I have been confused for some while by what seems to be a contradiction between the principles of gravitation expounded by Galileo and those expounded by Newton - both of which we know to be correct, so I must be missing something here! Galileo established that two objects - regardless of differences in size, mass or shape - will, discounting air resistance , fall to the ground at the same rate. We had this principle impressively demonstrated by astronaut Dave Scott during his moon walk on the Apollo 15 mission when, taking advantage of the absence of any atmosphere on the moon, he simultaneously dropped a feather and a hammer onto the moon's surface.
On the other hand Newton's formulation of the force of gravitational attraction between any two objects states that this is proportional to the product of the mass of BOTH objects. SO it seems to me that if the strength of gravitational attraction between the ground and a hammer on the one hand and of a feather on the other, depends at least partly on the mass of those two objects, as well as the mass of the earth (or the moon) then the calculation of those forces ought to produce very different figures in each case.
Where am I going wrong here?