History contributions by Sir Isaac Newton and Gottfried Wilhelm Leibniz in calculus
Gottfried Wilhelm Leibniz and Isaac Newton were geniuses who lived quite different lives and invented quite different versions of the infinitesimal calculus, each to suit his own interests and purposes. Newton discovered his fundamental ideas in 1664–1666, while a student at Cambridge University. During a good part of these years the University was closed due to the plague, and Newton worked at his family home in Woolsthorpe, Lincolnshire. However, his ideas were not published until 1687. Leibniz, in France and Germany, on the other hand, began his own breakthroughs in 1675, publishing in 1684. The importance of publication is illustrated by the fact that scientific communication was still sufficiently uncoordinated that it was possible for the work of Newton and Leibniz to proceed independently for many years without reciprocal knowledge and input. Disputes about the priority of their discoveries raged for centuries, fed by nationalistic tendencies in England and Germany.
Newton made contributions to all branches of mathematics then studied, but is especially famous for his solutions to the contemporary problems in analytical geometry of drawing tangents to curves (differentiation) and defining areas bounded by curves (integration). Not only did Newton discover that these problems were inverse to each other, but he discovered general methods of resolving problems of curvature, embraced in his "method of fluxions" and "inverse method of fluxions", respectively equivalent to Leibniz's later differential and integral calculus. Newton used the term "fluxion" (from Latin meaning "flow") because he imagined a quantity "flowing" from one magnitude to another. Fluxions were expressed algebraically, as Leibniz's differentials were, but Newton made extensive use also (especially in the Principia) of analogous geometrical arguments.
According to Leibniz's notebooks, a critical...