Commentary on Proposition IX.5¶

If the product $a^3b$ is a cube number, $b$ is cube. By IX. 3, the product $a^3 \cdot a^3$ is a cube. And

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

The first two terms are cubes, and therefore “similar solids”; therefore there are two mean proportionals between them. [VIII. 19]

Therefore there are two mean proportionals between $a^3$, $b$. [VIII. 8]

And $a^3$ is a cube: therefore $b$ is a cube number. [VIII. 23]