Timeline of calculus and mathematical analysis

A timeline of calculus and mathematical analysis.

500BC to 1600

• 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,
• 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,
• 665 - Brahmagupta discovers a second order Taylor interpolation for [math]\displaystyle{ \sin (x+\epsilon) }[/math],
• 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 the paraboloid,
• 11th century - Ibn al-Haytham derived a formula for the sum of fourth powers which allowed him to calculate the volume of a paraboloid,
• 12th century - Sharaf al-Din al-Tusi finds the minimum and maximum values of cubic functions,
• 12th century - Bhāskara II discovers a rule equivalent to Rolle's theorem for [math]\displaystyle{ \sin x }[/math],
• 14th century - Nicole Oresme proves of the divergence of the harmonic series,
• 14th century - Madhava discovers the power series expansion for [math]\displaystyle{ \sin x }[/math], [math]\displaystyle{ \cos x }[/math], [math]\displaystyle{ \arctan x }[/math] and [math]\displaystyle{ \pi/4 }[/math],
• 14th century - Parameshvara discovers a third order Taylor interpolation for [math]\displaystyle{ \sin (x+\epsilon) }[/math],
• 1501 - Nilakantha Somayaji writes the Tantrasamgraha, which contains the Madhava's discoveries,
• 1548 - Francesco Maurolico attemptes 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,

17th century

• 1606 - Luca Valerio applies methods of Archimedes to find volumes and centres of gravity of solid bodies,
• 1609 - Johannes Kepler computes the integral [math]\displaystyle{ \int_0^\theta \sin x\ dx = 1-\cos \theta }[/math],
• 1611 - Thomas Harriot discovers an interpolation formula similar to Newton's interpolation formula,
• 1615 - Johannes Kepler publishes Nova stereometria doliorum,
• 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 [math]\displaystyle{ \int_0^a x^{n}\ dx=\frac{1}{n+1}a^{n+1} }[/math],
• 1647 - Grégoire de Saint-Vincent discovers that the area under a hyperbola represented a logarithm,
• 1650 - Pietro Mengoli proves of the divergence of the harmonic series,
• 1654 - Johannes Hudde discovers the power series expansion for [math]\displaystyle{ \ln(1+x) }[/math],
• 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 [math]\displaystyle{ \sin x }[/math] and [math]\displaystyle{ \cos x }[/math] (originally discovered by Madhava),
• 1670 - Isaac Barrow publishes Lectiones Geometricae,
• 1671 - James Gregory rediscovers the power series expansion for [math]\displaystyle{ \arctan x }[/math] and [math]\displaystyle{ \pi/4 }[/math] (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 [math]\displaystyle{ \int }[/math] 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.

18th century

• 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,

19th century

• 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,
• 1873 - Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points,

20th century

• 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,