Commentary on Proposition IX.4

Back to Proposition IX.4.

[Folio 0393 | p. 387]

The product of two cubes, say $a^3 \cdot b^3$, is a cube. For

$$a^3 : b^3 = a^3 \cdot a^3 : a^3 \cdot b^3. \hfill \text{[VII. 17]}$$

And two mean proportionals fall between $a^3$, $b^3$ which are similar solid numbers. [VIII. 19]

Therefore two mean proportionals fall between $a^3 \cdot a^3$, $a^3 \cdot b^3$. [VIII. 8]

But $a^3 \cdot a^3$ is a cube: [IX. 3]

therefore $a^3 \cdot b^3$ is a cube. [VIII. 23]