The visualization of quadratures in the mystery of corollary 3 to proposition 41 of Newton's \textit{Principia}
The orbits for an inverse cube force are the only concrete examples given by Newton to his general solution of the inverse problem of central forces in his Principia. Seven years later, at a request of D. Gregory, he explained his use of auxiliary central conics in this Corollary 3. This was discussed analytically by \textit{H. W. Turnbull} [The correspondence of Isaac Newton Vol. 3, 348-354, C. U. P. (1974)]. The author offers a more detailed geometrical presentation to show that the mysterious conic sections are useful in providing visualizations of the orbits.
scientific article; zbMATH DE number 636022