If from a medial area a rational area be subtracted, there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.
若从中项面积减去一个有理面积，则余下面积的边是两种无理线段之一：要么是中项线的第一余线，要么是与有理面积合成中项全线的线段。
本页以“有理面积减中项面积得两无理线”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。
For from the medial area BC let the rational area BD be subtracted. I say that the “side” of the remainder EC becomes one of two irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.
设中项面积BC减去有理面积BD，余下面积为EC。
For let a rational straight line FG be set out, and let the areas be similarly applied. It follows then that FH is rational and incommensurable in length with FG, while KF is rational and commensurable in length with FG; therefore FH, FK are rational straight lines commensurable in square only; [X. 13] therefore KH is an apotome, and FK the annex to it.
取有理线段FG，类似地应用面积，则FH为有理且与FG长度不可公度，KF为有理且与FG长度可公度，故FH与FK是仅平方可公度的有理线段。
[X. 73] Now the square on HF is greater than the square on FK either by the square on a straight line commensurable with HF or by the square on a straight line incommensurable with it. If then the square on HF is greater than the square on FK by the square on a straight line commensurable with HF, while the annex FK is commensurable in length with the rational straight line FG set out, KH is a second apotome.
因此KH是一条余线，FK是其附加线。HF上的正方形大于FK上的正方形，要么等于与HF可公度线段上的正方形，要么等于与HF不可公度线段上的正方形。
[X. Deff. III. 2] But FG is rational; so that the “side” of LH, that is, of EC, is a first apotome of a medial straight line.
若HF上的正方形大于FK上的正方形等于与HF可公度线段上的正方形，且附加线FK与所设有理线段FG长度可公度，则KH是第二余线；由于FG是有理的，故LH（即EC）的边是中项线的第一余线。