For from the straight line AB let there be subtracted the straight line BC which is incommensurable in square with the whole and fulfils the given conditions. [X. 33] I say that the remainder AC is the irrational straight line called minor. For, since the sum of the squares on AB, BC is rational, while twice the rectangle AB, BC is medial, therefore the squares on AB, BC are incommensurable with twice the rectangle AB, BC; and, convertendo, the squares on AB, BC are incommensurable with the remainder, the square on AC. [II. 7, X. 16] But the squares on AB, BC are rational; therefore the square on AC is irrational; therefore AC is irrational.
从一条线段AB减去与整体平方不可公度的线段BC，且满足给定条件，则剩余线段AC是称为次线的无理线段。
本页以“第十卷第七十六命题”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。
因为AB与BC上的正方形之和是有理的，而两倍AB与BC所成矩形是中项线，所以AB与BC上的正方形之和与两倍矩形不可公度。
由转换比例，AB与BC上的正方形之和与剩余部分，即AC上的正方形，不可公度。
但AB与BC上的正方形之和是有理的，因此AC上的正方形是无理的。
所以AC是无理线段。