Commentary on Proposition VIII.6

Back to Proposition VIII.6.

[Folio 0362 | p. 356]

Let $a, b, c \dots k$ be a geometrical progression in which $a$ does not measure $b$. Suppose, if possible, that $a$ measures some term of the series, as $f$. Take $x, y, z, u, v, w$ the least numbers in the ratio $a, b, c, d, e, f$. Since

$$x : y = a : b,$$

and $a$ does not measure $b$, $x$ does not measure $y$; therefore $x$ cannot be unity. And, ex aequali,

$$x : w = a : f.$$

Now $x$, $w$ are prime to one another. [VIII. 3] Therefore $a$ does not measure $f$.

We can of course prove that an intermediate term, as $b$, does not measure a later term $f$ by using the series $b, c, d, e, f$ and remembering that, since $b : c = a : b$, $b$ does not measure $c$.