Commentary on Proposition V.3

Back to Proposition V.3.

[Folio 0147 | p. 141]

Heiberg remarks of the use of ex aequali in the enunciation of this proposition that, strictly speaking, it has no reference to the definition (17) of a ratio ex aequali. But the uses of the expression here and in the definition are, I think, sufficiently parallel, as may be seen thus. The proposition asserts that, if

$$na, nb \text{ are equimultiples of } a, b,$$

and if

$$m.na, m.nb \text{ are equimultiples of } na, nb,$$

then $m.na$ is the same multiple of $a$ that $m.nb$ is of $b$. Clearly the proposition can be extended by taking further equimultiples of the last equimultiples and so on; and we can prove that

$$p.q\dots m.na \text{ is the same multiple of } a \text{ that } p.q\dots m.nb \text{ is of } b,$$

where the series of numbers $p.q\dots m.n$ is exactly the same in both expressions; and ex aequali ($\delta\iota\text{'} \text{ }\ἴ\sigma\omicron\upsilon$) expresses the fact that the equimultiples are at the same distance from $a$, $b$ in the series $na$, $m.na \dots$ and $nb$, $m.nb \dots$ respectively.

[Folio 0148 | p. 142]

Here again the proof breaks $m$ into its units, and then breaks $n$ into its units; and we are practically shown that the multiple of $a$ arrived at, viz. $m.na$, is the multiple denoted by the product of the numbers $m$, $n$, i.e. the $(mn)$th multiple, or in other words that

$$m.na = mn.a.$$