If a number by multiplying itself make a cube number, it will itself also be cube.
若一个数自乘得到立方数，则该数本身也是立方数。
A 自乘得立方数 B；令 A 乘 B 得 C。三段水平堆叠：1:A = A:B = B:C，故 A、B 间有两项比例中项，A 也是立方数。
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For let the number A by multiplying itself make the cube number B; I say that A is also cube. For let A by multiplying B make C. Since, then, A by multiplying itself has made B, and by multiplying B has made C, therefore C is cube. And, since A by multiplying itself has made B, therefore A measures B according to the units in itself.
设数A自乘得立方数B，则A乘B得C。因A自乘得B，A乘B得C，故C为立方数。
But the unit also measures A according to the units in it. Therefore, as the unit is to A, so is A to B. [VII. Def. 20] And, since A by multiplying B has made C, therefore B measures C according to the units in A.
因A自乘得B，故A依自身单位数度量B；而单位亦依A中单位数度量A。故单位比A等于A比B。
But the unit also measures A according to the units in it. Therefore, as the unit is to A, so is B to C. [VII. Def. 20] But, as the unit is to A, so is A to B; therefore also, as A is to B, so is B to C. And, since B, C are cube, they are similar solid numbers.
因A乘B得C，故B依A中单位数度量C；而单位亦依A中单位数度量A。故单位比A等于B比C。
Therefore there are two mean proportional numbers between B, C. [VIII. 19] And, as B is to C, so is A to B. Therefore there are two mean proportional numbers between A, B also.
因单位比A等于A比B且等于B比C，故A比B等于B比C。又B、C为立方数，故为相似立体数，其间有二比例中项；而A比B等于B比C，故A、B间亦有二比例中项，从而A为立方数。