Galileo and the law of inertia

I thought it was interesting because although Cohen and Einstein and Infeld agree that Galileo's work on uniform and circular motion was a significant discovery and laid the groundwork for Newton's articulation of the law of inertia, they disagree in how significant Galileo's contribution really was. Cohen is very careful to illustrate how Galileo's conception of uniform circular motion is not inertia because it needs something mediating to keep it going and Newton's idea implies a infinite universe, which Galileo disagrees with (118-125). Einstein and Infeld equate Galileo and Newton's discoveries and credit Galileo as the true discoverer of this idea (8-9). What do we think? How does this example show us the difficulty of who gets credit for what within the evolution of physics?