第10卷命题 23 · 与中项线可公度的线段是中项线

Let A be medial, and let B be commensurable with A; I say that B is also medial. For let a rational straight line CD be set out, and to CD let the rectangular area CE equal to the square on A be applied, producing ED as breadth; therefore ED is rational and incommensurable in length with CD. [X. 22] And let the rectangular area CF equal to the square on B be applied to CD, producing DF as breadth. Since then A is commensurable with B, the square on A is also commensurable with the square on B.

设A是中项线，B与A可公度。取有理线段CD，在CD上作矩形CE等于A上的正方形，得宽ED；则ED是有理线且与CD长度不可公度。

But EC is equal to the square on A, and CF is equal to the square on B; therefore EC is commensurable with CF. And, as EC is to CF, so is ED to DF; [VI. 1] therefore ED is commensurable in length with DF. [X. 11] But ED is rational and incommensurable in length with DC; therefore DF is also rational [X. Def. 3] and incommensurable in length with DC.

在CD上作矩形CF等于B上的正方形，得宽DF。因A与B可公度，故A上的正方形与B上的正方形可公度；又EC等于A上的正方形，CF等于B上的正方形，故EC与CF可公度。

[X. 13] Therefore CD, DF are rational and commensurable in square only. But the straight line the square on which is equal to the rectangle contained by rational straight lines commensurable in square only is medial; [X. 21] therefore the side of the square equal to the rectangle CD, DF is medial. And B is the side of the square equal to the rectangle CD, DF; therefore B is medial.

由EC比CF等于ED比DF，得ED与DF长度可公度。但ED是有理线且与DC长度不可公度，故DF也是有理线且与DC长度不可公度。

PORISM. From this it is manifest that an area commensurable with a medial area is medial. [And in the same way as was explained in the case of rationals [Lemma following X. 18] it follows, as regards medials, that a straight line commensurable in length with a medial straight line is called medial and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also.

因此CD、DF是有理线且仅平方可公度。而等于仅平方可公度的有理线所成矩形的正方形边是中项线，故B是中项线。