The force of a body moving in a circle derived from an impact

Newton imagined that a square circumscribes a circular path and that a body follows a square path inside the circle rebounding at the four points where the circle touches the outer square. From the geometry of the square he was able to compare the force of one impact, in which the component of the body’s motion perpendicular to the side it strikes is reversed, to the force of the body’s motion and then to compare the force of the four reflections the total force in one circuit, to the force of the body’s motion. He then generalized the result to polygons of increasing number of sides.