If two similar plane numbers by multiplying one another make some number, the product will be square.
如果两个相似平面数相乘得到某个数，那么乘积是平方数。
相似平面数 A、B 以水平线段堆叠表示；C = A·B 与 D = A·A 是它们的乘积；A 与 B 之间有比例中项，故 D 与 C 之间也有比例中项，所以 C 为平方数。
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Let A, B be two similar plane numbers, and let A by multiplying B make C; I say that C is square. For let A by multiplying itself make D.
设A、B是两个相似平面数，A乘以B得C。
Therefore D is square.
令A自乘得D，则D是平方数。
Since then A by multiplying itself has made D, and by multiplying B has made C, therefore, as A is to B, so is D to C. [VII. 17] And, since A, B are similar plane numbers, therefore one mean proportional number falls between A, B.
因为A自乘得D，A乘B得C，所以A比B等于D比C（VII.17）。
[VIII. 18] But, if numbers fall between two numbers in continued proportion, as many as fall between them, so many also fall between those which have the same ratio; [VIII. 8] so that one mean proportional number falls between D, C also.
由于A、B是相似平面数，它们之间有一个比例中项（VIII.18），因此D与C之间也有一个比例中项（VIII.8），故C是平方数。
第九卷的素数和完全数传统，是数论史长卷 math-meta/topic-number-theory-history 的古希腊起点之一。