Commentary on Proposition VII.33

Back to Proposition VII.33.

[Folio 0341 | p. 335]

Given any numbers $a, b, c, \dots$, to find the least numbers that are in the same ratio.

Euclid’s method is the obvious one, and the result is verified by reductio ad absurdum.

We will, like Euclid, take three numbers only, $a, b, c$.

Let $g$, their greatest common measure, be found [VII. 3], and suppose that

$$a = mg, \text{ i.e. } gm, \quad \text{[VII. 16]}$$

$$b = ng, \text{ i.e. } gn,$$

$$c = pg, \text{ i.e. } gp.$$

It follows, by VII. Def. 20, that

$$m : n : p = a : b : c.$$

$m, n, p$ shall be the numbers required.

For, if not, let $x, y, z$ be the least numbers in the same ratio as $a, b, c$, being less than $m, n, p$.

Therefore

$$a = kx \text{ (or } xk \text{, VII. 16),}$$

$$b = ky \text{ (or } yk \text{),}$$

$$c = kz \text{ (or } zk \text{),} \quad \text{[VII. 20]}$$

where $k$ is some integer.

Thus

$$mg = a = xk.$$

Therefore

$$m : x = k : g. \quad \text{[VII. 19]}$$

And $m > x$; therefore $k > g$.

Since then $k$ measures $a, b, c$, it follows that $g$ is not the greatest common measure: which contradicts the hypothesis.

Therefore etc.

It is to be observed that Euclid merely supposes that $x, y, z$ are smaller numbers than $m, n, p$ in the ratio of $a, b, c$; but, in order to justify the next inference, which apparently can only depend on VII. 20, $x, y, z$ must also be assumed to be the least numbers in the ratio of $a, b, c$.

The inference from the last proportion that, since $m > x$, $k > g$ is supposed by Heiberg to depend upon VII. 13 and V. 14 together. I prefer to regard Euclid as making the inference quite independently of Book V. E.g., the proportion could just as well be written

$$x : m = g : k,$$

when the definition of proportion in Book VII. (Def. 20) gives all that we want, since, whatever proper fraction $x$ is of $m$, the same proper fraction is $g$ of $k$.