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Based on Propostion 6 of his Principia, Newton's geometrical derivation inPropositions 10 and 11 for the radial dependence of the two central forces thatlead to elliptical orbits is notoriously difficult. An alternate and moretransparent derivation is obtained by applying the affine transformation of acircle into an ellipse.

A simple derivation of the two force laws for elliptic orbits from Proposition 6 in Newton's Principia