第8卷命题 12 · 两立方数间有二比例中项

Let A, B be cube numbers, and let C be the side of A, and D of B; I say that between A, B there are two mean proportional numbers, and A has to B the ratio triplicate of that which C has to D. For let C by multiplying itself make E, and by multiplying D let it make F; let D by multiplying itself make G, and let the numbers C, D by multiplying F make H, K respectively. Now, since A is a cube, and C its side, and C by multiplying itself has made E, therefore C by multiplying itself has made E and by multiplying E has made A. For the same reason also D by multiplying itself has made G and by multiplying G has made B.

设A、B为立方数，C为A的边，D为B的边。C自乘得E，C乘D得F，D自乘得G；C、D分别乘F得H、K。

And, since C by multiplying the numbers C, D has made E, F respectively, therefore, as C is to D, so is E to F. [VII. 17] For the same reason also, as C is to D, so is F to G. [VII. 18] Again, since C by multiplying the numbers E, F has made A, H respectively, therefore, as E is to F, so is A to H. [VII. 17] But, as E is to F, so is C to D.

因为A是立方数且C是其边，C自乘得E，C乘E得A；同理D自乘得G，D乘G得B。

Therefore also, as C is to D, so is A to H. Again, since the numbers C, D by multiplying F have made H, K respectively, therefore, as C is to D, so is H to K. [VII. 18] Again, since D by multiplying each of the numbers F, G has made K, B respectively, therefore, as F is to G, so is K to B. [VII. 17] But, as F is to G, so is C to D; therefore also, as C is to D, so is A to H, H to K, and K to B.

由C乘C、D得E、F，故C比D等于E比F；同理C比D等于F比G。又C乘E、F得A、H，故E比F等于A比H，结合前式得C比D等于A比H。

Therefore H, K are two mean proportionals between A, B. I say next that A also has to B the ratio triplicate of that which C has to D. For, since A, H, K, B are four numbers in proportion, therefore A has to B the ratio triplicate of that which A has to H.

由C、D乘F得H、K，故C比D等于H比K；由D乘F、G得K、B，故F比G等于K比B，结合前式得C比D等于A比H、H比K、K比B。因此H、K为A、B间的两个比例中项，且A比B等于C比D的三次比。