第10卷命题 26 · 两中项面之差非有理面

For, if possible, let the medial area AB exceed the medial area AC by the rational area DB, and let a rational straight line EF be set out; to EF let there be applied the rectangular parallelogram FH equal to AB, producing EH as breadth, and let the rectangle FG equal to AC be subtracted; therefore the remainder BD is equal to the remainder KH. But DB is rational; therefore KH is also rational. Since, then, each of the rectangles AB, AC is medial, and AB is equal to FH, and AC to FG, therefore each of the rectangles FH, FG is also medial.

设中项面AB超过中项面AC的差为有理面DB，并取定有理线段EF。

And they are applied to the rational straight line EF; therefore each of the straight lines HE, EG is rational and incommensurable in length with EF. [X. 22] And, since [DB is rational and is equal to KH, therefore] KH is [also] rational; and it is applied to the rational straight line EF; therefore GH is rational and commensurable in length with EF. [X. 20] But EG is also rational, and is incommensurable in length with EF; therefore EG is incommensurable in length with GH.

在EF上作矩形FH等于AB，宽为EH；减去矩形FG等于AC，则余量BD等于余量KH，且KH为有理。

[X. 13] And, as EG is to GH, so is the square on EG to the rectangle EG, GH; therefore the square on EG is incommensurable with the rectangle EG, GH. [X. 11] But the squares on EG, GH are commensurable with the square on EG, for both are rational; and twice the rectangle EG, GH is commensurable with the rectangle EG, GH, for it is double of it; [X. 6] therefore the squares on EG, GH are incommensurable with twice the rectangle EG, GH; [X. 13] therefore also the sum of the squares on EG, GH and twice the rectangle EG, GH, that is, the square on EH [II. 4], is incommensurable with the squares on EG, GH. [X. 16] But the squares on EG, GH are rational; therefore the square on EH is irrational.

由于AB、AC均为中项面，故FH、FG亦为中项面，且贴于有理线段EF上，因此HE、EG均为有理且与EF长度不可公度。

[X. Def. 4] Therefore EH is irrational. But it is also rational: which is impossible.

KH为有理且贴于EF上，故GH为有理且与EF可公度；但EG与EF不可公度，故EG与GH不可公度，进而推出EH为无理，与EH为有理矛盾。