If as many numbers as we please beginning from an unit be in continued proportion, the third from the unit will be square, as will also those which successively leave out one; the fourth will be cube, as will also all those which leave out two; and the seventh will be at once cube and square, as will also those which leave out five.
若从单位开始任意多个数成连比例，则从单位起第三个数是平方数，且所有每隔一个的数也是平方数；第四个数是立方数，且所有每隔两个的数也是立方数；第七个数同时是平方数和立方数，且所有每隔五个的数也是如此。
从单位开始连比例 A、B、C、D、E、F；B 为第三项是平方数，C 为第四项是立方数，F 为第七项同时为平方与立方。
正文图形由校订坐标生成；点、线、角、圆可与证明和问答联动。
Let there be as many numbers as we please, A, B, C, D, E, F, beginning from an unit and in continued proportion; I say that B, the third from the unit, is square, as are also all those which leave out one; C, the fourth, is cube, as are also all those which leave out two; and F, the seventh, is at once cube and square, as are also all those which leave out five. For since, as the unit is to A, so is A to B, therefore the unit measures the number A the same number of times that A measures B. [VII. Def. 20] But the unit measures the number A according to the units in it; therefore A also measures B according to the units in A. Therefore A by multiplying itself has made B; therefore B is square.
设从单位开始的连比例数为A, B, C, D, E, F。因为单位比A等于A比B，所以单位量尽A的次数等于A量尽B的次数，而单位按A中的单位量尽A，故A按自身单位量尽B，即A自乘得B，因此B是平方数。
And, since B, C, D are in continued proportion, and B is square, therefore D is also square. [VIII. 22] For the same reason F is also square. Similarly we can prove that all those which leave out one are square.
由于B, C, D成连比例且B是平方数，根据第八卷命题22，D也是平方数。同理可证所有每隔一个的数都是平方数。
I say next that C, the fourth from the unit, is cube, as are also all those which leave out two. For since, as the unit is to A, so is B to C, therefore the unit measures the number A the same number of times that B measures C. But the unit measures the number A according to the units in A; therefore B also measures C according to the units in A. Therefore A by multiplying B has made C.
因为单位比A等于B比C，所以单位量尽A的次数等于B量尽C的次数，而单位按A中的单位量尽A，故B按A中的单位量尽C，即A乘B得C。又A自乘得B，故A乘B得C，因此C是立方数。
Since then A by multiplying itself has made B, and by multiplying B has made C, therefore C is cube. And, since C, D, E, F are in continued proportion, and C is cube, therefore F is also cube. [VIII. 23] But it was also proved square; therefore the seventh from the unit is both cube and square.
由于C, D, E, F成连比例且C是立方数，根据第八卷命题23，F也是立方数。又已证F是平方数，故从单位起第七个数同时是平方数和立方数。