Commentary on Proposition VII.9

Back to Proposition VII.9.

[Folio 0316 | p. 310]

If $a = \frac{1}{n}b$ and $c = \frac{1}{n}d$, then, whatever fraction (“part” or “parts”) $a$ is of $c$, the same fraction will $b$ be of $d$.

Dividing $b$ into each of its parts equal to $a$, and $d$ into each of its parts equal to $c$, it is clear that, whatever fraction one of the parts $a$ is of one of the parts $c$, the same fraction is any other of the parts $a$ of any other of the parts $c$.

And the number of the parts $a$ is equal to the number of the parts $c$, viz. $n$.

Therefore, by VII. 5, 6, $na$ is the same fraction of $nc$ that $a$ is of $c$, i.e. $b$ is the same fraction of $d$ that $a$ is of $c$.