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Commentary on Proposition VII.12¶

$$ \begin{aligned} \text{If} \quad & a : a' = b : b' = c : c' = \dots, \\ \text{then each ratio is equal to} \quad & (a + b + c + \dots) : (a' + b' + c' + \dots). \end{aligned} $$

The proposition corresponds to V. 12, and the enunciation is word for word the same with that of V. 12 except that ἀριθμός takes the place of μέγεθος.

Again the proof merely connects the arithmetical definition of proportion (VII. Def. 20) with the results of VII. 5, 6, which are quoted as true for any number of numbers, and not merely for two numbers as in the enunciations of VII. 5, 6.