Commentary on Proposition VII.38

Back to Proposition VII.38.

[Folio 0349 | p. 343]

This proposition is practically a restatement of the preceding proposition. It asserts that, if $b$ is $\frac{1}{m}\text{th}$ part of $a$, i.e., if

$$b = \frac{1}{m} a,$$

then

$$m\text{ measures }a.$$

We have

$$b = \frac{1}{m} a,$$

and

$$1 = \frac{1}{m} m.$$

Therefore $1, m, b, a$, satisfy the enunciation of VII. 15, and therefore $m$ measures $a$ the same number of times as $1$ measures $b$, or

$$m = \frac{1}{b} a.$$