How many major thirds are there in an octave on a musical scale?

Going back in history, before the discovery of logarithms, some theorists used Euclid's algorithm to find the answer to this question.

A rational approximation ${m\over n}$ for the relationship between 5/4, the musical interval called the major third, and the octave 2/1, is given by $$ \left({5\over 4}\right)^m \approx \left({2\over 1}\right)^n, $$ where $m$ and $n$ are integers. Using Euclid's algorithm show that ${m\over n}={28\over 9}$ gives a first approximation and find three closer rational approximations.