The early application of the calculus to the inverse square force problem

Abstract

The translation of Newton's geometrical Propositions in the Principia into the language of the differential calculus in the form developed by Leibniz and his followers has been the subject of many scholarly articles and books. One of the most vexing problems in this translation concerns the transition from the discrete polygonal orbits and force impulses in Prop. 1 to the continuous orbits and forces in Prop. 6. Newton justified this transition by lemma 1 on prime and ultimate ratios which was a concrete formulation of a limit, but it took another century before this concept was established on a rigorous mathematical basis. This difficulty was mirrored in the newly developed calculus which dealt with differentials that vanish in this limit, and therefore were considered to be fictional quantities by some mathematicians. Despite these problems, early practitioners of the differential calculus like Jacob Hermann, Pierre Varignon, and Johann Bernoulli succeeded without apparent difficulties in applying the differential calculus to the solution of the fundamental problem of orbital motion under the action of inverse square central forces. By following their calculations and describing some essential details that have been ignored in the past, I clarify the reason why the lack of rigor in establishing the continuum limit was not a practical problem.

Barrow and Leibniz on the fundamental theorem of the calculus

“…But the assignable triangle T BC consists of the axis, the ordinate, and the tangent, and therefore contains the angle between the direction of the curve (or its tangent) and the axis or base, that is, the inclination of the curve at a given point C. (Struik 1969, 283) The tangent line to the curve C(C) at C is T C, and GL and GC are sides of its "characteristic triangle," GLC, satisfying a "given mutual relation" specified below. But this triangle which Leibniz called inassignable 26 , does not appear in the proof of Leibniz's theorem, while another characteristic triangle, CE C, where C (not indicated in Leibniz's diagram, Fig. 2, is the intersection of the tangent line T C with the extension of (H)(F ), turns out to be relevant to Leibniz's proof.…”

“…1, with Newton's diagram for his earliest geometrical proof of the fundamental theorem of the calculus (Guicciardini, 2009, 184). 26 Leibniz chose the Latin word inassignabilis for the characteristic triangle, because its sides are differentials which do not have an assignable magnitude. this law of tangency which he formulated, instead, in terms of the subtangent T F , where T (not shown in Fig.…”

Lost in translation? Reading Newton on inverse-cube trajectories

“…The quadrature of the circle that we have just reviewed is of great importance for the treatment of inverse-cube orbits, as is apparent if we consider the integrals (16). Further, Newton could extend the quadrature of the circle to the unit hyperbola…”

A simple derivation of the two force laws for elliptic orbits from Proposition 6 in Newton's Principia

“…C Maxwell [5] who in turn attributed it to Sir William Hamilton. At the start of 18th century, however, able mathematicians like Jacob Hermann, Pierre Varignon and Johann Bernoulli were able to express Newton's relation for central force in Proposition 6 as a differential equation for the orbit in Cartesian coordinates, while Gotfried Leibniz obtained such an equation in polar coordinates [6].…”

“…Substituting cosθ = (r/a)cosθ + , and r/a = (1 − 2 )/(1 + cosθ) for the equation of the ellipse in polar coordinates, Eq. 4, one finds (6) δθ = λa r δθ…”