In Aristotle's theory of motion, projectiles were pushed along by an external force which was transmitted through the air. His medieval successors internalized this force in the projectile itself and called it "impetus." This impetus caused the object to move in a straight line until it was expended, at which point the object fell straight to the ground. While objects projected through small distances may appear to behave in this manner, under closer inspection and when viewing projectiles traveling greater distances, it becomes clear that projectiles do not behave in this manner.
During the Renaissance, the focus, especially in the arts, was on representing as accurately as possible the real world whether on a 2 dimensional surface or a solid such as marble or granite. This required two things. The first was new methods for drawing or painting, e.g., perspective. The second, relevant to this topic, was careful observation.
With the spread of cannon in warfare, the study of projectile motion had taken on greater importance, and now, with more careful observation and more accurate representation, came the realization that projectiles did not move the way Aristotle and his followers had said they did: the path of a projectile did not consist of two consecutive straight line components but was instead a smooth curve. 
Now someone needed to come up with a method to determine if there was a special curve a projectile followed. But measuring the path of a projectile was not easy.
Using an inclined plane, Galileo had performed experiments on uniformly accelerated motion, and he now used the same apparatus to study projectile motion. He placed an inclined plane on a table and provided it with a curved piece at the bottom which deflected an inked bronze ball into a horizontal direction. The ball thus accelerated rolled over the table-top with uniform motion and then fell off the edge of the table Where it hit the floor, it left a small mark. The mark allowed the horizontal and vertical distances traveled by the ball to be measured. 
By varying the ball's horizontal velocity and vertical drop, Galileo was able to determine that the path of a projectile is parabolic.
A page from Galileo's notebooks, showing an experiment such as the one described here. See Stillman Drake, Galileo's Notes on Motion, monograph 5, Annali dell'Istituto e Museo di Storia della Scienza (Florence, 1979), p. 79.