If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.
如果两个数之间有一个等比中项，那么这两个数是相似面数。
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For let one mean proportional number C fall between the two numbers A, B; I say that A, B are similar plane numbers. Let D, E, the least numbers of those which have the same ratio with A, C, be taken; [VII. 33] therefore D measures A the same number of times that E measures C. [VII. 20] Now, as many times as D measures A, so many units let there be in F; therefore F by multiplying D has made A, so that A is plane, and D, F are its sides.
设两数A、B之间有一等比中项C，取与A、C有相同比的最小数组D、E，则D量尽A的次数等于E量尽C的次数。
Again, since D, E are the least of the numbers which have the same ratio with C, B, therefore D measures C the same number of times that E measures B. [VII. 20] As many times, then, as E measures B, so many units let there be in G; therefore E measures B according to the units in G; therefore G by multiplying E has made B. Therefore B is plane, and E, G are its sides.
设D量尽A的次数为F，则F乘以D得A，故A是面数，边为D、F。
Therefore A, B are plane numbers. I say next that they are also similar. For, <*> since F by multiplying D has made A, and by multiplying E has made C, therefore, as D is to E, so is A to C, that is, C to B.
同理，因D、E与C、B有相同比，故D量尽C的次数等于E量尽B的次数，设此次数为G，则G乘以E得B，故B是面数，边为E、G。
[VII. 17] Again, <*> since E by multiplying F, G has made C, B respectively, therefore, as F is to G, so is C to B. [VII. 17] But, as C is to B, so is D to E; therefore also, as D is to E, so is F to G. And alternately, as D is to F, so is E to G.
由F乘D得A、乘E得C，得D比E等于A比C即C比B；又E乘F、G得C、B，得F比G等于C比B，故D比E等于F比G，更比得D比F等于E比G，因此A、B的边成比例，故为相似面数。