In this paper I concentrate on two points which are, I hope, of interest not only to musicologists but also to historians of ideas in other fields. I deal mainly with the works of the two Galileis, but of course mention the views of other musical theorists, in particular of Zarlino, since a large part of Vincenzo Galilei's writings is directed against him. This brings me to the first of my two general points: that polemical writings often present to the historian peculiar difficulties of interpretation, especially when the two adversaries, on the one hand, genuinely hate each other, and, on the other, may in fact agree on the main subject under discussion. This was, I believe, the case in the controversy between Vincenzo Galilei and Zarlino, and the resultant dishonesty and evasion of crucial problems makes very tricky the task of disentangling their true thought.