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

• Published in: Archive for history of exact sciences Vol. 64; no. 3; pp. 269 - 300
• Main Author: Nauenberg, M.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer 01.05.2010; Springer-Verlag
• Subjects: 18th century; Algebra; Astronomy; Calculus; Cartesian coordinates; Classic(al) mechanics; Conic sections; Curves; Differential calculus; Differentials; History; History and Philosophical Foundations of Physics; History of Mathematical Sciences; History of Science; History of science and technology; Mathematical integrals; Mathematical sciences and techniques; Mathematics; Mathematics and Statistics; Mechanics; Mechanics. Acoustics; Observations and Techniques; Philosophy of Science; Physical sciences and techniques; Spherical coordinates; Triangles; Century 18; Central Force; Conic Section; Differential Calculus; Seventeenth Century; Early Application; Differential calculus; Orbit; Bernoulli (J.); Mechanics; Varignon (P.); Isaac Newton; Principia; Leibniz (G. W.); Planetary motion; Celestial mechanics
• ISSN: 0003-9519; 1432-0657
• DOI: 10.1007/s00407-009-0056-z

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.