Editing Isaac Newton (section)

=== Calculus ===
Newton's work has been said "to distinctly advance every branch of mathematics then studied".{{sfn|Ball|1908|p=319}} His work on calculus, usually referred to as fluxions, began in 1664, and by 20 May 1665 as seen in a manuscript, Newton "had already developed the calculus to the point where he could compute the tangent and the curvature at any point of a continuous curve".<ref>{{Cite book |last1=Press |first1=S. James |url=https://books.google.com/books?id=aAJYCwAAQBAJ&pg=PA88 |title=The Subjectivity of Scientists and the Bayesian Approach |last2=Tanur |first2=Judith M. |date=2016 |publisher=Dover Publications, Inc |isbn=978-0-486-80284-8 |edition= |location= |pages=88}}</ref> Another manuscript of October 1666, is now published among Newton's mathematical papers.<ref>{{cite book |last1=Newton |first1=Isaac |editor1-last=Whiteside |editor1-first=Derek Thomas |title=The Mathematical Papers of Isaac Newton Volume 1 from 1664 to 1666 |date=1967 |publisher=Cambridge University Press |isbn=978-0-521-05817-9 |page=400 |chapter-url=https://archive.org/details/MathematicsIsaacNewtonVol1_1664-66Whiteside1967/MathematicsIsaacNewtonVol1_1664-66Whiteside1967_144x75/page/400/mode/1up |chapter=The October 1666 tract on fluxions}}</ref> His work ''[[De analysi per aequationes numero terminorum infinitas]]'', sent by [[Isaac Barrow]] to [[John Collins (mathematician)|John Collins]] in June 1669, was identified by Barrow in a letter sent to Collins that August as the work "of an extraordinary genius and proficiency in these things".{{sfn|Gjertsen|1986|p=149}} Newton later [[Leibniz–Newton calculus controversy|became involved in a dispute]] with [[Gottfried Wilhelm Leibniz]] over priority in the development of calculus. Both are now credited with independently developing calculus, though with very different [[mathematical notation]]s. However, it is established that Newton came to develop calculus much earlier than Leibniz.<ref name=":2">{{Cite book |last=Newman |first=James Roy |url=https://archive.org/details/world1ofmathemati00newm/page/58 |title=The World of Mathematics: A Small Library of the Literature of Mathematics from Aʻh-mosé the Scribe to Albert Einstein |publisher=Simon and Schuster |year=1956 |page=58}}</ref><ref>{{Cite book |last=Grattan-Guinness |first=Ivor |author-link=Ivor Grattan-Guinness |url=https://books.google.com/books?id=oej5DwAAQBAJ&pg=PA4 |title=From the Calculus to Set Theory 1630-1910: An Introductory History |publisher=[[Princeton University Press]] |year=1980 |isbn=978-0-691-07082-7 |location= |pages=4, 49–51 |language=en}}</ref>{{Sfn|Hall|1980|pp=1, 15, 21}} The notation of Leibniz is recognized as the more convenient notation, being adopted by continental European mathematicians, and after 1820, by British mathematicians.<ref>{{cite book |author1=H. Jerome Keisler |url=https://books.google.com/books?id=8NTCAgAAQBAJ&pg=PA903 |title=Elementary Calculus: An Infinitesimal Approach |publisher=Dover Publications |year=2013 |isbn=978-0-486-31046-6 |edition=3rd |page=903}}</ref>

Historian of science [[A. Rupert Hall]] notes that while Leibniz deserves credit for his independent formulation of calculus, Newton was undoubtedly the first to develop it, stating:{{Sfn|Hall|1980|pp=15, 21}}{{blockquote|But all these matters are of little weight in comparison with the central truth, which has indeed long been universally recognized, that Newton was master of the essential techniques of the calculus by the end of 1666, almost exactly nine years before Leibniz . . . Newton’s claim to have mastered the new infinitesimal calculus long before Leibniz, and even to have written — or at least made a good start upon — a publishable exposition of it as early as 1671, is certainly borne out by copious evidence, and though Leibniz and some of his friends sought to belittle Newton’s case, the truth has not been seriously in doubt for the last 250 years.}}Hall further notes that in ''Principia'', Newton was able to "formulate and resolve problems by the integration of differential equations" and "in fact, he anticipated in his book many results that later exponents of the calculus regarded as their own novel achievements."{{Sfn|Hall|1980|p=30}} Hall notes Newton's rapid development of calculus in comparison to his contemporaries, stating that Newton "well before 1690 . . . had reached roughly the point in the development of the calculus that Leibniz, the two Bernoullis, L’Hospital, Hermann and others had by joint efforts reached in print by the early 1700s".{{Sfn|Hall|1980|p=136}}

Despite the convenience of Leibniz's notation, it has been noted that Newton's notation could also have developed multivariate techniques, with his dot notation still widely used in [[physics]]. Some academics have noted the richness and depth of Newton's work, such as physicist [[Roger Penrose]], stating "in most cases Newton’s geometrical methods are not only more concise and elegant, they reveal deeper principles than would become evident by the use of those formal methods of calculus that nowadays would seem more direct." Mathematician [[Vladimir Arnold]] states "Comparing the texts of Newton with the comments of his successors, it is striking how Newton’s original presentation is more modern, more understandable and richer in ideas than the translation due to commentators of his geometrical ideas into the formal language of the calculus of Leibniz."<ref>{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=u0NBDwAAQBAJ&pg=PA48 |title=Newton – Innovation And Controversy |publisher=[[World Scientific Publishing]] |year=2017 |isbn=9781786344045 |pages=48–49}}</ref>

His work extensively uses calculus in geometric form based on limiting values of the ratios of vanishingly small quantities: in the ''Principia'' itself, Newton gave demonstration of this under the name of "the method of first and last ratios"<ref>Newton, ''Principia'', 1729 English translation, [https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA41 p.&nbsp;41] {{Webarchive|url=https://web.archive.org/web/20151003114205/https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA41 |date=3 October 2015 }}.</ref> and explained why he put his expositions in this form,<ref>Newton, ''Principia'', 1729 English translation, [https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA54 p.&nbsp;54] {{Webarchive|url=https://web.archive.org/web/20160503022921/https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA54 |date=3 May 2016 }}.</ref> remarking also that "hereby the same thing is performed as by the method of indivisibles."<ref name="Newton 1850">{{Cite book |last=Newton |first=Sir Isaac |url=https://books.google.com/books?id=N-hHAQAAMAAJ&pg=PA506 |title=Newton's Principia: The Mathematical Principles of Natural Philosophy |date=1850 |publisher=Geo. P. Putnam |pages=506–507 |access-date=}}</ref> Because of this, the ''Principia'' has been called "a book dense with the theory and application of the infinitesimal calculus" in modern times<ref>{{Cite book |last=Truesdell |first=Clifford |author-link=Clifford Truesdell |url=https://archive.org/details/essaysinhistoryo0000true/page/99 |title=Essays in the History of Mechanics |publisher=[[Springer-Verlag]] |year=1968 |pages=99}}</ref> and in Newton's time "nearly all of it is of this calculus."<ref>In the preface to the Marquis de L'Hospital's ''Analyse des Infiniment Petits'' (Paris, 1696).</ref> His use of methods involving "one or more orders of the infinitesimally small" is present in his ''De motu corporum in gyrum'' of 1684<ref>Starting with [[De motu corporum in gyrum#Contents|De motu corporum in gyrum]], see also [https://books.google.com/books?id=uvMGAAAAcAAJ&pg=RA1-PA2 (Latin) Theorem 1] {{Webarchive|url=https://web.archive.org/web/20160512135306/https://books.google.com/books?id=uvMGAAAAcAAJ&pg=RA1-PA2 |date=12 May 2016 }}.</ref> and in his papers on motion "during the two decades preceding 1684".<ref>Whiteside, D.T., ed. (1970). "The Mathematical principles underlying Newton's Principia Mathematica". ''Journal for the History of Astronomy''. '''1'''. Cambridge University Press. pp.&nbsp;116–138.</ref>

[[File:Sir Isaac Newton by Sir Godfrey Kneller, Bt.jpg|thumb|upright=0.75|Newton in 1702 by [[Godfrey Kneller]]]]

Newton had been reluctant to publish his calculus because he feared controversy and criticism.{{sfn|Stewart|2009|p=107}} He was close to the Swiss mathematician [[Nicolas Fatio de Duillier]]. In 1691, Duillier started to write a new version of Newton's ''Principia'', and corresponded with Leibniz.{{sfn|Westfall|1980|pp=538–539}} In 1693, the relationship between Duillier and Newton deteriorated and the book was never completed.{{sfn|Westfall|1994|p=108}} Starting in 1699, Duillier accused Leibniz of plagiarism.<ref>{{Cite journal |last=Palomo |first=Miguel |date=2 January 2021 |title=New insight into the origins of the calculus war |url=https://www.tandfonline.com/doi/full/10.1080/00033790.2020.1794038 |journal=Annals of Science |volume=78 |issue=1 |pages=22–40 |doi=10.1080/00033790.2020.1794038 |pmid=32684104 |issn=0003-3790}}</ref> Mathematician [[John Keill]] accused Leibniz of plagiarism in 1708 in the Royal Society journal, thereby deteriorating the situation even more.{{Sfn|Iliffe|Smith|2016|pp=|p=414}} The dispute then broke out in full force in 1711 when the Royal Society proclaimed in a study that it was Newton who was the true discoverer and labelled Leibniz a fraud; it was later found that Newton wrote the study's concluding remarks on Leibniz. Thus began the bitter controversy which marred the lives of both men until Leibniz's death in 1716.{{sfn|Ball|1908|p=356}}

Newton is credited with the [[Binomial theorem#Newton's generalized binomial theorem|generalised binomial theorem]], valid for any exponent. He discovered [[Newton's identities]], [[Newton's method]], classified [[cubic plane curve]]s ([[polynomials]] of degree three in two [[variable (mathematics)|variables]]), is a founder of the theory of [[Cremona transformation|Cremona transformations]],<ref name=":20" /> made substantial contributions to the theory of [[finite differences]], with Newton regarded as "the single most significant contributor to finite difference [[interpolation]]", with many formulas created by Newton.<ref>{{Cite book |last=Roy |first=Ranjan |author-link=Ranjan Roy |url=https://books.google.com/books?id=KyYhEAAAQBAJ&pg=PA190 |title=Series and Products in the Development of Mathematics |publisher=[[Cambridge University Press]] |year=2021 |isbn=978-1-108-70945-3 |edition=2nd |volume=I |location=Cambridge |pages=190–191 }}</ref> He was the first to state [[Bézout's theorem]], and was also the first to use fractional indices and to employ [[coordinate geometry]] to derive solutions to [[Diophantine equations]]. He approximated [[series (mathematics)|partial]] sums of the [[harmonic series (mathematics)|harmonic series]] by [[logarithms]] (a precursor to [[Euler's summation formula]]) and was the first to use [[power series]] with confidence and to revert power series. He introduced the [[Puiseux series|Puisseux series]].<ref name=":172">{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=ipA4DwAAQBAJ&pg=PA45 |title=Newton and the Great World System |date=2017 |publisher=[[World Scientific Publishing]] |isbn=978-1-78634-372-7 |pages=45 |language=en |doi=10.1142/q0108}}</ref> He originated the [[Newton–Cotes formulas|Newton-Cotes formulas]] for [[numerical integration]].{{Sfn|Iliffe|Smith|2016|pp=382–394, 411}} Newton's work on infinite series was inspired by [[Simon Stevin]]'s decimals.<ref>{{Cite journal |last1=Błaszczyk |first1=P. |last2=Katz |first2=M.&nbsp;G. |author-link2=Mikhail Katz |last3=Sherry |first3=D. |display-authors=1 |date=March 2013 |title=Ten misconceptions from the history of analysis and their debunking |journal=[[Foundations of Science]] |volume=18 |issue=1 |pages=43–74 |arxiv=1202.4153 |doi=10.1007/s10699-012-9285-8 |s2cid=119134151}}</ref> He also initiated the field of [[calculus of variations]], being the first to clearly formulate and correctly solve a problem in the field, that being [[Newton's minimal resistance problem]], which he posed and solved in 1685, and then later published in ''Principia'' in 1687.<ref>{{Cite book |last=Goldstine |first=Herman H. |author-link=Herman Goldstine |url=https://books.google.com/books?id=_iTnBwAAQBAJ&pg=PA7 |title=A History of the Calculus of Variations from the 17th Through the 19th Century |date=1980 |publisher=Springer New York |isbn=978-1-4613-8106-8 |series= |location= |pages=7–21}}</ref> It is regarded as one of the most difficult problems tackled by variational methods prior to the twentieth century.<ref name=":02">{{cite arXiv |last=Ferguson |first=James |title=A Brief Survey of the History of the Calculus of Variations and its Applications |date=2004 |arxiv=math/0402357}}</ref> He then used calculus of variations in his solving of the [[brachistochrone curve]] problem in 1697, which was posed by [[Johann Bernoulli]] in 1696, thus he pioneered the field with his work on the two problems.<ref name=":17">{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=ipA4DwAAQBAJ&pg=PA36 |title=Newton and the Great World System |date=2017 |publisher=[[World Scientific Publishing]] |isbn=978-1-78634-372-7 |pages=36–39 |language=en |doi=10.1142/q0108}}</ref> He was also a pioneer of [[vector calculus|vector analysis]], as he demonstrated how to apply the parallelogram law for adding various physical quantities and realized that these quantities could be broken down into components in any direction.<ref name=":173">{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=ipA4DwAAQBAJ&pg=PA26 |title=Newton and the Great World System |date=2017 |publisher=[[World Scientific Publishing]] |isbn=978-1-78634-372-7 |pages=26 |language=en |doi=10.1142/q0108}}</ref>