Commentary on Proposition VII.37

Back to Proposition VII.37.

[Folio 0348 | p. 342]

If $b$ measures $a$, then $\frac{1}{b}\text{th}$ of $a$ is a whole number.

Let $a = m \cdot b$.

Now $m = m \cdot 1$.

Thus $1, m, b, a$ satisfy the enunciation of VII. 15; therefore $m$ measures $a$ the same number of times that $1$ measures $b$.

But $1$ is $\frac{1}{b}\text{th}$ part of $b$; therefore $m$ is $\frac{1}{b}\text{th}$ part of $a$.