Between two square numbers there is one mean proportional number, and the square has to the square the ratio duplicate of that which the side has to the side.
在两个平方数之间有一个比例中项，且第一个平方数与第二个平方数之比是它们的边之比的二次比。
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Let A, B be square numbers, and let C be the side of A, and D of B; I say that between A, B there is one mean proportional number, and A has to B the ratio duplicate of that which C has to D. For let C by multiplying D make E. Now, since A is a square and C is its side, therefore C by multiplying itself has made A.
设A、B为平方数，C为A的边，D为B的边。
For the same reason also D by multiplying itself has made B. Since then C by multiplying the numbers C, D has made A, E respectively, therefore, as C is to D, so is A to E. [VII. 17] For the same reason also, as C is to D, so is E to B.
令C乘以D得E。因为A是平方数且C是其边，故C自乘得A；同理D自乘得B。
[VII. 18] Therefore also, as A is to E, so is E to B. 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 C has to D.
由于C乘C、D得A、E，故C比D等于A比E（VII.17）；同理C比D等于E比B（VII.18）。因此A比E等于E比B，故A、B之间有一个比例中项。
For, since A, E, B are three numbers in proportion, therefore A has to B the ratio duplicate of that which A has to E. [V. Def. 9] But, as A is to E, so is C to D.
又因A、E、B成连比例，故A比B等于A比E的二次比（V.定义9），而A比E等于C比D，所以A比B等于C比D的二次比。