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.