Newton called his calculus "the science of fluxions". It is Leibniz, however, who gave the new discipline its name.
Today, both Newton and Leibniz are given credit for developing calculus independently.
A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. Newton derived his results first, but Leibniz published first. When Newton and Leibniz first published their results, there was great controversy over which mathematician (and therefore which country) deserved credit. By Newton's time, the fundamental theorem of calculus was known. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. Leibniz and Newton are usually both credited with the invention of calculus. Unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. His contribution was to provide a clear set of rules for manipulating infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms. He is now regarded as an independent inventor of and contributor to calculus. These ideas were systematized into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.
Infinitesimals to derive chain rule series#
In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, and many other problems discussed in his Principia Mathematica (1687).
In his publications, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1670.
Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The formal study of calculus combined Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The ideas were similar to Archimedes' in The Method, but this treatise was lost until the early part of the twentieth century. In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking." -John von Neumann Modern "The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance.
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