What shape did Galileo prove a projectile's trajectory follows?

Galileo proved that a projectile’s trajectory follows a parabola, assuming uniform gravity and negligible air resistance. This result appeared in his Two New Sciences, published in 1638.

His key insight was that projectile motion combines two independent movements: steady horizontal motion and uniformly accelerated vertical motion caused by gravity. When these are combined mathematically, the projectile’s height varies with the square of its horizontal distance, producing a parabolic curve.

The result was revolutionary because earlier theories often treated motion as a single, undivided process. Galileo’s analysis helped establish mathematical physics and influenced later work by Isaac Newton. The answer applies to ordinary terrestrial distances; over very long ranges, changing gravity, Earth’s curvature, and air resistance make the real path differ slightly from an ideal parabola.