Galileo's Theory of Motion
Galileo was to a large part responsible for the birth of the modern scientific method. He relied on careful, systematic observations and measurements from which conclusions could be drawn. His theories could be tested by others independently in experiments far away, maybe resting on different principles. If they agreed to within experimental error, that theory was supported, else more investigation was required. Many scientists prior to Galileo avoided experiment, drawing conclusions using thought and reason, though in practice, they started from what was intuitively obvious or common sense.
Galileo it was the was reported to have shown the acceleration due to gravity is the same for all bodies by dropping a one pound and a ten pound weight of the leaning tower of pisa.
They reached the ground at the same time, disproving one of the foundations of Aristotlean motion, that heavier bodies fall faster.
Galileo also brought new rigour to science with his use of maths, putting his theory of motion on a theoretical footing and this greatly supported his theories.