第9卷命题 9 · 等比数列中平方与立方传递性

And, if the number after the unit be cube, all the rest will also be cube. Let there be as many numbers as we please, A, B, C, D, E, F, beginning from an unit and in continued proportion, and let A, the number after the unit, be square; I say that all the rest will also be square. Now it has been proved that B, the third from the unit, is square, as are also all those which leave out one; [IX. 8] I say that all the rest are also square. For, since A, B, C are in continued proportion, and A is square, therefore C is also square.

设从单位开始的连比例数A, B, C, D, E, F，且A是平方数。由IX.8已证，从单位起每隔一个的数（如B、D等）是平方数。

[VIII. 22] Again, since B, C, D are in continued proportion, and B is square, D is also square. [VIII. 22] Similarly we can prove that all the rest are also square. Next, let A be cube; I say that all the rest are also cube.

由于A、B、C成连比例且A是平方数，根据VIII.22，C也是平方数。同理，B、C、D成连比例且B是平方数，则D也是平方数。类似可证其余所有数都是平方数。

Now it has been proved that C, the fourth from the unit, is cube, as also are all those which leave out two; [IX. 8] I say that all the rest are also cube. For, since, as the unit is to A, so is A to B, therefore the unit measures A the same number of times as A measures B. But the unit measures A according to the units in it; therefore A also measures B according to the units in itself; therefore A by multiplying itself has made B. And A is cube.

设A是立方数。由IX.8已证，从单位起每隔两个的数（如C、F等）是立方数。

But, if a cube number by multiplying itself make some number, the product is cube. [IX. 3] Therefore B is also cube. And, since the four numbers A, B, C, D are in continued proportion, and A is cube, D also is cube.

由于单位比A等于A比B，故单位量尽A的次数等于A量尽B的次数。单位按自身单位量尽A，故A也按自身单位量尽B，即A自乘得B。A是立方数，由IX.3，立方数自乘得立方数，故B是立方数。又A、B、C、D成连比例且A是立方数，由VIII.23，D也是立方数。类似可证其余所有数都是立方数。