Newton’s Principia and Inverse-Square Orbits

Author(s): M. Nauenburg, R. Weinstock

*The College Mathematics Journal*, Vol. 25, No. 3 (May, 1994), pp. 212-222.

Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2687650

Tags: Nonlinear differential equations, numerical methods for differential equations

Abstract. This paper details an argument in which author M. Nauenburg opposes R. Weinstock’s views about Newton’s Principia and Inverse-Square Orbits. Weinstock had published an article in 1982 in which he stated three claims: First, that Newton did not offer any proof that inverse-square forces led to conic section orbits. Second, that nowhere in Newton’s Principia was it proved that there existed a central force law requiring a particle to move in a conic section with a focus as its center. Third, that ($\gamma$), as shown, only meant that if the inverse-square orbit was a conic section, then that conic was uniquely determined by initial conditions. Nauenburg then offered proof to show that Weinstock’s views were incorrect. To refute the first claim, Nauenburg stated that the reader is meant to understand that Props. 11-13 of Newton’s Principia are special examples of an instance alluded to earlier in the works, which described the construction of the orbit of a particle given on a planar geometrical curve. To refute the second claim, Nauenburg showed that Newton proved in Prop. 2 of the Principia that such a central force did in fact exist, and later described a way to evaluate its magnitude. To destroy Weinstock’s final claim, Nauenburg revealed that Weinstock had not read the second edition of Newton’s Principia, in which Newton provided an explicit expression for the centripetal force and a uniqueness theorem for ($\gamma$). Weinstock later responded, saying that Nauenburg had misread his works, and urged readers to verify for themselves. Lance Hetrick, Stephen F. Austin State University, May 3, 2017.