Commentary on Proposition VII.14¶
Back to Proposition VII.14.
$$ \begin{aligned} \text{If} \quad & a : b = d : e, \\ \text{and} \quad & b : c = e : f, \\ \text{then, } \textit{ex aequali}, \quad & a : c = d : f; \end{aligned} $$
and the same is true however many successive numbers are so related. The proof is simplicity itself.
$$ \begin{aligned} \text{By VII. 13, alternately,} \quad & a : d = b : e, \\ \text{and} \quad & b : e = c : f. \\ \text{Therefore} \quad & a : d = c : f, \\ \text{and, again alternately,} \quad & a : c = d : f. \end{aligned} $$
Observe that this simple method cannot be used to prove the corresponding proposition for magnitudes, V. 22, although V. 22 has been preceded by the two propositions in that Book corresponding to the propositions used here, viz. V. 16 and V. 11. The reason of this is that this method would only prove V. 22 for six magnitudes all of the same kind, whereas the magnitudes in V. 22 are not subject to this limitation.
Heiberg remarks in a note on VII. 19 that, while Euclid has proved several propositions of Book V. over again, by a separate proof, for numbers, he has neglected to do so in certain cases; e.g., he often uses V. 11 in these propositions of Book VII., V. 9 in VII. 19, V. 7 in the same proposition, and so on. Thus Heiberg would apparently suppose Euclid to use V. 11 in the last step of the present proof (Ratios which are the same with the same ratio are also the same with one another). I think it preferable to suppose that Euclid regarded the last step as axiomatic; since, by the definition of proportion, the first number is the same multiple or the same part or the same parts of the second that the third is of the fourth: the assumption is no more than an assumption that the numbers or proper fractions which are respectively equal to the same number or proper fraction are equal to one another.
Though the proposition is only proved of six numbers, the extension to as many as we please (as expressed in the enunciation) is obvious.