Calculus, originally calledinfinitesimal calculus, is a mathematical discipline focused onlimits,continuity,derivatives,integrals, andinfinite series. Many elements of calculus appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India. Infinitesimal calculus was developed in the late 17th century byIsaac Newton andGottfried Wilhelm Leibniz independently of each other. An argument over priority led to theLeibniz–Newton calculus controversy which continued until the death of Leibniz in 1716. The development of calculus and its uses within the sciences have continued to the present.

Inmathematics education,calculus denotes courses of elementarymathematical analysis, which are mainly devoted to the study offunctions and limits. The wordcalculus isLatin for "small pebble" (thediminutive ofcalx, meaning "stone"), a meaning which stillpersists in medicine. Because such pebbles were used for counting out distances,^[1] tallying votes, and doingabacus arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton.^[2]

In addition to the differential calculus and integral calculus, the term is also used widely for naming specific methods of calculation. Examples of this includepropositional calculus in logic, thecalculus of variations in mathematics,process calculus in computing, and thefelicific calculus in philosophy.

The ancient period introduced some of the ideas that led tointegral 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 EgyptianMoscow 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.[3]Babylonians may have discovered thetrapezoidal rule while doing astronomical observations ofJupiter.^[4][5]

From the age of Greek mathematics,Eudoxus (c. 408–355 BC) used themethod of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, whileArchimedes (c. 287–212 BC)developed this idea further, inventingheuristics which resemble the methods of integral calculus.^[6] Greek mathematicians are also credited with a significant use ofinfinitesimals.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 theparadoxes which they seemingly create.

Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat in hisThe Quadrature of the Parabola,The Method, andOn the Sphere and Cylinder.^[7] 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 17th century that the method was formalized byCavalieri as themethod of Indivisibles and eventually incorporated byNewton into a general framework ofintegral 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.^[8]

Themethod of exhaustion was independently invented inChina byLiu Hui in the 4th century AD in order to find the area of a circle.^[9] In the 5th century,Zu Chongzhi established a method that would later be calledCavalieri's principle to find the volume of asphere.^[10]

In the Middle East,Hasan Ibn al-Haytham, Latinized as Alhazen (c. 965 – c. 1040 AD) derived a formula for the sum offourth powers. He determined the equations to calculate the area enclosed by the curve represented by${\displaystyle y=x^k}$ (which translates to the integral${\displaystyle \int x^{k}dx}$ in contemporary notation), for any given non-negative integer value of${\displaystyle k}$.^[11] He used the results to carry out what would now be called anintegration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of aparaboloid.^[12]Roshdi Rashed has argued that the 12th century mathematicianSharaf al-Dīn al-Tūsī must have used the derivative of cubic polynomials in hisTreatise on Equations. Rashed's conclusion has been contested by other scholars, who argue that he could have obtained his results by other methods which do not require the derivative of the function to be known.^[13]

Bhāskara suggested the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'.^[14] There is evidence of an early form ofRolle's theorem in his work, though it was stated without a modern formal proof.^[15][16] In his astronomical work, Bhāskara gives a result that looks like a precursor to infinitesimal methods: if${\displaystyle x\approx y}$ then${\displaystyle \sin (y)-\sin (x)\approx (y-x)\cos (y)}$. This can be interpreted as the discovery that cosine is the derivative of sine,although he did not develop the notion of a derivative.^[16]

Some ideas on calculus later appeared in Indian mathematics, at theKerala school of astronomy and mathematics.^[12]Madhava of Sangamagrama in the 14th century, and later mathematicians of the Kerala school, stated components of calculus such as theTaylor series andinfinite series approximations.^[17] They considered series equivalent to the Maclaurin expansions of⁠${\displaystyle \sin (x)}$⁠,⁠${\displaystyle \cos (x)}$⁠, and⁠${\displaystyle \arctan (x)}$⁠ more than two hundred years before they were studied in Europe. According toVictor J. Katz they were not able to "combine many differing ideas under the two unifying themes of thederivative and theintegral, show the connection between the two, and turn calculus into the great problem-solving tool we have today."^[12]

The mathematical study of continuity was revived in the 14th century by theOxford Calculators and French collaborators such asNicole Oresme. They proved the "Mertonmean 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.^[18]

Johannes Kepler's workStereometrica Doliorum published in 1615 formed the basis of integral calculus.^[19] Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.^[20]

A significant work was a treatise inspired by Kepler's methods^[20] published in 1635 byBonaventura Cavalieri on hismethod of indivisibles. He argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. He discoveredCavalieri's quadrature formula which gave the area under the curvesxⁿ of higher degree. This had previously been computed in a similar way for the parabola by Archimedes inThe Method, but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. 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.

Torricelli extended Cavalieri's work to other curves such as thecycloid, 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.^[21] Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature.

In the 17th century, European mathematiciansIsaac Barrow,René Descartes,Pierre de Fermat,Blaise Pascal,John Wallis and others discussed the idea of aderivative. In particular, inMethodus ad disquirendam maximam et minima and inDe tangentibus linearum curvarum distributed in 1636, Fermat introduced the concept ofadequality, which represented equality up to an infinitesimal error term.^[22] This method could be used to determine the maxima, minima, and tangents to various curves and was closely related to differentiation.^[23]

Isaac Newton would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents".^[24]

The formal study of calculus brought together Cavalieri's infinitesimals with thecalculus of finite differences developed in Europe at around the same time, and Fermat's adequality. The combination was achieved byJohn Wallis,Isaac Barrow, andJames Gregory, the latter two proving predecessors to the secondfundamental theorem of calculus around 1670.^[25][26]

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, that integrals can be computed using any of a function's antiderivatives.^[27][28]

The first full proof of the fundamental theorem of calculus was given byIsaac Barrow.^{[29]: p.61 when arc ME ~ arc NH at point of tangency F fig.26 [30]}

One prerequisite to the establishment of a calculus of functions of areal variable involved finding anantiderivative for therational function${\displaystyle f(x)=\frac{1}{x}.}$ This problem can be phrased asquadrature of the rectangular hyperbolaxy = 1. In 1647Gregoire de Saint-Vincent noted that the required functionF satisfied${\displaystyle F(st)=F(s)+F(t),}$ so that ageometric sequence became, underF, anarithmetic sequence.A. A. de Sarasa associated this feature with contemporary algorithms calledlogarithms that economized arithmetic by rendering multiplications into additions. SoF was first known as thehyperbolic logarithm. AfterEuler exploited e = 2.71828..., andF was identified as theinverse function of theexponential function, it became thenatural logarithm, satisfying${\displaystyle \frac{dF}{dx}=\frac{1}{x}.}$

The first proof ofRolle's theorem was given byMichel Rolle in 1691 using methods developed by the Dutch mathematicianJohann van Waveren Hudde.^[31] The mean value theorem in its modern form was stated byBernard Bolzano andAugustin-Louis Cauchy (1789–1857) also after the founding of modern calculus. Important contributions were made by Barrow,Huygens, and many others.

Barrow has been credited by some authors as having invented calculus, however, Swiss mathematicianFlorian Cajori notes that while Barrow did work out a set of "geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus", he did not create "what by common agreement of mathematicians has been designated by the term differential and integral calculus", and further notes that "Two processes yielding equivalent results are not necessarily the same." Cajori finishes with stating that "The invention rightly belongs to Newton and Leibniz."^[32]

BeforeNewton andLeibniz, the word "calculus" referred to any body of mathematics, but in the following years, "calculus" became a popular term for a field of mathematics based upon their insights.^[33] 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 ofintegral calculus.

By the middle of the 17th century, European mathematics had changed its primary repository of knowledge. In comparison to the last century which maintainedHellenistic mathematics as the starting point for research, Newton, Leibniz and their contemporaries increasingly looked towards the works of more modern thinkers.^[34]

Newton came to calculus as part of his investigations inphysics andgeometry. He viewed calculus as the scientific description of the generation of motion andmagnitudes. In comparison, Leibniz focused on thetangent problem and came to believe that calculus was ametaphysical explanation of change. Importantly, the core of their insight was the formalization of the inverse properties between theintegral and thedifferential 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.^[35]

Newton completed no definitive publication formalizing hisfluxional calculus; rather, many of his mathematical discoveries were transmitted through correspondence, smaller papers or as embedded aspects in his other definitive compilations, such as thePrincipia andOpticks. Newton would begin his mathematical training as the chosen heir ofIsaac Barrow inCambridge. His aptitude was recognized early and he quickly learned the current theories. By 1664 Newton had made his first important contribution by advancing thebinomial theorem, which he had extended to include fractional and negativeexponents. Newton succeeded in expanding the applicability of the binomial theorem by applying the algebra of finite quantities in an analysis ofinfinite series. He showed a willingness to view infinite series not only as approximate devices, but also as alternative forms of expressing a term.^[36]

Newton formulated his calculus between 1664 and 1666,^[37][38][39] later describing them as, "the prime of my age for invention and minded mathematics and [natural] philosophy more than at any time since." A manuscript dated May 20, 1665 showed that 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."^[40] It was during his plague-induced isolation that the first written conception offluxionary calculus was recorded in the unpublishedDe Analysi per Aequationes Numero Terminorum Infinitas. In this paper, he determined the area under acurve 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 ofx andy. He then reasoned that theinfinitesimal increase in the abscissa will create a new formula wherex =x +o (importantly,o is the letter, not thedigit 0). He then recalculated the area with the aid of the binomial theorem, removed all quantities containing the lettero and re-formed an algebraic expression for the area. Significantly, Newton would then "blot out" the quantities containingo because terms "multiplied by it will be nothing in respect to the rest".

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, thefundamental 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".

In an effort to give calculus a more rigorous explication and framework, Newton compiled in 1671 theMethodus Fluxionum et Serierum Infinitarum. In this book, Newton's strictempiricism shaped and defined his fluxional calculus. He exploitedinstantaneousmotion and infinitesimals informally. He used math as amethodological 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. As with many of his works, Newton delayed publication.Methodus Fluxionum was not published until 1736.^[41]

Newton attempted to avoid the use of the infinitesimal by forming calculations based onratios of changes. In theMethodus Fluxionum he defined the rate of generated change as afluxion, which he represented by a dotted letter, and the quantity generated he defined as afluent. For example, if${\displaystyle x}$ and${\displaystyle y}$ are fluents, then${\displaystyle \dot{x}}$ and${\displaystyle \dot{y}}$ are their respective fluxions. This revised calculus of ratios continued to be developed and was maturely stated in the 1676 textDe 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:^[42]

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

Newton developed his fluxional calculus in an attempt to evade the informal use of infinitesimals in his calculations.

HistorianA. Rupert Hall noted Newton's rapid development of calculus in comparison to 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".^[43] Hall also mentions that while Newton did not employ fluxions inPrincipia, he was able to solve problems by using methods equivalent to 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."^[44]

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 apolymath, and his intellectual interests and achievements involvedmetaphysics,law,economics,politics,logic, andmathematics. In order to understand Leibniz's reasoning in calculus his background should be kept in mind. Particularly, hismetaphysics which described the universe as aMonadology, 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".^[45]

In 1672, Leibniz met the mathematicianHuygens who convinced Leibniz to dedicate significant time to the study of mathematics. By 1673 he had progressed to readingPascal'sTraité des sinus du quart de cercle and it was during his largelyautodidactic research that Leibniz said "a light turned on". Like Newton, Leibniz saw the tangent as a ratio but declared it as simply the ratio betweenordinates andabscissas. He continued this reasoning to argue that theintegral 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 over the course of his career used several approaches in addition to an approach usinginfinitesimals, Leibniz made this the cornerstone of his notation and calculus.^[46][47]

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 and his earlier plans to form a precise logicalsymbolism became evident. Eventually, Leibniz denoted the infinitesimal increments of abscissas and ordinatesdx anddy, and the summation of infinitely many infinitesimally thin rectangles as along s (∫ ), which became the present integral symbol${\displaystyle \int }$.

While Leibniz's notation is used by modern mathematics, his logical base was different from our current one. Leibniz embraced infinitesimals and wrote extensively so as, "not to make of the infinitely small a mystery, as had Pascal."^[48] According toGilles 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").^[49] Alternatively, he defines them as, "less than any given quantity". For Leibniz, the world was an aggregate of infinitesimal points 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.^[50]

The rise of calculus stands out as a unique moment in mathematics. Calculus is the mathematics 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.

Historically, there was much debate over whether it was Newton or Leibniz who first "invented" calculus. This argument, theLeibniz 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 oftangents by the time Leibniz became interested in the question.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 ofplagiarism.

The priority dispute had an effect of separating English-speaking mathematicians from those in continental Europe for many years. Only in the 1820s, due to the efforts of theAnalytical Society, didLeibnizian 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 offluents andfluxions".

While neither of the two offered convincing logical foundations for their calculus according to mathematicianCarl B. Boyer, Newton was the closer of the two. Newton's best attempt was inPrincipia, where he introduced his idea of “prime and ultimate ratios” and came “extraordinarily close to thelimit concept,” although he continued to describe the derivative as “a ratio of velocities” rather than as a singlereal number—which would not be fully defined until the late nineteenth century.^[51] In contrast, according to Boyer, the calculus of Leibniz was from a "logical point of view, distinctly inferior to that of Newton, for it never transcended the view of${\displaystyle \frac{dy}{dx}}$ as a quotient of infinitely small changes or differences iny andx."^[51] However, heuristically, it was a success, despite being a "failure" from a logical point of view.^[51]

The work of both Newton and Leibniz is reflected in the notation used today. Newton introduced the notation${\displaystyle \dot{f}}$ for thederivative of a functionf.^[52] Leibniz introduced the symbol${\displaystyle \int }$ for theintegral and wrote thederivative of a functiony of the variablex as${\displaystyle \frac{dy}{dx}}$, both of which are still in use.

Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal andintegral calculus was written in 1748 byMaria Gaetana Agnesi.^[53][54]

Thecalculus of variations began with the work ofIsaac Newton, such as withNewton's minimal resistance problem,^[55] which Newton formulated and solved in 1685, and later published in hisPrincipia in 1687,^[56] and which was the first problem in the field to be formulated and correctly solved, and was one of the most difficult problems tackled by variational methods prior to the twentieth century.^[56][57][58] It was followed by thebrachistochrone curve problem ofJohann Bernoulli (1696), which Bernoulli solved, using the principle of least time, but not the calculus of variations, whereas Newton did to solve it in 1697, thus he pioneered the field with his work on the two problems.^[58] The problem was similar to one raised byGalileo Galilei in 1638, but he did not solve the problem explicitly nor did he use the methods based on calculus.^[57] It immediately occupied the attention ofJakob Bernoulli butLeonhard Euler first elaborated the subject. His contributions began in 1733, and hisElementa Calculi Variationum gave to the science its name.Joseph Louis Lagrange contributed extensively to the theory, andAdrien-Marie Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. To this discrimination Brunacci (1810),Carl Friedrich Gauss (1829),Siméon Denis Poisson (1831),Mikhail Vasilievich Ostrogradsky (1834), andCarl Gustav Jacob Jacobi (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved byAugustin Louis Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850),Otto Hesse (1857),Alfred Clebsch (1858), and Carll (1885), but perhaps the most important work of the century is that ofKarl Weierstrass. His course on the theory may be asserted to be the first to place calculus on a firm and rigorous foundation.

Antoine Arbogast (1800) was the first to separate the symbol of operation from that of quantity in a differential equation.Francois-Joseph Servois (1814) seems to have been the first to give correct rules on the subject.Charles James Hargreave (1848) applied these methods in his memoir on differential equations, andGeorge Boole freely employed them.Hermann Grassmann andHermann Hankel made great use of the theory, the former in studyingequations, the latter in his theory ofcomplex numbers.

Niels Henrik Abel seems to have been the first to consider in a general way the question as to whatdifferential equations can be integrated in a finite form by the aid of ordinary functions, an investigation extended byLiouville.Cauchy early undertook the general theory of determiningdefinite integrals, and the subject has been prominent during the 19th century.Frullani integrals,David Bierens de Haan's work on the theory and his elaborate tables,Lejeune Dirichlet's lectures embodied inMeyer's treatise, and numerous memoirs ofLegendre,Poisson,Plana,Raabe,Sohncke,Schlömilch,Elliott,Leudesdorf andKronecker are among the noteworthy contributions.

Eulerian integrals were first studied byEuler and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows:

${\displaystyle {\int }_0^1{x}^{n-1}(1-x{)}^{n-1}dx}$

${\displaystyle {\int }_0^{\mathrm{\infty }}{e}^{-x}{x}^{n-1}dx}$

although these were not the exact forms of Euler's study.

Ifn is a positiveinteger:

${\displaystyle {\int }_0^{\mathrm{\infty }}{e}^{-x}{x}^{n-1}dx=(n-1)!,}$

but the integral converges for all positive real${\displaystyle n}$ and defines ananalytic continuation of thefactorial function to all of thecomplex plane except for poles at zero and the negative integers. To it Legendre assigned the symbol${\displaystyle \mathrm{\Gamma }}$, and it is now called thegamma function. Besides being analytic over positive reals${\displaystyle {\mathbb{R}}^{+}}$,${\displaystyle \mathrm{\Gamma }}$ also enjoys the uniquely defining property that${\displaystyle \log \mathrm{\Gamma }}$ isconvex, which aesthetically justifies this analytic continuation of the factorial function over any other analytic continuation. To the subjectLejeune Dirichlet has contributed an important theorem (Liouville, 1839), which has been elaborated byLiouville,Catalan,Leslie Ellis, and others.Raabe (1843–44), Bauer (1859), andGudermann (1845) have written about the evaluation of${\displaystyle \mathrm{\Gamma }(x)}$ and${\displaystyle \log \mathrm{\Gamma }(x)}$. Legendre's great table appeared in 1816.

The application of theinfinitesimal calculus to problems inphysics andastronomy was contemporary with the origin of the science. All through the 18th century these applications were multiplied, until at its closeLaplace andLagrange had brought the whole range of the study of forces into the realm of analysis. ToLagrange (1773) we owe the introduction of the theory of the potential into dynamics, although the name "potential function" and the fundamental memoir of the subject are due toGreen (1827, printed in 1828). The name "potential" is due toGauss (1840), and the distinction between potential and potential function toClausius. With its development are connected the names ofLejeune Dirichlet,Riemann,von Neumann,Heine,Kronecker,Lipschitz,Christoffel,Kirchhoff,Beltrami, and many of the leading physicists of the century.

It is impossible in this article 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,Lamé,Saint-Venant, andClebsch on theelasticity of three-dimensional bodies;Fourier onheat diffusion;Fresnel onlight;Maxwell,Helmholtz, andHertz onelectricity; Hansen, Hill, andGyldén onastronomy; Maxwell onspherical harmonics;Lord Rayleigh onacoustics; and the contributions of Lejeune Dirichlet,Weber,Kirchhoff,F. Neumann,Lord Kelvin,Clausius,Bjerknes,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.

Furthermore, infinitesimal calculus was introduced into the social sciences, starting withNeoclassical economics. Today, it is a valuable tool in mainstream economics.

• Analytic geometry
• History of logarithms
• Nonstandard calculus

• Calinger, Ronald (1999).A Contextual History of Mathematics. Prentice-Hall.ISBN 978-0-02-318285-3.OCLC 40479696.
• Reyes, Mitchell (2004). "The Rhetoric in Mathematics: Newton, Leibniz, the Calculus, and the Rhetorical Force of the Infinitesimal".Quarterly Journal of Speech.90 (2):159–184.doi:10.1080/0033563042000227427.S2CID 145802382.

• Grattan-Guinness, Ivor (2000).The Rainbow of Mathematics : A History of the Mathematical Sciences. W.W. Norton.ISBN 978-0-393-32030-5.OCLC 44155819.Zbl 0876.01003.. Ch. 5 The age of trigonometry : Europe, 1540–1660 and Ch. 6 The calculus and its consequences, 1660–1750.
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• Roero, C.S. (2005)."Gottfried Wilhelm Leibniz, first three papers on the calculus (1684, 1686, 1693)". InGrattan-Guinness, I. (ed.).Landmark writings in Western mathematics 1640–1940. Elsevier. pp. 46–58.ISBN 978-0-444-50871-3.Zbl 1090.01002.
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