# The New Infinitesimal Calculus and the Leibnizian Origins of Analytical Mechanics

# The New Infinitesimal Calculus and the Leibnizian Origins of Analytical Mechanics

This chapter looks at the Leibnizian sources for analytical mechanics by first clarifying the relationship of the calculus to Newton's work in the Principia, and then to the history of analytical mechanics in France. It then narrates the history of the reception of the Leibnizian calculus in France, and the differences between it and Newton's "fluxional calculus," which he developed at the same time and suggested without actually deploying it in his Principia. Central to this story is the formation around the original Leibnizian calculus of the "Malebranche Circle – Father Nicolas Malebranche together with Johann Bernoulli, who introduced Leibniz's work into France, along with the Marquis Guillaume de l'Hôpital and Perre Varignon. This group, the book argues, exerted the key intellectual influence in originating analytical mechanics. The chapter also explores the epistemological conundrums attendant to the new infinitesimal analysis, and the seventeenth-century history of contestation over infinitesimalist mathematics overall. It concludes by positioning those in the Malebranche Circle as mathematical modernizers encouraging innovation, and figures like Newton as geometrical traditionalists suspicious of the new analysis.

*Keywords:*
Gottfried Wilhelm von Leibniz, Johann Bernoulli, Father Nicolas Malebranche, The Malebranche Circle, The Marquis Guillaume de l'Hôpital, infinitesimal analysis, differential and integral calculus, fluxional calculus, infinitesimals, synthetic geometry versus mathematical analysis