Commentary on Proposition VII.22

Back to Proposition VII.22.

[Folio 0330 | p. 324]

If $a : b$ is “in its lowest terms,” $a$, $b$ are prime to one another. Again the proof is indirect. If $a$, $b$ are not prime to one another, they have some common measure $c$, and

$$a = mc, \quad b = nc.$$

Therefore

$$m : n = a : b. \qquad \text{[VII. 17 or 18]}$$

But $m$, $n$ are less than $a$, $b$ respectively, so that $a : b$ is not in its lowest terms : which is contrary to the hypothesis.

Therefore etc.