Modern light on ancient feud: Robert Hooke and Newton's graphical method

TL;DR

This paper investigates the historical dispute between Hooke and Newton over the graphical method used for planetary orbits, using modern symplectic integrator analysis to assess Hooke's potential use of Newton's method.

Contribution

It applies modern symplectic integrator theory to analyze historical claims, clarifying whether Hooke could have used Newton's graphical method to obtain elliptical orbits.

Findings

Hooke likely did not use Newton's graphical method for orbit construction.

The analytical orbit derived from Newton's method does not match Hooke's drawing.

Modern analysis challenges claims of Hooke's use of Newton's graphical approach.

Abstract

The feud between Robert Hooke and Isaac Newton has remained ongoing even after 300 years, over whether Newton should have acknowledged Hooke's influence on his graphical method of constructing planet orbits, the celebrated Proposition 1, Theorem 1 of the $Principia$. The drama has escalated in recent decades, with a claim that Hooke may have used the same method and obtained an elliptical orbit for a linear force, a feat that some considered Newton never did for the inverse-square force. Modern understanding of Newton's graphical method as a symplectic integrator can now shed light on whether this claim is creditable. This work, based on knowing the Hamiltonian of the symplectic integrator, deduced the analytical orbit corresponding to Newton's graphical construction. A detailed comparison between this analytical orbit and Hooke's drawing shows that it is unlikely that Hooke had used Newton's graphical method and obtained the correct orbit.