The Mathematics and Physics of Diderot. I. On Pendulums and Air Resistance

Abstract

In this article Denis Diderot's Fifth Memoir of 1748 on the problem of a pendulum damped by air resistance is discussed. Diderot wrote the Memoir in order to clarify an assumption Newton made without further justification in the first pages of the Principia in connection with an experiment to verify the Third Law of Motion using colliding pendulums. To explain the differences between experimental and theoretical values of momentum in the collision experiments he conducted Newton assumed that the bob was retarded by an air resistance $F_R$ proportional to the velocity $v$. By giving Newton's arguments a mathematical scaffolding and recasting his geometrical reasoning in the language of differential calculus, Diderot provides a step-by-step solution guide to the problem and proposes experiments to settle the question about the appropriate form of $F_R$, which for Diderot quadratic in $v$, that is $F_R \sim v^2$. The solution of Diderot is presented in full detail and his results are compared to those obtained from a Lindstedt-Poincare approximation for an oscillator with quadratic damping. It is shown that, up to a prefactor, both coincide. Some results that one can derive from his approach are presented and discussed for the first time. Experimental evidence to support Diderot's or Newton's claims is discussed together with the limitations of their solutions. Some misprints in the original memoir are pointed out.