The visualization of quadratures in the mystery of corollary 3 to proposition 41 of Newton's Principia

When Newton gave his general solution to the inverse problem of central forces in Proposition 41 of Book I of his Principia he offered only one concrete example, the orbits for an inverse cube force with the initial velocity perpendicular to the line to the force center. This example is contained in Corollary 3 to Proposition 41. As it stands in the Principia, Corollary 3 is essentially inaccessible. An explanation of the corollary in a manuscript written by Newton seven years after the publication of the Principia made the corollary accessible to persons with a knowledge of Latin and with significant training in mathematics and physics. This explanation was “evidently written” at the request of the mathematician David Gregory, so Gregory qualifies as one person for whom the corollary became accessible. But the corollary remained inaccessible to most people interested in the history of science. An English translation and notes written by Turnbull in modern times made Corollary 3 much more accessible, but even this further explanation did not go into sufficient detail. The heart of the mystery of Corollary 3 concerns what we call “the visualization of quadratures.” In this paper we go beyond Turnbull's notes and provide a detailed analysis which makes minimal requirements on the technical background of readers. It is our hope that in this way this single important example of Newton's pathbreaking work on the inverse problem will be rendered accessible to the majority of historians of science