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| :''This is a sub-article to [[Calculus]] and [[History of mathematics]].''
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| '''Calculus''', historically known as '''[[infinitesimal calculus]]''', is a [[mathematics|mathematical]] discipline focused on [[limit (mathematics)|limits]], [[function (mathematics)|functions]], [[derivative]]s, [[integral]]s, and [[infinite series]]. Ideas leading up to the notions of [[function (mathematics)|function]], [[derivative]], and [[integral]] were developed throughout the 17th century, but the decisive step was made by [[Isaac Newton]] and [[Gottfried Leibniz]]. Publication of Newton's main treatises took many years, whereas Leibniz published first ([[Nova_Methodus_pro_Maximis_et_Minimis|''Nova methodus'']], 1684) and the whole subject was subsequently marred by [[Leibniz and Newton calculus controversy|a priority dispute between the two inventors of calculus]].
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| ==Precursors of calculus==
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| ===Ancient===
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| [[Image:Archimedes pi.svg|thumb|right|300px|Archimedes used the method of exhaustion to compute the area inside a circle]]
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| The ancient period introduced some of the ideas that led to [[integral]] calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volumes and areas, one goal of integral calculus, can be found in the [[Egyptian mathematics|Egyptian]] [[Moscow Mathematical Papyrus|Moscow papyrus]] (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning.<ref>{{cite book | first1=Morris | last1=Kline | title=Mathematical thought from ancient to modern times | volume=1 | publisher=Oxford University Press | isbn=978-0-19-506135-2 | pages=18–21 }}</ref>
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| From the age of [[Greek mathematics]], [[Eudoxus of Cnidus|Eudoxus]] (c. 408−355 BC) used the [[method of exhaustion]], which foreshadows the concept of the limit, to calculate areas and volumes, while [[Archimedes]] (c. 287−212 BC) [[Archimedes' use of infinitesimals|developed this idea further]], inventing [[heuristics]] which resemble the methods of integral calculus.<ref>Archimedes, ''Method'', in ''The Works of Archimedes'' ISBN 978-0-521-66160-7</ref> The [[method of exhaustion]] was later reinvented in [[Chinese mathematics|China]] by [[Liu Hui]] in the 3rd century AD in order to find the area of a circle.<ref>{{cite journal
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| |series=Chinese studies in the history and philosophy of science and technology|volume=130|title=A comparison of Archimdes' and Liu Hui's studies of circles|first1=Liu|last1=Dun|first2=Dainian|last2=Fan|first3=Robert Sonné|last3=Cohen|publisher=Springer|year=1966|isbn=0-7923-3463-9|page=279|url=http://books.google.com/books?id=jaQH6_8Ju-MC}}, [http://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 Chapter , p. 279]</ref> In the 5th century AD, [[Zu Chongzhi]] established a method that would later be called [[Cavalieri's principle]] to find the volume of a [[sphere]].<ref>{{cite book|title=Calculus: Early Transcendentals|edition=3|first1=Dennis G.|last1=Zill|first2=Scott|last2=Wright|first3=Warren S.|last3=Wright|publisher=Jones & Bartlett Learning|year=2009|isbn=0-7637-5995-3|page=xxvii|url=http://books.google.com/books?id=R3Hk4Uhb1Z0C}}, [http://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27 Extract of page 27]</ref> [[Greek mathematics|Greek mathematicians]] are also credited with a significant use of [[infinitesimal]]s. [[Democritus]] is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, [[Zeno of Elea]] discredited infinitesimals further by his articulation of the [[Zeno's paradoxes|paradoxes]] which they create.
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| [[Archimedes]] of [[Syracuse, Sicily|Syracuse]] developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. (See ''[[The Quadrature of the Parabola|Archimedes' Quadrature of the Parabola]]'', ''[[Archimedes use of infinitesimals|The Method]]'', ''Archimedes on Spheres & Cylinders''.<ref>MathPages — [http://mathpages.com/home/kmath343.htm Archimedes on Spheres & Cylinders]</ref>) It should not be thought that infinitesimals were put on a rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the time of [[Isaac Newton|Newton]] that these methods were incorporated into a general framework of [[integral calculus]]. Archimedes was the first to find the tangent to a curve, other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a point's motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together, thereby finding the tangent to the curve.<ref>{{cite book|first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of Mathematics |edition=2nd |publisher=Wiley |year=1991 |isbn=0-471-54397-7 |chapter=Archimedes of Syracuse|pages=127|quote=Greek mathematics sometimes has been described as essentially static, with little regard for the notion of variability; but Archimedes, in his study of the spiral, seems to have found the tangent to a curve through kinematic considerations akin to differential calculus. Thinking of a point on the spiral {{nowrap|1=''r'' = ''aθ''}} as subjected to a double motion — a uniform radial motion away from the origin of coordinates and a circular motion about the origin — he seems to have found (through the parallelogram of velocities) the direction of motion (hence of the tangent to the curve) by noting the resultant of the two component motions. This appears to be the first instance in which a tangent was found to a curve other than a circle.<br/>Archimedes' study of the spiral, a curve that he ascribed to his friend [[Conon of Alexandria]], was part of the Greek search for the solution of the three famous problems.}}</ref> The pioneers of the calculus such as [[Isaac Barrow]] and [[Johann Bernoulli]] were diligent students of Archimedes; see for instance C. S. Roero (1983).
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| ===Medieval===
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| [[Alexander the Great]]'s invasion of northern India brought Greek trigonometry, using the [[Chord (geometry)|chord]], to India where the sine, cosine, and tangent were conceived. Indian mathematicians gave a semi-rigorous method of differentiation of some trigonometric functions. In the Middle East, [[Alhazen]] derived a formula for the sum of [[fourth power]]s. He used the results to carry out what would now be called an [[Integral|integration]], where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a [[paraboloid]].<ref name=katz>Katz, V. J. 1995. "Ideas of Calculus in Islam and India." ''Mathematics Magazine'' (Mathematical Association of America), 68(3):163-174.</ref> In the 14th century, Indian mathematician [[Madhava of Sangamagrama]] and the [[Kerala school of astronomy and mathematics]] stated components of calculus such as the [[Taylor series]] and [[infinite series]] approximations.<ref>[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html Indian mathematics<!-- Bot generated title -->]</ref> However, they were not able to combine many differing ideas under the two unifying themes of the [[derivative]] and the [[integral]], show the connection between the two, and turn calculus into the great problem-solving tool we have today.<ref name=katz/>
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| The mathematical study of continuity was revived in the 14th century by the [[Oxford Calculators]] and French collaborators such as [[Nicole Oresme]]. They proved the "Merton [[mean speed theorem]]": that a uniformly accelerated body travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body.<ref>{{cite book|first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of the Calculus and Its Conceptual Development |publisher=Dover |year=1959 |isbn=978-0-486-60509-8 |chapterurl=http://books.google.com/books?id=KLQSHUW8FnUC&pg=PA79&lpg=PP1#v=onepage&q&f=false |chapter=III. Medieval Contributions |pages=79–89 |url=http://books.google.com/books?id=KLQSHUW8FnUC}}</ref>
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| ==Pioneers of modern calculus==
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| In the 17th century, European mathematicians [[Isaac Barrow]], [[René Descartes]], [[Pierre de Fermat]], [[Blaise Pascal]], [[John Wallis]] and others discussed the idea of a [[derivative]]. In particular, in ''Methodus ad disquirendam maximam et minima'' and in ''De tangentibus linearum curvarum'', Fermat developed an [[adequality]] method for determining maxima, minima, and tangents to various curves that was closely related to differentiation.<ref name=Pellegrino>{{cite web | last = Pellegrino | first = Dana | title=Pierre de Fermat | url=http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html| accessdate=2008-02-24}}</ref> [[Isaac Newton]] would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."<ref name=Simmons>{{cite book | last = Simmons | first = George F. | title = Calculus Gems: Brief Lives and Memorable Mathematics | publisher = Mathematical Association of America | year = 2007 | page = 98 | isbn = 0-88385-561-5}}</ref>
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| On the integral side, [[Bonaventura Cavalieri|Cavalieri]] developed his [[method of indivisibles]] in the 1630s and 1640s, providing a more modern form of the ancient Greek [[method of exhaustion]],{{Disputed-inline|Two different methods|date=December 2011}} and computing [[Cavalieri's quadrature formula]], the area under the curves ''x''<sup>''n''</sup> of higher degree, which had previously only been computed for the parabola, by Archimedes. [[Evangelista Torricelli|Torricelli]] extended this work to other curves such as the [[cycloid]], and then the formula was generalized to fractional and negative powers by Wallis in 1656. In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly.<ref name=quadrature>{{Cite web | last1 = Paradís | first1 = Jaume | last2 = Pla | first2 = Josep | last3 = Viader | first3 = Pelagrí | title=Fermat’s Treatise On Quadrature: A New Reading | url=http://www.econ.upf.edu/docs/papers/downloads/775.pdf | accessdate=2008-02-24 | postscript = <!--None-->}}</ref> Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature. [[James Gregory (astronomer and mathematician)|James Gregory]], influenced by Fermat's contributions both to tangency and to quadrature, was then able to prove a restricted version of the second [[fundamental theorem of calculus]] in the mid-17th century.{{Citation needed|date=December 2011}} The first full proof of the [[fundamental theorem of calculus]] was given by [[Isaac Barrow]].<ref>{{cite book |title=The geometrical lectures of Isaac Barrow, translated, with notes and proofs, and a discussion on the advance made therein on the work of his predecessors in the infinitesimal calculus |publisher=Open Court |location=Chicago |year=1916 |url=http://www.archive.org/details/geometricallectu00barruoft }}</ref>
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| Newton and Leibniz, building on this work, independently developed the surrounding theory of [[infinitesimal calculus]] in the late 17th century. Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts. Newton provided some of the most important applications to physics, especially of [[integral calculus]].
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| The first proof of Rolle's theorem was given by [[Michel Rolle]] in 1691 using methods developed by the Dutch mathematician [[Johann van Waveren Hudde]].<ref>{{cite book
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| |title=A Transition to Advanced Mathematics: A Survey Course
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| |first1=William
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| |last1=Johnston
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| |first2=Alex
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| |last2=McAllister
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| |publisher=Oxford University Press US
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| |year=2009
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| |isbn=0-19-531076-4
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| |page=333
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| |url=http://books.google.com/books?id=LV21vHwnkpIC}}, [http://books.google.com/books?id=LV21vHwnkpIC&pg=PA333 Chapter 4, p. 333]
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| </ref> The mean value theorem in its modern form was stated by [[Bernard Bolzano]] and [[Augustin-Louis Cauchy]] (1789–1857) also after the founding of modern calculus. Important contributions were also made by [[Isaac Barrow|Barrow]], [[Christiaan Huygens|Huygens]], and many others.
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| ==Newton and Leibniz==
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| [[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|right|[[Isaac Newton]]]]
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| [[File:Gottfried Wilhelm von Leibniz.jpg|thumb|right|[[Gottfried Leibniz]]]]
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| Before [[Sir Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]], the word “calculus” was a general term used to refer to any body of mathematics, but in the following years, "calculus" became a popular term for a field of mathematics based upon their insights.<ref>{{harvnb|Reyes|2004|p=160}}</ref> The purpose of this section is to examine Newton and Leibniz’s investigations into the developing field of [[infinitesimal calculus]]. Specific importance will be put on the justification and descriptive terms which they used in an attempt to understand calculus as they themselves conceived it.
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| By the middle of the 17th century, European mathematics had changed its primary repository of knowledge. In comparison to the last century which maintained [[Hellenistic]] mathematics as the starting point for research, Newton, Leibniz and their contemporaries increasingly looked towards the works of more modern thinkers.<ref>Such as Kepler, Descartes, Fermat, Pascal and Wallis. {{harvnb|Calinger|1999|p=556}}</ref> Europe had become home to a burgeoning mathematical community and with the advent of enhanced institutional and organizational bases a new level of organization and academic integration was being achieved. Importantly, however, the community lacked formalism; instead it consisted of a disordered mass of various methods, techniques, [[Mathematical notation|notation]]s, [[theory|theories]], and [[paradox]]es.
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| Newton came to calculus as part of his investigations in [[physics]] and [[geometry]]. He viewed calculus as the scientific description of the generation of motion and [[magnitude (mathematics)|magnitude]]s. In comparison, Leibniz focused on the [[tangent problem]] and came to believe that calculus was a [[metaphysics|metaphysical]] explanation of change. Importantly, the core of their insight was the formalization of the inverse properties between the [[integral]] and the [[differential of a function]]. This insight had been anticipated by their predecessors, but they were the first to conceive calculus as a system in which new rhetoric and descriptive terms were created.<ref>Foremost among these was [[Isaac Barrow|Barrow]] who had created formulas for specific cases and Fermat who created a similar definition for the derivative. For more information; Boyer 184</ref> Their unique discoveries lay not only in their imagination, but also in their ability to synthesize the insights around them into a universal algorithmic process, thereby forming a new mathematical system.
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| {{See also|Leibniz–Newton calculus controversy}}
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| ===Newton===
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| Newton completed no definitive publication formalizing his Fluxional Calculus; rather, many of his mathematical discoveries were transmitted through correspondence, smaller papers or as embedded aspects in his other definitive compilations, such as the ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' and ''[[Opticks]]''. Newton would begin his mathematical training as the chosen heir of [[Isaac Barrow]] in [[Cambridge]]. His incredible aptitude was recognized early and he quickly learned the current theories. By 1664 Newton had made his first important contribution by advancing the [[binomial theorem]], which he had extended to include fractional and negative [[Exponentiation|exponent]]s. Newton succeeded in expanding the applicability of the binomial theorem by applying the algebra of finite quantities in an analysis of [[infinite series]]. He showed a willingness to view infinite series not only as approximate devices, but also as alternative forms of expressing a term.<ref>{{harvnb|Calinger|1999|p=610}}</ref>
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| Many of Newton’s critical insights occurred during the plague years of 1665-1666<ref>{{cite web|last=Newton|first=Isaac|title=Waste Book|url=http://cudl.lib.cam.ac.uk/view/MS-ADD-04004/|accessdate=10 January 2012}}</ref> which he later described as, “the prime of my age for invention and minded mathematics and [natural] philosophy more than at any time since.” It was during his plague-induced isolation that the first written conception of [[Fluxionary Calculus]] was recorded in the unpublished ''[[De analysi per aequationes numero terminorum infinitas |De Analysi per Aequationes Numero Terminorum Infinitas]]''. In this paper, Newton determined the area under a [[curve]] by first calculating a momentary rate of change and then extrapolating the total area. He began by reasoning about an indefinitely small triangle whose area is a function of ''x'' and ''y''. He then reasoned that the [[infinitesimal]] increase in the abscissa will create a new formula where {{nowrap|1=''x'' = ''x'' + ''o''}} (importantly, ''o'' is the letter, not the [[Numerical digit|digit]] 0). He then recalculated the area with the aid of the binomial theorem, removed all quantities containing the letter ''o'' and re-formed an algebraic expression for the area. Significantly, Newton would then “blot out” the quantities containing ''o'' because terms “multiplied by it will be nothing in respect to the rest”.
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| At this point Newton had begun to realize the central property of inversion. He had created an expression for the area under a curve by considering a momentary increase at a point. In effect, the [[fundamental theorem of calculus]] was built into his calculations. While his new formulation offered incredible potential, Newton was well aware of its logical limitations at the time. He admits that “errors are not to be disregarded in mathematics, no matter how small” and that what he had achieved was “shortly explained rather than accurately demonstrated.”
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| In an effort to give calculus a more rigorous explication and framework, Newton compiled in 1671 the ''[[Methodus Fluxionum et Serierum Infinitarum]]''. In this book, Newton’s strict [[empiricism]] shaped and defined his [[Fluxional Calculus]]. He exploited [[instantaneous velocity|instantaneous]] [[Motion (physics)|motion]] and infinitesimals informally. He used math as a [[methodological]] tool to explain the physical world. The base of Newton’s revised Calculus became continuity; as such he redefined his calculations in terms of continual flowing motion. For Newton, variable magnitudes are not aggregates of infinitesimal elements, but are generated by the indisputable fact of motion.
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| Newton attempted to avoid the use of the infinitesimal by forming calculations based on [[ratio]]s of changes. In the ''Methodus Fluxionum'' he defined the rate of generated change as a [[Method of Fluxions|fluxion]], which he represented by a dotted letter, and the quantity generated he defined as a fluent. For example, if <math>{x}</math> and <math>{y}</math> are fluents, then <math>\dot{x}</math> and <math>\dot{y}</math> are their respective fluxions. This revised calculus of ratios continued to be developed and was maturely stated in the 1676 text ''[[De Quadratura Curvarum]]'' where Newton came to define the present day derivative as the ultimate ratio of change, which he defined as the ratio between evanescent increments (the ratio of fluxions) purely at the moment in question. Essentially, the ultimate ratio is the ratio as the increments vanish into nothingness. Importantly, Newton explained the existence of the ultimate ratio by appealing to motion;
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| “For by the ultimate velocity is meant that, with which the body is moved, neither before it arrives at its last place, when the motion ceases nor after but at the very instant when it arrives... the ultimate ratio of evanescent quantities is to be understood, the ratio of quantities not before they vanish, not after, but with which they vanish”<ref>''Principia'', [[Florian Cajori]] 8</ref>
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| Newton developed his Fluxional Calculus in an attempt to evade the informal use of infinitesimals in his calculations.
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| ===Leibniz===
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| {{unreferenced section|date=March 2012}}
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| While Newton began development of his fluxional calculus in 1665-1666 his findings did not become widely circulated until later. In the intervening years Leibniz also strove to create his calculus. In comparison to Newton who came to math at an early age, Leibniz began his rigorous math studies with a mature intellect. He was a [[polymath]], and his intellectual interests and achievements involved [[metaphysics]], [[law]], [[economics]], [[politics]], [[logic]], and [[mathematics]]. In order to understand Leibniz’s reasoning in calculus his background should be kept in mind. Particularly, his [[metaphysics]] which considered the world as an infinite aggregate of indivisible [[Monad (Greek philosophy)|monad]]s.{{Dubious|World as infinite aggregate|date=February 2012}} and his plans of creating a precise formal logic whereby, “a general method in which all truths of the reason would be reduced to a kind of calculation.”
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| In 1672 Leibniz met the mathematician [[Christiaan Huygens|Huygens]] who convinced Leibniz to dedicate significant time to the study of mathematics. By 1673 he had progressed to reading [[Blaise Pascal|Pascal]]’s ''[[Traité des Sinus du Quarte Cercle]]'' and it was during his largely [[autodidactic]] research that Leibniz said "a light turned on" {{cn|date=December 2012}}. Like Newton, Leibniz, saw the tangent as a ratio but declared it as simply the ratio between [[ordinate]]s and [[abscissa]]s. He continued this reasoning to argue that the [[integral]] was in fact the sum of the ordinates for infinitesimal intervals in the abscissa; in effect, the sum of an infinite number of rectangles. From these definitions the inverse relationship or differential became clear and Leibniz quickly realized the potential to form a whole new system of mathematics. Where Newton shied away from the use of [[infinitesimal]]s, Leibniz made it the cornerstone of his notation and calculus.
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| In the manuscripts of 25 October to 11 November 1675, Leibniz recorded his discoveries and experiments with various forms of notation. He was acutely aware of the notational terms used{{attribution needed|date=March 2012}} and his earlier plans to form a precise logical [[symbol]]ism became evident. Eventually, Leibniz denoted the infinitesimal increments of abscissas and ordinates ''dx'' and ''dy'', and the summation of infinitely many infinitesimally thin rectangles as a [[long s#Modern usage|long s]] (∫ ), which became the present integral symbol <math>\scriptstyle\int</math>.
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| While Leibniz's notation is used by modern mathematics, his logical base was different than our current one.{{citation needed|date=March 2012}} Leibniz embraced infinitesimals and wrote extensively so as, “not to make of the infinitely small a mystery, as had Pascal.”{{Cite quote|date=March 2012}} According to [[Deleuze]], Leibniz's zeroes "are nothings, but they are not absolute nothings, they are nothings respectively" (quoting Leibniz' text "Justification of the calculus of infinitesimals by the calculus of ordinary algebra.") <ref name=Deleuze>{{cite web|last=Deleuae|first=Gilles|title=DELEUZE / LEIBNIZ Cours Vincennes - 22/04/1980|url=http://www.webdeleuze.com/php/texte.php?cle=53&groupe=Leibniz&langue=2|accessdate=30 April 2013}}</ref> Alternatively, he defines them as, “less than any given quantity.”{{Cite quote|date=March 2012}} For Leibniz, the world was an aggregate of infinitesimal points{{Dubious|World as infinite aggregate|date=February 2012}} and the lack of scientific proof for their existence did not trouble him. Infinitesimals to Leibniz were ideal quantities of a different type from appreciable numbers. The truth of continuity was proven by existence itself. For Leibniz the principle of continuity and thus the validity of his Calculus was assured. Three hundred years after Leibniz's work, [[Abraham Robinson]] showed that using infinitesimal quantities in calculus could be given a solid foundation.
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| ===Legacy===
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| The rise of Calculus stands out as a unique moment in mathematics. Calculus is the math of motion and change, and as such, its invention required the creation of a new mathematical system. Importantly, Newton and Leibniz did not create the same Calculus and they did not conceive of modern Calculus. While they were both involved in the process of creating a mathematical system to deal with variable quantities their elementary base was different. For Newton, change was a variable quantity over time and for Leibniz it was the difference ranging over a sequence of infinitely close values. Notably, the descriptive terms each system created to describe change was different.
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| Historically, there was much debate over whether it was Newton or Leibniz who first "invented" calculus. This argument, the [[Leibniz and Newton calculus controversy]], involving Leibniz, who was German, and the Englishman Newton, led to a rift in the European mathematical community lasting over a century. Leibniz was the first to publish his investigations; however, it is well established that Newton had started his work several years prior to Leibniz and had already developed a theory of [[tangent]]s by the time Leibniz became interested in the question.
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| <!-- The following is not true as stated: they didn't resolve the priority dispute, rather, they brought Leibniz style calculus to English universities. Incorporated below.
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| Much of the credit for the resolution of the conflict goes to the [[Analytical Society]].
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| Much of the controversy centers on the question whether Leibniz had seen certain early manuscripts of Newton before publishing his own memoirs on the subject. Newton began his work on calculus no later than 1666, and Leibniz did not begin his work until 1673. Leibniz visited England in 1673 and again in 1676, and was shown some of Newton's unpublished writings. He also corresponded with several English scientists (as well as with Newton himself), and may have gained access to Newton's manuscripts through them.
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| It is not known how much this may have influenced Leibniz. The initial accusations were made by students and supporters of the two great scientists at the turn of the century, but after 1711 both of them became personally involved, accusing each other of [[plagiarism]].
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| The priority dispute had an effect of separating English-speaking mathematicians from those in the continental Europe for many years. Only in the 1820s, due to the efforts of the [[Analytical Society]], did [[Leibnizian analytical calculus]] become accepted in England. Today, both Newton and Leibniz are given credit for independently developing the basics of calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the science of [[fluent (mathematics)|fluent]]s and [[fluxion (disambiguation)|fluxion]]s".
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| The work of both Newton and Leibniz is reflected in the notation used today. Newton introduced the notation <math>\dot{f}</math> for the [[derivative (mathematics)|derivative]] of a function ''f''.<ref>The use of prime to denote the [[derivative]], <math> f'\left(x\right),</math> is due to Lagrange.</ref> Leibniz introduced the symbol <math>\int</math> for the [[integral]] and wrote the [[derivative]] of a function ''y'' of the variable ''x'' as <math>\frac{dy}{dx}</math>, both of which are still in use.
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| ==Integrals==
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| [[Niels Henrik Abel]] seems to have been the first to consider in a general way the question as to what [[differential expression]]s can be integrated in a finite form by the aid of ordinary functions, an investigation extended by [[Joseph Liouville|Liouville]]. [[Augustin Louis Cauchy|Cauchy]] early undertook the general theory of determining [[definite integral]]s, and the subject has been prominent during the 19th century. [[Giuliano Frullani|Frullani]]'s theorem (1821), [[Bierens de Haan]]'s work on the theory (1862) and his elaborate tables (1867), [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]]'s lectures (1858) embodied in [[Friedrich Wilhelm Franz Meyer|Meyer]]'s treatise (1871), and numerous memoirs of [[Adrien-Marie Legendre|Legendre]], [[Siméon Denis Poisson|Poisson]], [[Giovanni Antonio Amedeo Plana|Plana]], [[Joseph Ludwig Raabe|Raabe]], [[Leonhard Sohncke|Sohncke]], [[Oscar Xavier Schlömilch|Schlömilch]], [[Edwin Bailey Elliott|Elliott]], [[Charles Leudesdorf|Leudesdorf]], and [[Leopold Kronecker|Kronecker]] are among the noteworthy contributions.
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| [[Euler integral (disambiguation)|Eulerian integrals]] were first studied by [[Leonhard Euler|Euler]] and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows:
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| :<math>\int_0^1 x^{n-1}(1 - x)^{n-1} \, dx</math>
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| :<math>\int_0^\infty e^{-x} x^{n-1} \, dx</math>
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| although these were not the exact forms of Euler's study.
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| If ''n'' is an integer, it follows that:
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| :<math>\int_0^\infty e^{-x}x^{n-1}dx = (n-1)!,</math>
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| but the integral converges for all positive real <math>n</math> and defines an [[analytic continuation]] of the [[factorial]] function to all of the [[complex plane]] except for poles at zero and the negative [[integers]]. To it Legendre assigned the symbol <math>\Gamma</math>, and it is now called the [[gamma function]]. Besides being analytic over the positive reals, <math>\Gamma</math> also enjoys the uniquely defining property that <math>\log \Gamma</math> is [[Convex function|convex]], which aesthetically justifies this analytic continuation of the factorial function over any other analytic continuation. To the subject [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]] has contributed an important theorem (Liouville, 1839), which has been elaborated by [[Joseph Liouville|Liouville]], [[Eugène Charles Catalan|Catalan]], [[Leslie Ellis]], and others. On the evaluation of <math>\Gamma (x)</math> and <math>\log \Gamma (x)</math> [[Joseph Ludwig Raabe|Raabe]] (1843–44), Bauer (1859), and [[Christoph Gudermann|Gudermann]] (1845) have written. Legendre's great table appeared in 1816.
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| ==Symbolic methods==
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| [[Symbolic method]]s may be traced back to [[Brook Taylor|Taylor]], and the much debated analogy between successive differentiation and ordinary exponentials had been observed by numerous writers before the 19th century. [[Antoine Arbogast|Arbogast]] (1800) was the first, however, to separate the symbol of operation from that of quantity in a differential equation. François (1812) and [[François Joseph Servois|Servois]] (1814) {{Citation needed|date=July 2007}} seem to have been the first to give correct rules on the subject. Hargreave (1848) applied these methods in his memoir on differential equations, and [[George Boole|Boole]] freely employed them. [[Hermann Grassmann|Grassmann]] and [[Hermann Hankel]] made great use of the theory, the former in studying [[equation]]s, the latter in his theory of [[complex number]]s.
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| ==Calculus of variations==
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| The [[calculus of variations]] may be said to begin with a problem of [[Johann Bernoulli]]'s (1696). It immediately occupied the attention of [[Jakob Bernoulli]] and the Marquis de [[Guillaume François Antoine, Marquis de l'Hôpital|l'Hôpital]], but [[Leonhard Euler|Euler]] first elaborated the subject. His contributions began in 1733, and his ''Elementa Calculi Variationum'' gave to the science its name. [[Joseph Louis Lagrange|Lagrange]] contributed extensively to the theory, and [[Adrien-Marie Legendre|Legendre]] (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. To this discrimination Brunacci (1810), [[Carl Friedrich Gauss|Gauss]] (1829), [[Siméon Denis Poisson|Poisson]] (1831), [[Mikhail Vasilievich Ostrogradsky|Ostrogradsky]] (1834), and [[Carl Gustav Jakob Jacobi|Jacobi]] (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by [[Augustin Louis Cauchy|Cauchy]] (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), [[Otto Hesse|Hesse]] (1857), [[Alfred Clebsch|Clebsch]] (1858), and Carll (1885), but perhaps the most important work of the century{{Peacock term|date=March 2012}} is that of [[Karl Weierstrass|Weierstrass]]. His course on the theory may be asserted{{By whom|date=March 2012}} that he was the first to place calculus on a firm and rigorous foundation. | |
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| ==Applications==
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| The application of the [[infinitesimal calculus]] to problems in [[physics]] and [[astronomy]] was contemporary with the origin of the science. All through the 18th century these applications were multiplied, until at its close [[Pierre-Simon Laplace|Laplace]] and [[Joseph Louis Lagrange|Lagrange]] had brought the whole range of the study of forces into the realm of analysis. To [[Joseph Louis Lagrange|Lagrange]] (1773) we owe the introduction of the theory of the potential into dynamics, although the name "[[Potential function (disambiguation)|potential function]]" and the fundamental memoir of the subject are due to [[George Green|Green]] (1827, printed in 1828). The name "[[potential]]" is due to [[Carl Friedrich Gauss|Gauss]] (1840), and the distinction between potential and potential function to [[Rudolf Julius Emanuel Clausius|Clausius]]. With its development are connected the names of [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]], [[Bernhard Riemann|Riemann]], [[John von Neumann|von Neumann]], [[Eduard Heine|Heine]], [[Leopold Kronecker|Kronecker]], [[Rudolf Lipschitz|Lipschitz]], [[Elwin Bruno Christoffel|Christoffel]], [[Gustav Kirchhoff|Kirchhoff]], [[Eugenio Beltrami|Beltrami]], and many of the leading physicists of the century. | |
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| It is impossible in this place to enter into the great variety of other applications of analysis to physical problems. Among them are the investigations of Euler on vibrating chords; [[Sophie Germain]] on elastic membranes; Poisson, [[Gabriel Lamé|Lamé]], [[Jean Claude Saint-Venant|Saint-Venant]], and [[Alfred Clebsch|Clebsch]] on the [[theory of elasticity|elasticity]] of three-dimensional bodies; [[Jean Baptiste Joseph Fourier|Fourier]] on [[heat]] diffusion; [[Augustin-Jean Fresnel|Fresnel]] on [[light]]; [[James Clerk Maxwell|Maxwell]], [[Hermann von Helmholtz|Helmholtz]], and [[Heinrich Rudolf Hertz|Hertz]] on [[electricity]]; Hansen, Hill, and [[Hugo Gyldén|Gyldén]] on [[astronomy]]; Maxwell on [[spherical harmonic]]s; [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]] on [[acoustics]]; and the contributions of Lejeune Dirichlet, [[Wilhelm Eduard Weber|Weber]], [[Gustav Kirchhoff|Kirchhoff]], [[Franz Ernst Neumann|F. Neumann]], [[William Thomson, 1st Baron Kelvin|Lord Kelvin]], [[Rudolf Julius Emanuel Clausius|Clausius]], [[Vilhelm Bjerknes|Bjerknes]], [[James MacCullagh|MacCullagh]], and Fuhrmann to physics in general. The labors of Helmholtz should be especially mentioned, since he contributed to the theories of dynamics, electricity, etc., and brought his great analytical powers to bear on the fundamental axioms of mechanics as well as on those of pure mathematics.
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| Furthermore, infinitesimal calculus was introduced into the social sciences, starting with [[Neoclassical economics]]. Today, it is a valuable tool in mainstream economics.
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| ==Non-European antecedents of the calculus==
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| ===Indian mathematics===
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| See [[Kerala school of astronomy and mathematics]]. See also [[Yuktibhāṣā]].
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| ===Islamic mathematics===
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| In the 11th century, when [[Ibn al-Haytham]] (known as ''Alhacen'' in Europe), an [[Islamic mathematics|Iraqi mathematician]] working in Egypt, devised what is now known as "Alhazen's problem", which leads to an [[Quartic equation|equation of the fourth degree]], in his ''[[Book of Optics]]''. While solving this problem, he was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a [[paraboloid]], and was able to generalize his result for the integrals of [[polynomial]]s up to the [[Quartic polynomial|fourth degree]]. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.<ref name=Katz>{{cite journal |doi=10.2307/2691411 |author=Katz, Victor J. |title=Ideas of Calculus in Islam and India |jstor=2691411 |journal=Mathematics Magazine |volume=68 |issue=3 |pages=163–174 |year=1995 }} [165–9, 173–4]</ref>
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| ==See also==
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| * [[Analytic geometry]]
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| * [[Non-standard calculus]]
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| ==Notes==
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| {{Reflist}}
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| ==Further reading==
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| *{{cite book |author=Roero, C.S. |chapterurl=http://books.google.com/books?id=UdGBy8iLpocC&pg=PA46#v=onepage&q&f=false |chapter=Gottfried Wilhelm Leibniz, first three papers on the calculus (1684, 1686, 1693) |editor=Grattan-Guinness, I. |title=Landmark writings in Western mathematics 1640–1940 |url=http://books.google.com/books?id=UdGBy8iLpocC |year=2005 |publisher=Elsevier |isbn=978-0-444-50871-3 |pages=46–58 |editor-link=Ivor Grattan-Guinness}}
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| *{{cite journal |author=Roero, C.S. |title=Jakob Bernoulli, attentive student of the work of Archimedes: marginal notes to the edition of Barrow |journal=Boll. Storia Sci. Mat. |volume=3 |issue=1 |pages=77–125 |year=1983 }}
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| * [[Carl Boyer|Boyer, Carl]]. The History of Calculus. New York: Dover Publications, 1949
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| *{{cite book |last=Calinger |first=Ronald |title=A Contextual History of Mathematics |publisher=Prentice-Hall |location=Toronto |year=1999 |ref=harv |isbn=0-02-318285-7}}
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| *{{cite journal |last=Reyes |first=Mitchell |title=The Rhetoric in Mathematics: Newton, Leibniz, the Calculus, and the Rhetorical Force of the Infinitesimal |journal=[[Quarterly Journal of Speech]] |volume=90 |pages=159–184 |year=2004 |ref=harv}}
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| * [[Ivor Grattan-Guinness|Grattan-Guinness, Ivor]]. ''The Rainbow of Mathematics: A History of the Mathematical Sciences'', Chapters 5 and 6, W. W. Norton & Company, 2000.
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| ==External links==
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| * [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html A history of the calculus in The MacTutor History of Mathematics archive], 1996.
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| * [http://www.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.htm Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis]
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| * [http://cudl.lib.cam.ac.uk/collections/newton Newton Papers, Cambridge University Digital Library]
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| * {{en icon}} {{ar icon}} [http://www.wdl.org/en/item/4327/ The Excursion of Calculus], 1772
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| [[Category:Articles with inconsistent citation formats]]
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| [[Category:History of calculus| ]]
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