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Commentary on Proposition VII.17¶

$b : c = ab : ac$.

In this case Euclid translates the language of measures into that of proportions, and the proof is exactly like that set out in the last note.

$$ \begin{aligned} \text{By VII. Def. 20,} \quad & 1 : a = b : ab, \\ \text{and} \quad & 1 : a = c : ac. \\ \text{Therefore} \quad & b : ab = c : ac, \\ \text{and, alternately,} \quad & b : c = ab : ac. \quad \text{[VII. 13]} \end{aligned} $$