Prompted by a post by Gene Callahan's recent post, I have thought more about Galileo's thought experiment concerning the two balls (of different mass) tied to each other. Galileo attempted to prove the inconsistency of Aristotelian physics by arguing that two balls tied to each other are a composite object of greater mass than either, so they must fall faster than either constituent ball, but at the same time the lighter ball has a natural tendency to fall slowly and so it will try to slow down the heavier ball. Based on this, Galileo concludes that the model is inconsistent and so the balls must accelerate at a mass-independent rate.
This is, of course, wrong in one very important sense: In a world with air resistance things do fall at mass-dependent rates! Yet, at the same time, we know that Galileo is right about purely gravitatinal effects. So, how did he get that right, and was it a coincidence?
Well, first, he got it right about purely gravitational effects because of an embedded assumption of inertia. He is assuming that the lighter ball is trying to retard the motion of the heavier ball, which means that he's assuming that it's harder to move if you're pulling something. While that's not a fully Newtonian model, it has a general idea of inertia in there. If you assume a mass-dependent resistance to changes in motion then adding mass to an object should slow its acceleration, which can (if the dependencies are the same) cancel the mass-dependent pull of the earth.
Galileo got this right, but it was partly a coincidence because these cancellations require that mass enter into inertial and gravitational effects in the same way, so that you can divide it out. Still, he got the qualitative idea. I wouldn't recommend acceptance of his theoretical arguments at a journal today, but I give him full marks for groping toward the right physical idea and making a major leap in 17th century physics.
The other reason Galileo got this right is that he made the Aristotelian assumptions explicit while keeping inertia implicit. When your suppositions lead to a contradiction it means that at least one supposition is wrong. If you only make one assumption explicit then you'll reject that assumption, rather than concluding that there's a tension between two assumptions and we need more investigation to figure out which is wrong. Galileo may not have realized to what extent he was invoking an assumption of inertia. I'd probably need to read Two New Sciences to see how well he understood inertia.