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We that show two body gravitational orbits may be plotted using a radialreference 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 thedifference between an inertial force varying inversely, with the cubed radiusand the gravitational force varying with an inversed squared radius. His radialmethod was severely criticised by Newton and supporters, who preferred therectilinear inertial frame, also claiming Leibniz failed to satisfy Newton'sthird law of gravity. We show that these two approaches are equivalent if thecentral body is much larger than that in orbit. Furthermore we justify theLeibniz least action approach using the semi-latus rectum of the orbit,characteristic of the eccentricity to calculate the Newtonian centripetalacceleration opposed by the inertial acceleration, with the net radialacceleration controlling radius also allowing estimation of angulardisplacement and the action for each interval. Numerically, our Leibniz modelcan be more accurate since it requires no assumption that linear vectorsadequately simulate a curvilinear trajectory. Our review of the Newtongravitational equation using centre of mass coordinates concludes that thethird law is satisfied for symmetry if the gravitational and inertial massesare taken separately as equating force expressions for gravity and radialinertia, directed to the centre of mass for radial inertia. This symmetryvalidates centre of mass radial coordinates for more accurate plotting oforbits including dual stars, possibly applicable for better understanding ofEinstein relativity. An advantage of radial coordinates is that orbitingcouples can also accept perturbations from other nearby bodies actingcentripetally, but not inertially.

Improved numerical plotting of elliptical orbits using radial action  coordinates -- has the symmetry of Leibniz radial theory based on inertia  versus gravity been ignored?