Some basic features of Newtonian exact science: mathematics and the disciplined creative imagination

An outstanding feature of Newton's scientific thought is the close interplay of mathematics and physical science. It is, no doubt, a mark of his extraordinary genius that he could exercise such skill in imagining and designing experiments, and in performing such experiments and drawing from them their theoretical significance. He also displayed a fertile imagination in speculating about the nature of matter (including its structure, the forces that might hold it together, and the causes of the interactions between varieties of matter). In the present context, my primary concern is with mathematics in relation to the physical sciences of dynamics and celestial mechanics, and not with these other aspects of Newton's scientific endeavors. We shall see that although Newton expressed the pious wish that optics might become a fully developed branch of mathematical science in the Newtonian style, this subject never achieved that state during his lifetime (see §3.11); hence Newton's optical researches are not given major consideration here.

The ‘principles of natural philosophy’ that Isaac Newton displayed and elaborated in his Principia are ‘mathematical principles’. His exploration of the properties of various motions under given conditions of forces is based on mathematics and not on experiment and induction. What is not so well known is that his essays in pure mathematics (analytic geometry and calculus) often tend to be couched in the language and principles of the physics of motion. This interweaving of dynamics and pure mathematics is also a characteristic feature of the science of the Principia.