Commentary on Proposition VIII.10

Back to Proposition VIII.10.

[Folio 0368 | p. 362]

If there be n geometric means between $1$ and $a$, and also between $1$ and $b$, there will be n geometric means between $a$ and $b$.

The proposition is the converse of the preceding.

The n means with the extremes form two geometric series of the form

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

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

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

By multiplying the last term in the first line by the first in the second, the last but one in the first line by the second in the second, and so on, we get the series

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

and we have the n means between $a$ and $b$.

It will be observed that, when Euclid says “For the same reason also, as D is to F to G, “ the reference is really to VII. 18 instead of VII. 17.

[Folio 0369 | p. 363]

He infers namely that $D \times F : F \times F = D : F$. But since, by VII. 16, the order of multiplication is indifferent, he is practically justified in saying “for the same reason.” The same thing occurs in later propositions.