第8卷命题 21 · 两等比中项数得相似立体数

For let two mean proportional numbers C, D fall between the two numbers A, B; I say that A, B are similar solid numbers. For let three numbers E, F, G, the least of those which have the same ratio with A, C, D, be taken; [VII. 33 or VIII. 2] therefore the extremes of them E, G are prime to one another. [VIII. 3] Now, since one mean proportional number F has fallen between E, G, therefore E, G are similar plane numbers. [VIII. 20] Let, then, H, K be the sides of E, and L, M of G. Therefore it is manifest from the theorem before this that E, F, G are continuously proportional in the ratio of H to L and that of K to M.

设A、B两数之间有两个等比中项C、D。取与A、C、D同比的最小三数E、F、G，则E、G互质。

Now, since E, F, G are the least of the numbers which have the same ratio with A, C, D, and the multitude of the numbers E, F, G is equal to the multitude of the numbers A, C, D, therefore, ex aequali, as E is to G, so is A to D. [VII. 14] But E, G are prime, primes are also least, [VII. 21] and the least measure those which have the same ratio with them the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures A the same number of times that G measures D. Now, as many times as E measures A, so many units let there be in N. Therefore N by multiplying E has made A. But E is the product of H, K; therefore N by multiplying the product of H, K has made A.

因E、G之间有一个等比中项F，故E、G是相似平面数。设H、K为E的边，L、M为G的边，则E、F、G连续成比例，比例等于H比L及K比M。

Therefore A is solid, and H, K, N are its sides. Again, since E, F, G are the least of the numbers which have the same ratio as C, D, B, therefore E measures C the same number of times that G measures B. Now, as many times as E measures C, so many units let there be in O. Therefore G measures B according to the units in O; therefore O by multiplying G has made B. But G is the product of L, M; therefore O by multiplying the product of L, M has made B.

因E、F、G是与A、C、D同比的最小三数，且个数相等，故由等比，E比G等于A比D。因E、G互质且最小，故E量尽A的次数等于G量尽D的次数。设此次数为N，则N乘E得A，故A为立体数，边为H、K、N。

Therefore B is solid, and L, M, O are its sides; therefore A, B are solid. I say that they are also similar. For since N, O by multiplying E have made A, C, therefore, as N is to O, so is A to C, that is, E to F. [VII. 18] But, as E is to F, so is H to L and K to M; therefore also, as H is to L, so is K to M and N to O. And H, K, N are the sides of A, and O, L, M the sides of B.

同理，E量尽C的次数等于G量尽B的次数，设此次数为O，则O乘G得B，故B为立体数，边为L、M、O。因N比O等于A比C即E比F，而E比F等于H比L及K比M，故H比L等于K比M等于N比O，因此A、B相似。