Geometry, the Calculus and the Use of Limits in Newton’s Principia

Abstract

Since its publication in 1687 the Prinipia originated much debate. In particular, during the first decades of the eighteenth century the mathematical methods employed by Newton were criticised or defended by the small number of experts who could read the magnum opus with sufficient competence.1 Under its classic façade the Principia hides a panoply of mathematical methods; series, infinitesimals, quadratures, geometric limit procedures, classical theories of conic sections and higher curves, interpolation techniques, and much more. How should the science of motion be mathematized? During Newton’s lifetime this question was still unanswered. It is only in the l730s, mainly thanks to the work of Euler, that the mathematical community became convinced, at first on the Continent, that the calculus, most notably differential equations, was the appropriate language for ‘dynamics’.2 Nowadays, a student of ‘Newtonian mechanics’ will find the language used in the post-Eulerian era somewhat familiar. On the contrary, the language of the Principia, burdened by geometrical diagrams, the theory of proportions, almost devoid of symbolical expressions, leaves our student, even a tenacious one, perplexed.