Commentary on Proposition VIII.15

Back to Proposition VIII.15.

If $a^3$ measures $b^3$, $a$ measures $b$; and vice versa. The proof is, mutatis mutandis, the same as for squares.

  1. $a^3$, $a^2b$, $ab^2$, $b^3$ are continuously proportional in the ratio of $a$ to $b$; and $a^3$ measures $b^3$.

Therefore $a^3$ measures $a^2b$; [VIII. 7]

and hence $a$ measures $b$.

  1. Since $a$ measures $b$, $a^3$ measures $a^2b$.

And, $a^3$, $a^2b$, $ab^2$, $b^3$ being continuously proportional, each term measures the succeeding term; therefore $a^3$ measures $b^3$.