If from a rational straight line there be subtracted a rational straight line commensurable with the whole in square only, the remainder is irrational; and let it be called an apotome.
如果从一条有理线段中减去另一条有理线段，后者与整个线段仅在平方上可公度，则余线段是无理的，称为余线。
本页以“从有理线段减有理线段得余线”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。
For from the rational straight line AB let the rational straight line BC, commensurable with the whole in square only, be subtracted; I say that the remainder AC is the irrational straight line called apotome. For, since AB is incommensurable in length with BC, and, as AB is to BC, so is the square on AB to the rectangle AB, BC, therefore the square on AB is incommensurable with the rectangle AB, BC.
设AB为有理线段，从中减去有理线段BC，BC与AB仅在平方上可公度。
[X. 11] But the squares on AB, BC are commensurable with the square on AB, [X. 15] and twice the rectangle AB, BC is commensurable with the rectangle AB, BC.
由于AB与BC长度不可公度，且AB比BC等于AB上的正方形比AB、BC所成矩形，故AB上的正方形与AB、BC所成矩形不可公度。
[X. 6] And, inasmuch as the squares on AB, BC are equal to twice the rectangle AB, BC together with the square on CA, [II. 7] therefore the squares on AB, BC are also incommensurable with the remainder, the square on AC.
但AB、BC上的正方形与AB上的正方形可公度，且二倍AB、BC所成矩形与AB、BC所成矩形可公度。
[X. 13, 16] But the squares on AB, BC are rational; therefore AC is irrational.
因AB、BC上的正方形等于二倍AB、BC所成矩形加AC上的正方形，故AB、BC上的正方形与AC上的正方形不可公度；而AB、BC上的正方形是有理的，所以AC是无理的，称为余线。