If a number by multiplying itself make a cube number, it will itself also be cube.

Ἐὰν ἀριθμὸς ἑαυτὸν πολλαπλασιάσας κύβον ποιῇ, καὶ αὐτὸς κύβος ἔσται. Ἀριθμὸς γὰρ ὁ Α ἑαυτὸν πολλαπλασιάσας κύβον τὸν Β ποιείτω: λέγω, ὅτι καὶ ὁ Α κύβος ἐστίν. | If a number by multiplying itself make a cube number, it will itself also be cube. 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. 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. 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. 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. [VIII. 8] |