第13卷命题 6 · 中末比分割得两余线

Let AB be a rational straight line, let it be cut in extreme and mean ratio at C, and let AC be the greater segment; I say that each of the straight lines AC, CB is the irrational straight line called apotome. For let BA be produced, and let AD be made half of BA. Since then the straight line AB has been cut in extreme and mean ratio, and to the greater segment AC is added AD which is half of AB, therefore the square on CD is five times the square on DA.

设AB为有理线段，被C分成中外比，且AC为较大段。

[XIII. 1] Therefore the square on CD has to the square on DA the ratio which a number has to a number; therefore the square on CD is commensurable with the square on DA. [X. 6] But the square on DA is rational, for DA is rational, being half of AB which is rational; therefore the square on CD is also rational; [X. Def. 4] therefore CD is also rational.

延长BA，取AD为AB的一半。由XIII.1，CD上的正方形是DA上的正方形的五倍，故CD上的正方形与DA上的正方形可公度。

And, since the square on CD has not to the square on DA the ratio which a square number has to a square number, therefore CD is incommensurable in length with DA; [X. 9] therefore CD, DA are rational straight lines commensurable in square only; therefore AC is an apotome. [X. 73] Again, since AB has been cut in extreme and mean ratio, and AC is the greater segment, therefore the rectangle AB, BC is equal to the square on AC. [VI. Def. 3, VI. 17] Therefore the square on the apotome AC, if applied to the rational straight line AB, produces BC as breadth.

DA为有理，故CD上的正方形为有理，从而CD为有理。但CD上的正方形与DA上的正方形之比不是平方数之比，故CD与DA长度不可公度，仅平方可公度，因此AC为余线。

But the square on an apotome, if applied to a rational straight line, produces as breadth a first apotome; [X. 97] therefore CB is a first apotome. And CA was also proved to be an apotome.

由VI.定义3及VI.17，AB与BC所成矩形等于AC上的正方形。将AC上的正方形应用于有理线段AB，所得宽BC为第一余线；已证AC为余线。