Editing Calculus (section)

=== Significance ===
While many of the ideas of calculus had been developed earlier in [[Greek mathematics|Greece]], [[Chinese mathematics|China]], [[Indian mathematics|India]], [[Islamic mathematics|Iraq, Persia]], and [[Japanese mathematics|Japan]], the use of calculus began in Europe, during the 17th century, when Newton and Leibniz built on the work of earlier mathematicians to introduce its basic principles.<ref name=":0">{{Cite book|title=Chinese studies in the history and philosophy of science and technology|date=1996 |publisher=Kluwer Academic Publishers|author1=Dainian Fan|author2=R. S. Cohen|isbn=0-7923-3463-9|location=Dordrecht|oclc=32272485}}</ref><ref name=":1">{{Cite book|title=Landmark writings in Western mathematics 1640–1940|date=2005 |publisher=Elsevier|editor-first1=I.|editor-last1=Grattan-Guinness|editor-link1=Ivor Grattan-Guinness |isbn=0-444-50871-6 |location=Amsterdam |oclc=60416766}}</ref><ref>{{Cite book|last=Kline |first=Morris|author-link=Morris Kline|title=Mathematical thought from ancient to modern times |volume=3|date=1990 |publisher=Oxford University Press|isbn=978-0-19-977048-9 |location=New York|oclc=726764443}}</ref> The Hungarian polymath [[John von Neumann]] wrote of this work,
{{blockquote|The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. 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.<ref>{{cite book|last=von Neumann |first=J. |author-link=John von Neumann |chapter=The Mathematician |editor-last=Heywood |editor-first=R. B. |title=The Works of the Mind |publisher=University of Chicago Press |year=1947 |pages=180–196}}  Reprinted in {{cite book|editor-last1=Bródy |editor-first1=F. |editor-last2=Vámos |editor-first2=T. |title=The Neumann Compendium |publisher=World Scientific Publishing Co. Pte. Ltd. |year=1995 |isbn=981-02-2201-7 |pages=618–626}}</ref>}}

Applications of differential calculus include computations involving [[velocity]] and [[acceleration]], the [[slope]] of a curve, and [[Mathematical optimization|optimization]].<ref name=":5" />{{Rp|pages=341–453}} Applications of integral calculus include computations involving area, [[volume]], [[arc length]], [[center of mass]], [[work (physics)|work]], and [[pressure]].<ref name=":5" />{{Rp|pages=685–700}} More advanced applications include [[power series]] and [[Fourier series]].

Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving [[division by zero]] or sums of infinitely many numbers. These questions arise in the study of [[Motion (physics)|motion]] and area. The [[ancient Greek]] philosopher [[Zeno of Elea]] gave several famous examples of such [[Zeno's paradoxes|paradoxes]]. Calculus provides tools, especially the [[Limit (mathematics)|limit]] and the [[infinite series]], that resolve the paradoxes.<ref>{{cite book|first=Eugenia |last=Cheng |author-link=Eugenia Cheng |title=Beyond Infinity: An Expedition to the Outer Limits of Mathematics |title-link=Beyond Infinity (mathematics book) |pages=206–210 |publisher=Basic Books |year=2017 |isbn=978-1-541-64413-7 |oclc=1003309980}}</ref>