If as many numbers as we please beginning from an unit be in continued proportion, and the number after the unit be not square, neither will any other be square except the third from the unit and all those which leave out one.
若从单位开始有任意多个数成连比例，且单位后的第一个数不是平方数，则除了第三个（从单位数起）以及每隔一个的数外，其余都不是平方数。又若单位后的第一个数不是立方数，则除了第四个以及每隔两个的数外，其余都不是立方数。
从单位开始连比例 A、B、C、D、E、F；若 A 不是平方数，则除第三项及每隔一个外无平方数；若 A 不是立方数，则除第四项及每隔两个外无立方数。
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And, if the number after the unit be not cube, neither will any other be cube except the fourth from the unit and all those which leave out two. 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, not be square; I say that neither will any other be square except the third from the unit <and those which leave out one>. For, if possible, let C be square. But B is also square; [IX. 8] [therefore B, C have to one another the ratio which a square number has to a square number]. And, as B is to C, so is A to B; therefore A, B have to one another the ratio which a square number has to a square number; [so that A, B are similar plane numbers].
设从单位开始的连比例数 A, B, C, D, E, F，且 A 不是平方数。假设 C 是平方数，则 B 也是平方数（IX.8），故 B 与 C 之比为平方数比平方数。
[VIII. 26, converse] And B is square; therefore A is also square: which is contrary to the hypothesis. Therefore C is not square. Similarly we can prove that neither is any other of the numbers square except the third from the unit and those which leave out one. Next, let A not be cube.
由于 B 比 C 等于 A 比 B，所以 A 与 B 之比也为平方数比平方数，从而 A 与 B 是相似面数（VIII.26逆）。但 B 是平方数，故 A 也是平方数，与假设矛盾，因此 C 不是平方数。类似可证除第三及每隔一个外无平方数。
I say that neither will any other be cube except the fourth from the unit and those which leave out two. For, if possible, let D be cube. Now C is also cube; for it is fourth from the unit. [IX. 8] And, as C is to D, so is B to C; therefore B also has to C the ratio which a cube has to a cube. And C is cube; therefore B is also cube.
再设 A 不是立方数。假设 D 是立方数，则 C 也是立方数（因是第四项，IX.8）。由于 C 比 D 等于 B 比 C，故 B 与 C 之比为立方数比立方数，而 C 是立方数，所以 B 也是立方数（VIII.25）。
[VIII. 25] And since, as the unit is to A, so is A to B, and 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 the cube number B. But, if a number by multiplying itself make a cube number, it is also itself cube. [IX. 6] Therefore A is also cube: which is contrary to the hypothesis. Therefore D is not cube.
因单位比 A 等于 A 比 B，且单位以 A 中的单位量 A，故 A 也以自身中的单位量 B，即 A 自乘得立方数 B。由 IX.6，若一数自乘得立方数，则该数本身也是立方数，故 A 是立方数，与假设矛盾。因此 D 不是立方数。