  We that show two body gravitational orbits may be plotted using a radial
reference frame rather than the customary Newtonian rectilinear inertial frame.
Infinitesimal calculus cofounder and continental contemporary of Newton,
Leibniz claimed that the second radial derivative could be found by taking the
difference between an inertial force varying inversely, with the cubed radius
and the gravitational force varying with an inversed squared radius. His radial
method was severely criticised by Newton and supporters, who preferred the
rectilinear inertial frame, also claiming Leibniz failed to satisfy Newton's
third law of gravity. We show that these two approaches are equivalent if the
central body is much larger than that in orbit. Furthermore we justify the
Leibniz least action approach using the semi-latus rectum of the orbit,
characteristic of the eccentricity to calculate the Newtonian centripetal
acceleration opposed by the inertial acceleration, with the net radial
acceleration controlling radius also allowing estimation of angular
displacement and the action for each interval. Numerically, our Leibniz model
can be more accurate since it requires no assumption that linear vectors
adequately simulate a curvilinear trajectory. Our review of the Newton
gravitational equation using centre of mass coordinates concludes that the
third law is satisfied for symmetry if the gravitational and inertial masses
are taken separately as equating force expressions for gravity and radial
inertia, directed to the centre of mass for radial inertia. This symmetry
validates centre of mass radial coordinates for more accurate plotting of
orbits including dual stars, possibly applicable for better understanding of
Einstein relativity. An advantage of radial coordinates is that orbiting
couples can also accept perturbations from other nearby bodies acting
centripetally, but not inertially.
