The Creation of Calculus

Leibniz himself in a 1703 letter to James Bernoulli and in his unpublished Historia et origo calculi differentialis, written in 1713 as an answer to the Commercium Epistolicum report issued by the Royal Society accusing him of plagiarizing Newton, dates his discovery of the calculus to the 1673-4 period when he first extended Pascal's method of computing the surface area of a sphere to general solids of revolution and then arrived at his method of transmutation. He also mentions work in this period using differences as a tool to sum numerical series as well as a new notation "invented by him at a later date" in which ʃ x and dx respectively represent the partial sums of a series (whose general term is x) and the difference of successive terms.¹ Still later (no precise date is given), he says that he realized that dx and dy represent infinitesimal differences of the abscissa and ordinate, since "the infinitely small lines occurring in diagrams were nothing else but the momentaneous differences of the variable lines."² The difference notation, originally applied to series, could then be used to denote the horizontal and vertical sides of the "characteristic triangle" which Leibniz says he discovered and named in 1673.³…