第8卷命题 18 · 相似平面数间有单中项

Let A, B be two similar plane numbers, and let the numbers C, D be the sides of A, and E, F of B. Now, since similar plane numbers are those which have their sides proportional, [VII. Def. 21] therefore, as C is to D, so is E to F. I say then that between A, B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to E, or D to F, that is, of that which the corresponding side has to the corresponding side. Now since, as C is to D, so is E to F, therefore, alternately, as C is to E, so is D to F.

设A、B为两个相似平面数，C、D为A的边，E、F为B的边。由相似平面数定义，C比D等于E比F。

[VII. 13] And, since A is plane, and C, D are its sides, therefore D by multiplying C has made A. For the same reason also E by multiplying F has made B. Now let D by multiplying E make G. Then, since D by multiplying C has made A, and by multiplying E has made G, therefore, as C is to E, so is A to G.

由比例交替性质，C比E等于D比F。因A由D乘C得，B由E乘F得，令D乘E得G。

[VII. 17] But, as C is to E, so is D to F; therefore also, as D is to F, so is A to G. Again, since E by multiplying D has made G, and by multiplying F has made B, therefore, as D is to F, so is G to B. [VII. 17] But it was also proved that, as D is to F, so is A to G; therefore also, as A is to G, so is G to B. Therefore A, G, B are in continued proportion.

由D乘C得A、乘E得G，得C比E等于A比G；又C比E等于D比F，故D比F等于A比G。

Therefore between A, B there is one mean proportional number. I say next that A also has to B the ratio duplicate of that which the corresponding side has to the corresponding side, that is, of that which C has to E or D to F. For, since A, G, B are in continued proportion, A has to B the ratio duplicate of that which it has to G. [V. Def. 9] And, as A is to G, so is C to E, and so is D to F.

由E乘D得G、乘F得B，得D比F等于G比B；结合前步得A比G等于G比B，故A、G、B成连比例。因此A与B之比等于C与E之比的二次比。