Commentary on Proposition VIII.8

Back to Proposition VIII.8.

[Folio 0364 | p. 358]

If $a : b = e : f$, and between $a$, $b$ there are any number of geometric means $c$, $d$, there will be as many such means between $e$, $f$.

Let $\alpha, \beta, \gamma, \dots, \delta$ be the least possible terms in the same ratio as $a$, $c$, $d, \dots b$. Then $\alpha, \delta$ are prime to one another, [VIII. 3]

and, ex aequali,

$$\begin{aligned} \alpha : \delta &= a : b\\ &= e : f. \end{aligned}$$

Therefore $e = m\alpha$, $f = m\delta$, where $m$ is some integer. [VII. 20]

Take the numbers

$$m\alpha,\quad m\beta,\quad m\gamma,\quad \dots\quad m\delta.$$

This is a series in the given ratio, and we have the same number of geometric means between $m\alpha$, $m\delta$, or $e$, $f$, that there are between $a$, $b$.