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Commentary on Proposition VIII.3¶

The proof consists in merely equating the given numbers to the terms of a series found in the manner of VIII. 2.

If $a, b, c, \dots k$ ($n$ terms) be a geometrical progression in the lowest terms having a given common ratio, the terms must respectively be of the form

$$a^{n-1},\, a^{n-2}\beta,\, \dots a^2\beta^{n-3},\, a\beta^{n-2},\, \beta^{n-1}$$

found by VIII. 2, where $a : \beta$ is the ratio $a : b$ expressed in its lowest terms, so that $a, \beta$ are prime to one another [VII. 22], and hence $a^{n-1}, \beta^{n-1}$ are prime to one another [VII. 27].

But the two series must be the same, so that

$$a = a^{n-1},\quad b = \beta^{n-1}$$