Excel Files for Newton’s Proposition V

Abstract

Newton in Principia gives us a mathematical method of finding the center of force for a body moving on an ellipse in Proposition V, Problem I. The same thinking can be applied also to the case of a hyperbola and also a parabola, only that in the last case the center of force is at infinite distance. For the first two cases there are 3 cases of possible forces: a) An force proportional to the distance from the center. For ellipse an attractive force for hyperbola a repulsive, b) a force proportional to the inverse of the square of the distance from the left focus, for ellipse an attractive and for hyperbola a repulsive force, c) an attractive force inversely proportional to the square of the distance, inversely proportional to the square of the distance for both the ellipse and the hyperbola. This method when applied to the case of circular orbits for which we can find the center of force with the same method: Newton studied a semicircular orbit with center of force at infinite distance, and the case of a central force whose center is located on the circular orbit or inside the circle studied the case of a spiral orbit. In each the law of the force was derived by using the law of areas.