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A timeline of ** calculus ** and ** mathematical analysis **.

- 5th century BC - The Zeno's paradoxes,
- 5th century BC - Antiphon attempts to square the circle,
- 5th century BC - Democritus finds the volume of cone is 1/3 of volume of cylinder,
- 4th century BC - Eudoxus of Cnidus develops the method of exhaustion,
- 3rd century BC - Archimedes displays geometric series in
*The Quadrature of the Parabola.*Archimedes also discovers a method which is similar to differential calculus.^{ [1] } - 3rd century BC - Archimedes develops a concept of the indivisibles—a precursor to infinitesimals—allowing him to solve several problems using methods now termed as integral calculus. Archimedes also derives several formulae for determining the area and volume of various solids including sphere, cone, paraboloid and hyperboloid.
^{ [2] } - Before 50 BC - Babylonian cuneiform tablets show use of the Trapezoid rule to calculate of the position of Jupiter.
^{ [3] } - 3rd century - Liu Hui rediscovers the method of exhaustion in order to find the area of a circle.
- 4th century - The Pappus's centroid theorem,
- 5th century - Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere.
- 600 - Liu Zhuo is the first person to use second-order interpolation for computing the positions of the sun and the moon.
^{ [4] } - 665 - Brahmagupta discovers a second order Taylor interpolation for ,
- 862 - The Banu Musa brothers write the
*"Book on the Measurement of Plane and Spherical Figures"*, - 9th century - Thābit ibn Qurra discusses the quadrature of the parabola and the volume of different types of conic sections.
^{ [5] } - 12th century - Bhāskara II discovers a rule equivalent to Rolle's theorem for ,
- 14th century - Nicole Oresme proves of the divergence of the harmonic series,
- 14th century - Madhava discovers the power series expansion for , , and
^{ [6] }^{ [7] }This theory is now well known in the Western world as the Taylor series or infinite series.^{ [8] } - 14th century - Parameshvara discovers a third order Taylor interpolation for ,
- 1445 - Nicholas of Cusa attempts to square the circle,
- 1501 - Nilakantha Somayaji writes the
*Tantrasamgraha*, which contains the Madhava's discoveries, - 1548 - Francesco Maurolico attempted to calculate the barycenter of various bodies (pyramid, paraboloid, etc.),
- 1550 - Jyeshtadeva writes the
*Yuktibhāṣā*, a commentary to Nilakantha's*Tantrasamgraha*, - 1560 - Sankara Variar writes the
*Kriyakramakari*, - 1565 - Federico Commandino publishes
*De centro Gravitati*, - 1588 - Commandino's translation of Pappus'
*Collectio*gets published, - 1593 - François Viète discovers the first infinite product in the history of mathematics,

- 1606 - Luca Valerio applies methods of Archimedes to find volumes and centres of gravity of solid bodies,
- 1609 - Johannes Kepler computes the integral ,
- 1611 - Thomas Harriot discovers an interpolation formula similar to Newton's interpolation formula,
- 1615 - Johannes Kepler publishes
*Nova stereometria doliorum*, - 1620 - Grégoire de Saint-Vincent discovers that the area under a hyperbola represented a logarithm,
- 1624 - Henry Briggs publishes
*Arithmetica Logarithmica*, - 1629 - Pierre de Fermat discovers his method of maxima and minima, precursor of the derivative concept,
- 1634 - Gilles de Roberval shows that the area under a cycloid is three times the area of its generating circle,
- 1635 - Bonaventura Cavalieri publishes
*Geometria Indivisibilibus*, - 1637 - René Descartes publishes
*La Géométrie*, - 1638 - Galileo Galilei publishes
*Two New Sciences*, - 1644 - Evangelista Torricelli publishes
*Opera geometrica*, - 1644 - Fermat's methods of maxima and minima published by Pierre Hérigone,
- 1647 - Cavalieri computes the integral ,
- 1647 - Grégoire de Saint-Vincent publishes
*Opus Geometricum*, - 1650 - Pietro Mengoli proves of the divergence of the harmonic series,
- 1654 - Johannes Hudde discovers the power series expansion for ,
- 1656 - John Wallis publishes
*Arithmetica Infinitorum*, - 1658 - Christopher Wren shows that the length of a cycloid is four times the diameter of its generating circle,
- 1659 - Second edition of Van Schooten's Latin translation of Descartes' Geometry with appendices by Hudde and Heuraet,
- 1665 - Isaac Newton discovers the generalized binomial theorem and develops his version of infinitesimal calculus,
- 1667 - James Gregory publishes
*Vera circuli et hyperbolae quadratura*, - 1668 - Nicholas Mercator publishes
*Logarithmotechnia*, - 1668 - James Gregory computes the integral of the secant function,
- 1670 - Isaac Newton rediscovers the power series expansion for and (originally discovered by Madhava),
- 1670 - Isaac Barrow publishes
*Lectiones Geometricae*, - 1671 - James Gregory rediscovers the power series expansion for and (originally discovered by Madhava),
- 1672 - René-François de Sluse publishes
*A Method of Drawing Tangents to All Geometrical Curves*, - 1673 - Gottfried Leibniz also develops his version of infinitesimal calculus,
- 1675 - Isaac Newton invents a Newton's method for the computation of roots of a function,
- 1675 - Leibniz uses the modern notation for an integral for the first time,
- 1677 - Leibniz discovers the rules for differentiating products, quotients, and the function of a function.
- 1683 - Jacob Bernoulli discovers the number e ,
- 1684 - Leibniz publishes his first paper on calculus,
- 1686 - The first appearance in print of the notation for integrals,
- 1687 - Isaac Newton publishes
*Philosophiæ Naturalis Principia Mathematica*, - 1691 - The first proof of Rolle's theorem is given by Michel Rolle,
- 1691 - Leibniz discovers the technique of separation of variables for ordinary differential equations,
- 1694 - Johann Bernoulli discovers the L'Hôpital's rule,
- 1696 - Guillaume de L'Hôpital publishes
*Analyse des Infiniment Petits*, the first calculus textbook, - 1696 - Jakob Bernoulli and Johann Bernoulli solve the brachistochrone problem, the first result in the calculus of variations.

- 1711 - Isaac Newton publishes
*De analysi per aequationes numero terminorum infinitas*, - 1712 - Brook Taylor develops Taylor series,
- 1722 - Roger Cotes computes the derivative of sine in his
*Harmonia Mensurarum*, - 1730 - James Stirling publishes
*The Differential Method*, - 1734 - George Berkeley publishes
*The Analyst*, - 1734 - Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations,
- 1735 - Leonhard Euler solves the Basel problem, relating an infinite series to π,
- 1736 - Newton's Method of Fluxions posthumously published,
- 1737 - Thomas Simpson publishes
*Treatise of Fluxions*, - 1739 - Leonhard Euler solves the general homogeneous linear ordinary differential equation with constant coefficients,
- 1742 - Modern definion of logarithm by William Gardiner,
- 1742 - Colin Maclaurin publishes
*Treatise on Fluxions*, - 1748 - Euler publishes
*Introductio in analysin infinitorum*, - 1748 - Maria Gaetana Agnesi discusses analysis in
*Instituzioni Analitiche ad Uso della Gioventu Italiana*, - 1762 - Joseph Louis Lagrange discovers the divergence theorem,
- 1797 - Lagrange publishes
*Théorie des fonctions analytiques*,

- 1807 - Joseph Fourier announces his discoveries about the trigonometric decomposition of functions,
- 1811 - Carl Friedrich Gauss discusses the meaning of integrals with complex limits and briefly examines the dependence of such integrals on the chosen path of integration,
- 1815 - Siméon Denis Poisson carries out integrations along paths in the complex plane,
- 1817 - Bernard Bolzano presents the intermediate value theorem — a continuous function which is negative at one point and positive at another point must be zero for at least one point in between,
- 1822 - Augustin-Louis Cauchy presents the Cauchy integral theorem for integration around the boundary of a rectangle in the complex plane,
- 1825 - Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths—he assumes the function being integrated has a continuous derivative, and he introduces the theory of residues in complex analysis,
- 1825 - André-Marie Ampère discovers Stokes' theorem,
- 1828 - George Green introduces Green's theorem,
- 1831 - Mikhail Vasilievich Ostrogradsky rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green,
- 1841 - Karl Weierstrass discovers but does not publish the Laurent expansion theorem,
- 1843 - Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem,
- 1850 - Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points,
- 1850 - George Gabriel Stokes rediscovers and proves Stokes' theorem,
- 1861 - Karl Weierstrass starts to use the language of epsilons and deltas,
- 1873 - Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points,

- 1908 - Josip Plemelj solves the Riemann problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky - Plemelj formulae,
- 1966 - Abraham Robinson presents non-standard analysis.
- 1985 - Louis de Branges de Bourcia proves the Bieberbach conjecture,

- Timeline of ancient Greek mathematicians – Timeline and summary of ancient Greek mathematicians and their discoveries
- Timeline of geometry
- Timeline of mathematical logic
- Timeline of mathematics

**Calculus** is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

In mathematics, an **integral** is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.

The number **π** is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulae across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, the **Taylor series** or **Taylor expansion** of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a **Maclaurin series** when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.

In mathematics, **differential calculus** is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

In mathematics, an **infinitesimal number** is a quantity that is closer to zero than any standard real number, but that is not zero. The word *infinitesimal* comes from a 17th-century Modern Latin coinage *infinitesimus*, which originally referred to the "infinity-th" item in a sequence.

Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite 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 by Isaac Newton and Gottfried Wilhelm Leibniz independently of each other. An argument over priority led to the Leibniz–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 day.

**Indian mathematics** emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, and Varāhamihira. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made significant early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.

In calculus, the **Leibniz integral rule** for differentiation under the integral sign states that for an integral of the form

**Mādhava of Sangamagrāma** (**Mādhavan**) was an Indian mathematician and astronomer who is considered as the founder of the Kerala school of astronomy and mathematics. One of the greatest mathematician-astronomers of the Late Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity".

The **Kerala school of astronomy and mathematics** or the **Kerala school** was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and its original discoveries seem to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called *Tantrasangraha*, and again in a commentary on this work, called *Tantrasangraha-vakhya*, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work *Yuktibhasa*, written in Malayalam, by Jyesthadeva, and also in a commentary on *Tantrasangraha*.

* Yuktibhāṣā*, also known as

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).

The Swiss mathematician **Leonhard Euler** (1707–1783) is among the most prolific and successful mathematicians in the history of the field. His seminal work had a profound impact in numerous areas of mathematics and he is widely credited for introducing and popularizing modern notation and terminology.

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

In mathematics, a **Madhava series** is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century Kerala by the mathematician and astronomer Madhava of Sangamagrama or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are:

- ↑ "History of the Calculus -- Differential and Integral Calculus".
*www.edinformatics.com*. Retrieved 2022-11-03. - ↑ Plummer, Brad (2006-08-09). "Modern X-ray technology reveals Archimedes' math theory under forged painting".
*Stanford University*. Retrieved 2022-11-03. - ↑ Ossendrijver, Mathieu (Jan 29, 2016). "Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph".
*Science*.**351**(6272): 482–484. doi:10.1126/science.aad8085. PMID 26823423. S2CID 206644971. - ↑ "On Squares, Rectangles, and Square Roots - Square roots in ancient Chinese mathematics | Mathematical Association of America".
*www.maa.org*. Retrieved 2022-11-03. - ↑ "Conic Sections: A Resource for Teachers and Students of Mathematics".
*jwilson.coe.uga.edu*. Retrieved 2022-11-03. - ↑ Weisstein, Eric W. "Taylor Series".
*mathworld.wolfram.com*. Retrieved 2022-11-03. - ↑ "The Taylor Series: an Introduction to the Theory of Functions of a Complex Variable".
*Nature*.**130**(3275): 188. August 1932. Bibcode:1932Natur.130R.188.. doi: 10.1038/130188b0 . ISSN 1476-4687. S2CID 4088442. - ↑ Saeed, Mehreen (2021-08-19). "A Gentle Introduction to Taylor Series".
*Machine Learning Mastery*. Retrieved 2022-11-03.

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