Why do projectiles follow a parabolic path according to Galileo Galilei?

Galileo Galilei determined that a projectile follows a parabolic path by recognizing that its motion is composed of two independent components acting simultaneously.1

He realized that a launched body undergoes:
* Vertical motion: The object experiences constant acceleration due to gravity, following his law of falling bodies, where the distance traveled varies as the square of the time.21
* Horizontal motion: The object maintains a constant speed, consistent with his law of inertia, as no force acts upon it in this direction.1

By mathematically combining these two independent motions—assuming air resistance is neglected—Galileo demonstrated that the resulting trajectory is a parabola. This conclusion was a significant departure from Aristotelian physics, which failed to correctly describe the mechanics of such motion.134