  The hodograph, i.e. the path traced by a body in velocity space, was
introduced by Hamilton in 1846 as an alternative for studying certain dynamical
problems. The hodograph of the Kepler problem was then investigated and shown
to be a circle, it was next used to investigate some other properties of the
motion. We here propose a new method for tracing the hodograph and the
corresponding configuration space orbit in Kepler's problem starting from the
initial conditions given and trying to use no more than the methods of
synthetic geometry in a sort of Newtonian approach. All of our geometric
constructions require straight edge and compass only.
