Why Galileo Performed this experiment
Galileo may have already known the answer to this experiment, but he probably did not know the theory. There is also the question among historians as to whether Galileo did perform this experiment or not. Both Drake and Naylor, two historians who studied Galileo, believe that Galileo did conduct the experiment and that it was not just a mathematical study. Naylor believes this because the values on his diagram are not nice round numbers and he would have made it easier on himself if he had used nice numbers. Plus in another one of his folios where he did only a mathematical study he used nice numbers. Galileo couldn't have done a mathematical study of falling bodies because he had no knowledge of gravitational acceleration. Drake believed he did the experiment because his numbers are very accurate and he believes he did very precise experiments. Also Galileo's notes were intended for personal use rather than for publishing.
Galileo may have just been interested in comparing curves and the symmetry (because he kept the horizontal distance constant) Galileo was familiar with parabolas and probably didn't calculate the equation, but he suspected they were parabolic and designed his experiment to test this.
What Naylor thinks Galileo did
Naylor believed that Galileo rolled a smooth metal shere down a straight smooth inclined groove. At the end of the groove the sphere falls off in a smooth cuve. For the same trajectory, he released it from the same point on the groove. To find other points on the trajectory he raised the level of the floor by putting boards or something else that he knew the height of and conducted the experiment again.
Galileo chose the horizontal distance (250 punti) and adjusted his other factors so he could keep this distance constant. Galileo probably did not care about the angle the plane made with horizontal.
What we did
We conducted the experiment much in the same way. We built a inclined plane that we could adjust the height and angle of. So instead of adjusting the "floor", we just lowered our plane.
Galileo didn't really do anything with his results, but there are equations that apply to parabolic trajectories in his notes:
AB/AC = (BL^2)/(CH^2) = (BM^2)/(CJ^2) = (BN^2)/(CK^2)
Our results were pretty close to what they whould have been. The farther the ball dropped, the better results we got (in terms of percent error). The trial where we dropped the ball 20 centimeters had the highest percent error mainly because the distances are so small that the smallest deviation from where the ball is supposed to drop will cause a large error. But our curves looked parabolic and for the most part fit the equations of parabloic motion. One of our curves was very off and does not quite fit parabolic motion, however there are many sources of error in our experiment so any of these could have caused problems.