Before Newton and Leibniz, the word “calculus” referred to any body of mathematics. Still, in the following years, “calculus” became a popular term for a field of mathematics based upon their insights. Newton and Leibniz, building on this work, independently developed the surrounding theory of infinitesimal calculus in the late 17th century. Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts. Newton provided some of the most important applications to physics, especially of integral calculus. The purpose of this section is to examine Newton and Leibniz’s investigations into the developing field of infinitesimal calculus. Specific importance will be put on the justification and descriptive terms which they used in an attempt to understand calculus as they conceived it.

By the middle of the 17th century, European mathematics had changed its primary repository of knowledge. In comparison to the last century which maintained Hellenistic mathematics as the starting point for research, Newton, Leibniz, and their contemporaries increasingly looked towards the works of more modern thinkers. Europe had become home to a burgeoning mathematical community, and with the advent of enhanced institutional and organizational bases, a new level of organization and academic integration was being achieved. Importantly, however, the community lacked formalism; instead, it consisted of a disordered mass of various methods, techniques, notations, theories, and paradoxes.

Newton came to calculus as part of his investigations in physics and geometry. He viewed calculus as the scientific description of the generation of motion and magnitudes. In comparison, Leibniz focused on the tangent problem and came to believe that calculus was a metaphysical explanation of change. Importantly, the core of their insight was the formalization of the inverse properties between the integral and the differential of a function. This insight had been anticipated by their predecessors, but they were the first to conceive calculus as a system in which new rhetoric and descriptive terms were created. Their unique discoveries lay not only in their imagination but also in their ability to synthesize the insights around them into a universal algorithmic process, thereby forming a new mathematical system.