To Explain the World: The Discovery of Modern Science

by Steven Weinberg

Galileo concluded that the trajectory is a parabola. Galileo does not describe this experiment in Two New Sciences, but instead gives the theoretical argument for a parabola. The crucial point, which turned out to be essential in Newton’s mechanics, is that each component of a projectile’s motion is separately subject to the corresponding component of the force acting on the projectile. Once a projectile rolls off a table edge or is shot out of a cannon, there is nothing but air resistance to change its horizontal motion, so the horizontal distance traveled is very nearly proportional to the time elapsed. On the other hand, during the same time, like any freely falling body, the projectile is accelerated downward, so that the vertical distance fallen is proportional to the square of the time elapsed. It follows that the vertical distance fallen is proportional to the square of the horizontal distance traveled. What sort of curve has this property? Galileo shows that the path of the projectile is a parabola, using Apollonius’ definition of a parabola as the intersection of a cone with a plane parallel to the cone’s surface. (See Technical Note 26.)