Galileo, who was born in 1564 (about forty years after Leonardo's death), is regarded by historians of science as the greatest thinker in the realm of motion and matter up to Newton's time. It is generally agreed that his practical demonstrations paved the way for Newton's own blend of experimental verification and mathematical integrity.
Galileo's work in this area was revolutionary because he was the first to devise repeatable experiments that showed that Aristotle's ideas were quite wrong. He is probably most famous for his use of the telescope, which destroyed the traditional ideas of how the solar system is constructed but equally important for the progress of science was his work in what became known as the science of dynamics.
Aristotle held that bodies were either intrinsically light or heavy and they fell at different velocities because of their innate tendency to seek their natural places. In 1590 the Flemish philosopher Simon Stevin had shown that light and heavy objects falling through a vacuum reach the ground simultaneously. Galileo repeated this experiment the following year (although probably not from the Leaning Tower of Pisa as tradition had it) using a cannonball and a musket-ball and showed that the two fall at equal speed if the resistance of air is ignored.
More importantly, Galileo suspected from this experiment that a falling body moves with a speed proportional to the time it has been falling. But, because the balls fall too quickly for the eye to measure their actual speed, he could not formulate a mathematical relationship between the speed of descent and the time it took. In order to find this relationship, he needed to conduct an experiment in which the speed of descent could be measured.
He quickly established that, ignoring friction, an object rolling own an inclined plane acquires the same speed as it would if it was falling vertically through the same distance. This enabled him to construct a series of experiments in which he let balls roll along inclined planes and measured the time of their journey and the speeds. This confirmed that the speed of a falling object does indeed increase with the time of the fall.
In a variation on this experiment, he allowed a ball to roll down an inclined plane and roll up another. In a further test, he allowed the ball to travel on beyond the slope along a horizontal path, where it continued steadily until slowed and eventually stopped by friction
It was these experiments that convinced Galileo that Aristotle's idea of the Unmoved Mover was false. Objects do not move because they are constantly being pushed or pulled: rather, they possess inertia - an innate tendency to move unless stopped.
This was a revolutionary notion, but his views on other questions concerning matter and energy also entitle Galileo to be seen as the first of the modernists. 
Galileo rejected Aristotle's idea of the four elements and subscribed to Democritus's atomic theory at least three decades before it began to make a reappearance in the schemes of Europe's leading thinkers, though he was unable to prove it. He also flew in the face of Aristotle's insistence that objects possess integrally all the properties we sense when we observe them, declaring:
"I feel myself impelled by necessity, as soon as I conceive a piece of matter or corporal substance, of conceiving that in its nature it is bounded and figured by such and such a figure, that in relation to others it is large or small, that it is in this or that place, in this or that time, that it is in motion or remains at rest, that it touches or does not touch another body, that it is single, few or many; in short by no imagination can a body be separated from such conditions. But that it must be white or red, bitter or sweet, sounding or mute, of a pleasant or unpleasant odour, I do not perceive my mind forced to acknowledge it accompanied by such conditions; so if the senses were not the escorts perhaps the reason or the imagination by itself would never have arrived at them. Hence I think that those tastes, odours, colours etc. on the side of the object in which they exist, are nothing else but mere names, but hold their residence solely in the sensitive body; so that if the animal were moved, every such quality would be abolished and annihilated.
So contrary to Aristotle, Galileo states categorically that there are distinct qualities of bodies. The first may be considered primary ties which are inseparable from and fundamental to the nature of the object in question - what twentieth-century scientists would ascribe to the atomic structure and chemical nature of an object. The others are secondary qualities, which are interpreted by the senses of the observer.
These revolutionary notions of Galileo's - ideas which have perhaps been swamped by his more famous discoveries in astronomy and dynamics - greatly influenced the French philosopher Rene Descartes, who for a time informed Newton's thinking on the subject of matter and the nature of the physical universe.