On the shapes of Newton's revolving orbits

Abstract

Newton's beautiful theorem on revolving orbits is described in propositions 43 and 44 of Principia . From Motte's translation revised by Cajori (see also Chandrasekhar, who first drew our attention to this theorem). 43. It is required to make a body move in a curve that revolves about the centre of force in the same manner as another body in the same curve at rest 4. The difference of the forces by which two bodies may be made to move equally, one in a fixed, the other in the same orbit revolving varies inversely as the cubes of their common altitudes (radii). Those who read these propositions without consulting the expositions that follow them are likely to believe that Newton was considering two orbits, the first in fixed axes and the second in uniformly rotating axes. This is not the case. Indeed, that could not be so because unless the orbits were circular, the second particle would not then sweep out equal areas in equal times relative to fixed axes, so the force could not be central. Newton's subtlety lies in choosing the rate of rotation of the axes proportional to the øof the particle in the fixed orbit, where ø is the azimuth. With that construction the second particle, when seen from fixed axes, sweeps in equal times equal areas proportional to those swept by the first. A Keplerian ellipse in fixed axes will generate the same ellipse in Newton's nonuniformly rotating axes when we add an inverse cube force. But what shape will that ellipse make relative to axes rotating uniformly at the same mean rate as Newton's?