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Commentary on Proposition VIII.2¶

To find a series of numbers in geometrical progression and in the least terms which have a given common ratio (understanding by that term the ratio of one term to the next).

Reduce the given ratio to its lowest terms, say, $a : b$. (This can be done by VII. 33.)

Then

$$a^n,\, a^{n-1}b,\, a^{n-2}b^2,\, \dots a^2 b^{n-2},\, a b^{n-1},\, b^n$$

is the required series of numbers if $(n + 1)$ terms are required.

That this is a series of terms with the given common ratio is clear from VII. 17, 18.

That the G.P. is in the smallest terms possible is proved thus.

$a, b$ are prime to one another, since the ratio $a : b$ is in its lowest terms. [VII. 22]

Therefore $a^2, b^2$ are prime to one another; so are $a^3, b^3$ and, generally, $a^n, b^n$. [VII. 27]

Whence the G.P. is in the smallest possible terms, by VIII. 1.

The Porism observes that, if there are $n$ terms in the series, the extremes are $(n - 1)\text{th}$ powers.