Commentary on Proposition X.15

Back to Proposition X.15.

[Folio 0043 | p. 39]

1. If $a$, $b$ be any two commensurable magnitudes, they are of the form $mc$, $nc$, where $c$ is a common measure of $a$, $b$ and $m$, $n$ some integers. It follows that

$$a + b = (m + n) c;$$

therefore $(a + b)$, being measured by $c$, is commensurable with both $a$ and $b$.

2. If $a + b$ is commensurable with either $a$ or $b$, say $a$, we may put $a + b = mc$, $a = nc$, where $c$ is a common measure of $(a + b)$, $a$, and $m$, $n$ are integers. Subtracting, we have

$$b = (m - n) c,$$

whence $b ⌢ a$.