Commentary on Proposition VIII.12

Back to Proposition VIII.12.

[Folio 0371 | p. 365]

The cube numbers $a^3$, $b^3$ being given, Euclid forms the products $a^2b$, $ab^2$ and then proves, as usual, by means of VII. 17, 18 that

$$a^3,\quad a^2b,\quad ab^2,\quad b^3$$

are in continued proportion.

He assumes that, since $a^3$ has to $b^3$ the ratio triplicate of $a^3 : a^2b$, the ratio $a^3 : b^3$ is triplicate of the ratio $a : b$ which is equal to $a^3 : a^2b$. This is again an obvious inference from VII. 14.