If from a medial straight line there be subtracted a medial straight line which is commensurable with the whole in square only, and which contains with the whole a rational rectangle, the remainder is irrational.
如果从中项线段中减去一条与之仅平方可通约且与之围成有理矩形的中项线段，则余线段为无理线段，称为中项线的第一余线。
本页以“中项线第一余线”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。
And let it be called a first apotome of a medial straight line. For from the medial straight line AB let there be subtracted the medial straight line BC which is commensurable with AB in square only and with AB makes the rectangle AB, BC rational; I say that the remainder AC is irrational; and let it be called a first apotome of a medial straight line.
设AB和BC均为中项线段，BC与AB仅平方可通约，且矩形AB·BC为有理。
For, since AB, BC are medial, the squares on AB, BC are also medial.
由于AB和BC是中项线，它们的平方也是中项面；但二倍矩形AB·BC是有理面，因此两平方之和与二倍矩形不可通约。
But twice the rectangle AB, BC is rational; therefore the squares on AB, BC are incommensurable with twice the rectangle AB, BC; therefore twice the rectangle AB, BC is also incommensurable with the remainder, the square on AC, [Cf. II. 7] since, if the whole is incommensurable with one of the magnitudes, the original magnitudes will also be incommensurable.
根据II.7，两平方之和减去二倍矩形得AC上的平方，而整体（两平方之和）与二倍矩形不可通约，故AC上的平方与二倍矩形也不可通约。
[X. 16] But twice the rectangle AB, BC is rational; therefore the square on AC is irrational; therefore AC is irrational.
二倍矩形AB·BC是有理面，所以AC上的平方是无理面，因此AC是无理线段，称为中项线的第一余线。