Commentary on Proposition IX.3

Back to Proposition IX.3.

[Folio 0392 | p. 386]

The product of $a^3$ into itself, or $a^3 \cdot a^3$, is a cube. For

$$1 : a = a : a^2 = a^2 : a^3.$$

Therefore between $1$ and $a^3$ there are two mean proportionals. Also

$$1 : a^3 = a^3 : a^3 \cdot a^3.$$

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

(It is true that VIII. 8 is only enunciated of two pairs of numbers, but the proof is equally valid if one number of one pair is unity.) And $a^3$ is a cube number: therefore $a^3 \cdot a^3$ is also cube. [VIII. 23]