Did Newton use his calculus in the Principia?

Abstract

A question that is often formulated by people interested in the history of mathematics is: Did Newton use his calculus in the Principia? This question comes very naturally to mind, since Newton discovered the calculus of fluxions before writing the Principia. It is just obvious to think that Newton had employed the calculus in order to mathematize his natural philosophy. However, very little trace of calculus techniques is to be found in the Principia, which are mostly written in ‘geometric style’. On the other hand, some propositions of the Principia are framed in a geometric language which appears to be very easily translatable into calculus concepts. Thus our question is very tricky. As a matter of fact, the question of the presence of calculus in the Principia has been debated since the 169Os, when Fontenelle stated that almost the whole work is about the differential calculus. The question played an important role in the muddled context of the priority dispute with Leibniz. Since then the opinions of mathematicians and Newtonian scholars have been very contradictory and our question seems still to be waiting for a definite answer. In order to achieve an understanding of Newton's use of calculus in his mugnum opus, we have to consider the exchange of information between a restricted group of adherents to the Newtonian school. By analyzing a group of manuscripts written by Newton and by some of his disciples (D. Gregory, Keill and Cotes), it is possible to show that in some propositions of the Principiu Newton did employ calculus techniques. More precisely it can be shown that he employed his highly algorithmic techniques for squaring curves. The answer to our question must be a resounding yes. The fact that Newton and his disciples knew how to apply the calculus to the science of motion and force implies a new question. Why did they keep the fluxional equations of motion almost entirely hidden? Their attitude is very puzzling, since the application of calculus to the science of motion appears to us as one of the main trends of the development of eighteenth century mathematics. In fact the Leibnizians pursued with enthusiasm a programme of application of the differential and integral calculus to dynamics, giving great publicity to it. On the contrary, the Newtonians, when dealing with natural philosophy, preferred to keep fluxions at the level of private exchange. In the last part of this paper I will address this new question, focusing on the different validation criteria active within the two schools. I will show that different values, expectations and research priorities determined a different approach towards the use of calculus methods in natural philosophy, and consequently an adoption in the two schools of different policies of publication.

Justificativas para o sucesso da matemática na descrição da natureza como demanda epistêmica no ensino de física

“…4 Nos Principia Newton utilizou geometria plana e cônica, embora em suas deduções ideias de infinitésimos estejam presentes. O uso do cálculo infinitesimal só passou a ser amplamente usado em mecânica ao longo da primeira metade do século 18 (GUICCIARDINI, 1988). Émilie du Châtelet fez a primeira tradução completa dos três livros dos Principia para o francês, incluindo uma seção de comentários, onde fundiu os três livros em um resumo de fácil compreensão para os estudiosos da época.…”

Simple and Intuitive Mathematics for Learning Elementary Physics

“…Without reliance on trigonometric identities or differential calculus, learners can be convinced at a glance by viewing the relationships in Fig. 1 in the order (1), (2), (3), (4). Other important information is provided by the upper-right diagram in Fig.…”

“…This situation is similar to Newton's Principia. Some propositions of the Principia are framed in a geometric language easily translatable into calculus concepts, although little of calculus techniques is found in the Principia [4].…”

What can numerical computation do for the history of science? (a study of an orbit drawn by Newton in a letter to Hooke)

“…This debate seems to be related to another debate in the history of physics: whether Newton had or had not used the analytic form of the calculus in his Principia, as opposed to the geometric (synthetic, in Newton's words) method. A story propagated by Newton himself is that the geometric propositions in the Principia were first obtained by him analytically, but he preferred to present them geometrically (Niccolò Guicciardini); in particular, Newton said that he used some method of curvature (Guicciardini 1998). Nauenberg (1994a) claims to have found what the method of curvature was.…”