Commentary on Proposition VII.16¶
Back to Proposition VII.16.
This proposition proves that, if any numbers be multiplied together, the order of multiplication is indifferent, or $ab = ba$.
It is important to get a clear understanding of what Euclid means when he speaks of one number multiplying another. VII. Def. 15 states that the effect of “$a$ multiplying $b$” is taking $a$ times $b$. We shall always represent “$a$ times $b$” by $ab$ and “$b$ times $a$” by $ba$. This being premised, the proof that $ab = ba$ may be represented as follows in the language of proportions.
$$ \begin{aligned} \text{By VII. Def. 20,} \quad & 1 : a = b : ab. \\ \text{Therefore, alternately,} \quad & 1 : b = a : ab. \quad \text{[VII. 13]} \\ \text{Again, by VII. Def. 20,} \quad & 1 : b = a : ba. \\ \text{Therefore} \quad & a : ab = a : ba, \\ \text{or} \quad & ab = ba. \end{aligned} $$
Euclid does not use the language of proportions but that of fractions or their equivalent measures, quoting VII. 15, a particular case of VII. 13 differently expressed, instead of VII. 13 itself.