If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second will also be cube.
若两数之比等于一个立方数比另一个立方数，且第一个数是立方数，则第二个数也是立方数。
本页以“立方数比与立方性”整体图解辅助阅读；点、线、角、圆索引已按命题文字和证明步骤校订，可与证明和问答联动。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
For let the two numbers A, B have to one another the ratio which the cube number C has to the cube number D, and let A be cube; I say that B is also cube. For, since C, D are cube, C, D are similar solid numbers.
设两数A、B之比等于立方数C比立方数D，且A是立方数。
Therefore two mean proportional numbers fall between C, D.
因C、D是立方数，故为相似立体数，由VIII.19，C、D之间有两个比例中项。
[VIII. 19] And, as many numbers as fall between C, D in continued proportion, so many will also fall between those which have the same ratio with them; [VIII. 8] so that two mean proportional numbers fall between A, B also.
由VIII.8，与C、D有相同比的两数之间，有同样多个连续比例数，故A、B之间也有两个比例中项。
Let E, F so fall.
设这两个比例中项为E、F，则A、E、F、B成连比例，且A为立方数，故B也为立方数。