If there be as many numbers as we please in continued proportion, and the extremes of them be prime to one another, the numbers are the least of those which have the same ratio with them.
Proof
Suppose a smaller set b1,…,bn in the same ratio existed.
By ex aequali (VII.14) the ratio of extremes b1:bn equals a1:an, so by VII.21 (least in a ratio are coprime) the original
a1, an would not be coprime — contradiction.