If two medial straight lines commensurable in square only and containing a rational rectangle be added together, the whole is irrational; and let it be called a first bimedial straight line.
若两条仅平方可通约且所成矩形为有理的中项线段相加，则整体为无理线，称之为第一双中项线。
本页以“第一双中项线之无理性质”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 let two medial straight lines AB, BC commensurable in square only and containing a rational rectangle be added together; I say that the whole AC is irrational.
设AB、BC为两条仅平方可通约且所成矩形为有理的中项线段，相加得AC。
For, since AB is incommensurable in length with BC, therefore the squares on AB, BC are also incommensurable with twice the rectangle AB, BC; [cf. X. 36, ll. 9-20] and, componendo, the squares on AB, BC together with twice the rectangle AB, BC, that is, the square on AC [II. 4], is incommensurable with the rectangle AB, BC.
因AB与BC长度不可通约，故AB、BC上的正方形与二倍矩形AB·BC不可通约。
[X. 16 ] But the rectangle AB, BC is rational, for, by hypothesis, AB, BC are straight lines containing a rational rectangle; therefore the square on AC is irrational; therefore AC is irrational.
由合比，AB、BC上的正方形之和加上二倍矩形AB·BC（即AC上的正方形）与矩形AB·BC不可通约。
但矩形AB·BC为有理，故AC上的正方形为无理，因此AC为无理线。