Commentary on Proposition VIII.9

Back to Proposition VIII.9.

[Folio 0366 | p. 360]

Suppose there are $n$ geometric means between $a$, $b$, two numbers prime to one another; there are the same number ($n$) of geometric means between $1$ and $a$ and between $1$ and $b$.

If $c, d\dots$ are the $n$ means between $a$, $b$,

$$a,\quad c,\quad d\dots b$$

are the least numbers in that ratio, since $a$, $b$ are prime to one another. [VIII. 1]

The terms are therefore respectively identical with

$$\alpha^{n+1},\quad \alpha^n\beta,\quad \alpha^{n-1}\beta^2\dots \alpha\beta^n,\quad \beta^{n+1},$$

where $\alpha$, $\beta$ is the common ratio in its lowest terms. [VIII. 2, Por.]

Thus

$$a = \alpha^{n+1},\quad b = \beta^{n+1}.$$

Now

$$1 : \alpha = \alpha : \alpha^2 = \alpha^2 : \alpha^3 \dots = \alpha^n : \alpha^{n+1},$$

and

$$1 : \beta = \beta : \beta^2 = \beta^2 : \beta^3 \dots = \beta^n : \beta^{n+1};$$

whence there are $n$ geometric means between $1$, $a$, and between $1$, $b$.