Lost in translation? Reading Newton on inverse-cube trajectories

This paper examines an annotation in Newton’s hand found by H. W. Turnbull in David Gregory’s papers in the Library of the Royal Society (London). It will be shown that Gregory asked Newton to explain to him how the trajectories of a body accelerated by an inverse-cube force are determined in a corollary in the Principia: an important topic for gravitation theory, since tidal forces are inverse cube. This annotation opens a window on the more hidden mathematical methods which Newton deployed in his magnum opus. The received view according to which the Principia are written in a geometric style with no help from calculus techniques must be revised.