If a cube number by multiplying any number make a cube number, the multiplied number will also be cube.
若一个立方数乘以某数得到立方数，则该乘数也是立方数。
立方数 A 乘 B 得立方数 C；令 D = A·A 也是立方数。A:B = D:C；D、C 同为立方数（相似立体数），故二者间有两项比例中项，从而 A、B 间也有，B 为立方数。
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For let the cube number A by multiplying any number B make the cube number C; I say that B is cube. For let A by multiplying itself make D; therefore D is cube.
设立方数A乘以数B得立方数C，则需证B是立方数。
[IX. 3] Now, since 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.
令A自乘得D，由IX.3知D是立方数。
[VII. 17] And since D, C are cube, they are similar solid numbers. Therefore two mean proportional numbers fall between D, C.
因A自乘得D，乘以B得C，故A比B等于D比C（VII.17）。
[VIII. 19] And, as D is to C, so is A to B; therefore two mean proportional numbers fall between A, B also.
由于D、C均为立方数，它们是相似立体数，故D、C间有两个比例中项（VIII.19）；又因A比B等于D比C，故A、B间也有两个比例中项（VIII.8），从而B是立方数。