The epic journey from Kepler's laws to Newton's law of universal   gravitation revisited
  In this article, we review the main results of Volume I of Newton's Principia which relates Kepler's law of planets and universal gravitation. In order to clarify the reasoning of Newton, elementary and simple proofs are given to inspire earnest young students. All proofs are based on either the symmetry of spheres or taking great advantage of Kepler's law of area which also reflects the symmetry of centripedal forces. 
Hai-Chau Chang and Wu-Yi Hsiang