Proceeding like Newton with a discrete time approach of motion and a
geometrical representation of velocity and acceleration, we obtain Kepler's
laws without solving differential equations. The difficult part of Newton's
work, when it calls for non trivial properties of ellipses, is avoided by the
introduction of polar coordinates. Then a simple reconsideration of Newton's
figure naturally leads to en explicit expression of the velocity and to the
equation of the trajectory. This derivation, which can be fully apprehended by
beginners at university (or even before) can be considered as a first
application of mechanical concepts to a physical problem of great historical
and pedagogical interest
