第10卷命题 89 · 求第五种余线

Let a rational straight line A be set out, and let CG be commensurable in length with A; therefore CG is rational. Let two numbers DF, FE be set out such that DE again has not to either of the numbers DF, FE the ratio which a square number has to a square number; and let it be contrived that, as FE is to ED, so is the square on CG to the square on GB. Therefore the square on GB is also rational; [X. 6] therefore BG is also rational.

设有理线段A，作CG与A长度可公度，故CG为有理。

Now since, as DE is to EF, so is the square on BG to the square on GC, while DE has not to EF the ratio which a square number has to a square number, therefore neither has the square on BG to the square on GC the ratio which a square number has to a square number; therefore BG is incommensurable in length with GC. [X. 9] And both are rational; therefore BG, GC are rational straight lines commensurable in square only; therefore BC is an apotome. [X. 73] I say next that it is also a fifth apotome.

取两数DF、FE，使DE与DF、FE均无平方数比，并令FE:ED = CG²:GB²，则GB²为有理，故GB亦为有理。

For let the square on H be that by which the square on BG is greater than the square on GC. Since then, as the square on BG is to the square on GC, so is DE to EF, therefore, convertendo, as ED is to DF, so is the square on BG to the square on H. [V. 19, Por.] But ED has not to DF the ratio which a square number has to a square number; therefore neither has the square on BG to the square on H the ratio which a square number has to a square number; therefore BG is incommensurable in length with H.

因DE:EF = BG²:GC²，且DE与EF无平方数比，故BG与GC长度不可公度，仅平方可公度，故BC为余线。

[X. 9] And the square on BG is greater than the square on GC by the square on H; therefore the square on GB is greater than the square on GC by the square on a straight line incommensurable in length with GB. And the annex CG is commensurable in length with the rational straight line A set out; therefore BC is a fifth apotome.

作H使BG² - GC² = H²，由比例反换得ED:DF = BG²:H²，因ED与DF无平方数比，故BG与H长度不可公度；且BG²大于GC²的量为与BG不可公度的H²，而附线CG与有理线段A长度可公度，故BC为第五种余线。