The instantaneous impulse construction as a formula for central force motion on an arbitrary plane curve with respect to an arbitrary force centre in the plane of that curve

Conclusion

We have shown that, via his instantaneous impulse construction applied to an arbitrary plane curve and arbitrary force centre in the plane of that curve, Newton had a prescription for the determination of a central force motion for a body moving on the arbitrary curve. In the Problems which followed Proposition VI Newton applied his centripetal force formula to a variety of curves and force centres. The conclusion of each problem was the determination of the centripetal force for the given curve and force centre, and the determination of the given orbit as one of the possible orbits for the centripetal force found. In the important Problem of the conic sections with the force centre at a focus, Newton determined the force to be inverse square, showed that a conic could be constructed for every initial condition (except of course where the initial velocity pointed to the force centre, in which case the body moves directly in a straight line to the force centre), and stated the uniqueness of the orbits; thus completing a proof that all the orbits of an inverse square force are conic sections.