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How did Galileo describe the path of a projectile?

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Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated. — analytic geometry The conic sections result from intersecting a plane with a double cone, as shown in the figure. There are three distinct families of conic sections: the ellipse (including the circle), the parabola (with one branch), and the hyperbola (with two branches).

Galileo Galilei described the path of a projectile as a parabola.12

He reached this conclusion by analyzing the motion as a combination of two simultaneous and independent components:
* Horizontal motion: Based on his principle of inertia, he determined that a projectile maintains a constant horizontal velocity in the absence of external forces.23
* Vertical motion: He recognized that the projectile is subject to the vertical force of gravity, resulting in uniform acceleration where the vertical displacement is proportional to the square of the elapsed time ( $t^{2}$ ).23

By mathematically combining these two independent motions, Galileo proved that the resulting trajectory is parabolic. He famously documented this discovery in his 1638 work, Dialogues Concerning the Two New Sciences, where he noted that while others had observed projectiles following curved paths, he was the first to prove that the specific curve is a parabola.23