Commentary on Proposition VII.4

Back to Proposition VII.4.

The meaning of the enunciation is of course that, if $a$, $b$ be two numbers of which $b$ is the less, then $b$ is either a submultiple or some proper fraction of $a$.

  1. If $a$, $b$ are prime to one another, divide each into its units; then $b$ contains $b$ of the same parts of which $a$ contains $a$. Therefore $b$ is “parts” or a proper fraction of $a$.

  2. If $a$, $b$ be not prime to one another, either $b$ measures $a$, in which case $b$ is a submultiple or “part” of $a$, or, if $g$ be the greatest common measure of $a$, $b$, we may put $a = mg$ and $b = ng$, and $b$ will contain $n$ of the same parts ($g$) of which $a$ contains $m$, so that $b$ is again “parts,” or a proper fraction, of $a$.