During his continued experiments with balls rolling relatively slowly down an inclined plane, Galileo discovered a mathematical ratio between the distance (d) traversed by a falling (uniformly accelerating) ball near the surface of the Earth, and the square of the elapsed time period of such acceleration (t2). After years of trial and error experimentation, observation, and computation, Galileo was able to assert that the exact rate of free-fall (gravitational acceleration) near the surface of the Earth is as follows: every object (regardless of its mass) uniformly increases in speed (accelerates) 9.8 meters per second for every second it free-falls (Figure 4A).
During the early 17th century, Galileo turned his attention to the motions of projectiles (i.e. cannonballs and arrows in flight). Aristotle had divided the motion of objects into two categories: natural and forced. According to Aristotle, natural motion occurred spontaneously without forces, such as an apple falling from a tree. Aristotle further conjectured that forced (horizontal) motion required a “mover” or force being imposed on the object, such as a man shooting an arrow or pushing a cart. But when the force was withdrawn, the object would return to its natural state: rest on the surface of the Earth. 
During his experiments with projectiles, Galileo reversed Aristotle’s concepts of natural and forced motions. He demonstrated that the tendency for projectiles (i.e. arrows and cannonballs) to continue at a uniform motion or speed through the air even after the force is withdrawn was actually the natural motion (Figure 4B), and he referred to this tendency as inertia or inertial motion.
Galileo also demonstrated that, except for the initial lateral force, projectiles are really just objects in free-fall, and that the vertical or downward accelerated motion of objects toward the Earth is actually a forced motion (Figure 4A). Finally, Galileo concluded that this combination of independent motions of a projectile near the surface of the Earth (lateral and vertical) form the geometrical shape of a parabola (Figure 4C).
Despite their many differences, there was one thing upon which both Aristotle and Galileo did agree: that was the ancient maxim that “ignorance of motion is ignorance of nature.”