What we were looking for in the problem Euclid's Algorithm and Musical
was, if you like, a 'ratio of ratios' but we were
not able to find that exactly. In that problem you are asked to
find rational approximations for the 'ratio of ratios' using
Euclid's algorithm. If the process terminates then you will have
found an exact 'ratio of ratios' but generally the process does
Here you are asked to prove that 'ratios of ratios' in this sense
are (nearly) always irrational.