”。
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For let the area AC be contained by the rational straight line AB and the fourth binomial AD divided into its terms at E, of which terms let AE be the greater; I say that the “side” of the area AC is the irrational straight line called major. For, since AD is a fourth 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 incommensurable with AE, and AE is commensurable in length with AB. [X. Deff. II. 4] Let DE be bisected at F, and let there be applied to AE a parallelogram, the rectangle AG, GE, equal to the square on EF; therefore AG is incommensurable in length with GE.
设面积AC由有理线段AB和第四条二项线AD围成，AD被分为两段AE和ED，其中AE较大。由于AD是第四条二项线，故AE、ED是有理线段且仅平方可通约，AE上的正方形比ED上的正方形大一个与AE不可通约的线段上的正方形，且AE与AB长度可通约。
[X. 18] Let GH, EK, FL be drawn parallel to AB, and let the rest of the construction be as before; it is then manifest that MO is the “side” of the area AC. It is next to be proved that MO is the irrational straight line called major. Since AG is incommensurable with EG, AH is also incommensurable with GK, that is, SN with NQ; [VI. 1, X. 11] therefore MN, NO are incommensurable in square.
取ED的中点F，在AE上作平行四边形AG·GE等于EF上的正方形，则AG与GE长度不可通约。作GH、EK、FL平行于AB，如前构造，则MO是面积AC的“边”。
And, since AE is commensurable with AB, AK is rational; [X. 19] and it is equal to the squares on MN, NO; therefore the sum of the squares on MN, NO is also rational. And, since DE is incommensurable in length with AB, that is, with EK, while DE is commensurable with EF, therefore EF is incommensurable in length with EK. [X. 13] Therefore EK, EF are rational straight lines commensurable in square only; therefore LE, that is, MR, is medial.
由于AG与EG不可通约，故AH与GK也不可通约，即SN与NQ不可通约，因此MN与NO平方不可通约。又因AE与AB可通约，故AK为有理线段，且等于MN、NO上的正方形之和，因此MN、NO上的正方形之和为有理。
[X. 21] And it is contained by MN, NO; therefore the rectangle MN, NO is medial. And the [sum] of the squares on MN, NO is rational, and MN, NO are incommensurable in square. But, if two straight lines incommensurable in square and making the sum of the squares on them rational, but the rectangle contained by them medial, be added together, the whole is irrational and is called major.
由于DE与AB长度不可通约，即与EK不可通约，而DE与EF可通约，故EF与EK长度不可通约，因此EK、EF是有理线段且仅平方可通约，故LE（即MR）为中项线，且等于MN·NO，因此矩形MN·NO为中项面。而MN、NO上的正方形之和为有理，且MN、NO平方不可通约，故由X.39，MN与NO之和为无理线段，称为“大线”。