For from the straight line AB let there be subtracted the straight line BC which is incommensurable in square with AB and fulfils the given conditions; [X. 34] I say that the remainder AC is the irrational straight line aforesaid. For, since the sum of the squares on AB, BC is medial, while twice the rectangle AB, BC is rational, therefore the squares on AB, BC are incommensurable with twice the rectangle AB, BC; therefore the remainder also, the square on AC, is incommensurable with twice the rectangle AB, BC. [II. 7, X. 16] And twice the rectangle AB, BC is rational; therefore the square on AC is irrational; therefore AC is irrational.
从线段AB减去与AB平方不可通约且满足给定条件的线段BC，则余线段AC是前述无理线段。
本页以“第十卷第七十七命题”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的平方和是中项线，而两倍AB与BC所成矩形是有理线，所以AB与BC的平方和与两倍AB、BC所成矩形不可通约。
根据第二卷第七命题和第十卷第十六命题，余线段AC上的正方形与两倍AB、BC所成矩形也不可通约。
两倍AB、BC所成矩形是有理线，因此AC上的正方形是无理的。
所以AC是无理线段。