A timeline of **Pure and Applied Mathematics**

- 2450 BC - Egypt, first systematic method for the approximative calculation of the circle on the basis of the Sacred Triangle 3-4-5,
- 1650 BC - Rhind Papyrus, copy of a lost scroll from around 1850 BC, the scribe Ahmes presents first known aproximate value of π at 3.16 and first attempt at squaring the circle.
- 530 BC - Pythagoras studies propositional geometry and vibrating lyre strings; his group discovers the irrationality of the square root of two,
- 370 BC - Eudoxus states the method of exhaustion for area determination,
- 350 BC - Aristotle discusses logical reasoning in
*Organon*, - 300 BC - Euclid in his
*Elements*studies geometry as an axiomatic system, proves the infinitude of prime numbers and presents the Euclidean algorithm; he states the law of reflection in*Catoptrics*, - 260 BC - Archimedes computes &pi to two decimal places using inscribed and circumscribed polygons and computes the area under a parabolic segment,
- 225 BC - Apollonius of Perga writes
*On Conic Sections*and names the ellipse, parabola, and hyperbola, - 200 BC ?240 BC - Eratosthenes uses his sieve algorithm to isolate prime numbers and finds the number of primes is infinite,
- 140 BC - Hipparchus develops the bases of trigonometry,
- 250 - Diophantus uses symbols for unknown numbers in terms of the syncopated algebra,
- 250 - Diophantus writes
*Arithmetica*the first systematic treatise on algebra, - 450 - Tsu Ch'ung-Chih and Tsu Kęng-Chih compute π to six decimal places,
- 550 - Hindu mathematicians give zero a numeral representation in a positional notation system,
- 628 - Brahmagupta writes
*Brahma- sphuta- siddhanta*, - 750 - Al-Khawarizmi - Considered father of modern algebra. First mathematician to work on the details of 'Arithmetic and Algebra of inheritance' besides the systematisation of the theory of linear and quadratic equations.
- 895 - Thabit ibn Qurra - The only surviving fragment of his original work contains a chapter on the solution and properties of cubic equations.
- 975 - Al-Batani - Extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tangent, secant and their reciprocals. Derived the formula: sin α = tan α / (1+tan˛ α) and cos α = 1 / (1 + tan˛ α).
- 1020 - Abul Wafa - Gave this famous formula: sin (α + β) = sin α cos β + sin β cos α. Also discussed the quadrature of the parabola and the volume of the paraboloid.
- 1030 - Ali Ahmed Nasawi - Develops the division of days into 24 hours, hours into 60 minutes and minutes into 60 seconds.
- 1070 - Omar Khayyam begins to write
*Treatise on Demonstration of Problems of Algebra*and classifies cubic equations. Invented the second and third degree of quadratic equations. - 1202 - Leonardo Fibonacci demonstrates the utility of Arabic numerals in his
*Book of the Abacus*, - 1424 - Ghiyath al-Kashi - computes π to sixteen decimal places using inscribed and circumscribed polygons,
- 1520 - Scipione dal Ferro develops a method for solving cubic equations,
- 1535 - Niccolo Tartaglia develops a method for solving cubic equations,
- 1540 - Lodovico Ferrari solves the quartic equation,
- 1596 - Ludolf van Ceulen computes π to twenty decimal places using inscribed and circumscribed polygons,
- 1614 - John Napier discusses Napierian logarithms in
*Mirifici Logarithmorum Canonis Descriptio*, - 1617 - Henry Briggs discusses decimal logarithms in
*Logarithmorum Chilias Prima*, - 1619 - René Descartes discovers analytic geometry,
- 1629 - Pierre de Fermat develops a rudimentary differential calculus,
- 1634 - Gilles de Roberval shows that the area under a cycloid is three times the area of its generating circle,
- 1637 - Pierre de Fermat claims to have proven Fermat's last theorem in his copy of Diophantus'
*Arithmetica*, - 1654 - Blaise Pascal and Pierre de Fermat create the theory of probability,
- 1655 - John Wallis writes
*Arithmetica Infinitorum*, - 1658 - Christopher Wren shows that the length of a cycloid is four times the diameter of its generating circle,
- 1665 - Isaac Newton invents his calculus,
- 1668 - Nicholas Mercator and William Brouncker discover an infinite series for the logarithm while attempting to calculate the area under a hyperbolic segment,
- 1671 - James Gregory discovers the series expansion for the inverse-tangent function,
- 1673 - Gottfried Leibniz invents his calculus,
- 1675 - Isaac Newton invents an algorithm for the computation of functional roots,
- 1691 - Gottfried Leibniz discovers the technique of separation of variables for ordinary differential equations,
- 1693 - Edmund Halley prepares the first mortality tables statistically relating death rate to age,
- 1696 - Guillaume de L'Hôpital states his rule for the computation of certain limits,
- 1696 - Jakob Bernoulli and Johann Bernoulli solve brachistochrone problem, the first result in the calculus of variations,
- 1706 - John Machin develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places,
- 1712 - Brook Taylor develops Taylor series,
- 1722 - Abraham De Moivre states De Moivre's theorem connecting trigonometric functions and complex numbers,
- 1724 - Abraham De Moivre studies mortality statistics and the foundation of the theory of annuities in
*Annuities on Lives*, - 1730 - James Stirling publishes
*The Differential Method*, - 1733 - Giovanni Gerolamo Saccheri studies what geometry would be like if Euclid's fifth postulate were false,
- 1733 - Abraham de Moivre introduces the normal distribution to approximate the binomial distribution in probability,
- 1734 - Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations,
- 1736 - Leonhard Euler solves the problem of the Seven bridges of Königsberg, in effect creating graph theory,
- 1739 - Leonhard Euler solves the general homogenous linear ordinary differential equation with constant coefficients,
- 1742 - Christian Goldbach conjectures that every even number greater than two can be expressed as the sum of two primes, now known as Goldbach's conjecture,
- 1748 - Maria Gaetana Agnesi discusses analysis in
*Instituzioni Analitiche ad Uso della Gioventu Italiana*, - 1761 - Thomas Bayes proves Bayes' theorem,
- 1762 - Joseph Louis Lagrange discovers the divergence theorem,
- 1789 - Jurij Vega improves Machin's formula and computes π to 140 decimal places,
- 1794 - Jurij Vega publishes
*Thesaurus Logarithmorum Completus*, - 1796 - Carl Friedrich Gauss presents a method for constructing a heptadecagon using only a compass and straightedge and also shows that only polygons with certain numbers of sides can be constructed,
- 1796 - Adrien-Marie Legendre conjectures the prime number theorem,
- 1797 - Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms,
- 1799 - Carl Friedrich Gauss proves that every polynomial equation has a solution among the complex numbers,
- 1805 - Adrien-Marie Legendre introduces the method of least squares for fitting a curve to a given set of observations,
- 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,
- 1824 - Niels Henrik Abel partially proves that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots,
- 1825 - Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths -- he assumes the function being integrated has a continuous derivative,
- 1825 - Augustin-Louis Cauchy introduces the theory of residues in complex analysis,
- 1825 - Johann Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre prove Fermat's last theorem for
*n*= 5, - 1825 - André-Marie Ampčre discovers Stokes' theorem,
- 1828 - George Green proves Green's theorem,
- 1829 - Nikolai Ivanovich Lobachevsky publishes his work on hyperbolic non-Euclidean geometry,
- 1831 - Mikhail Vasilievich Ostrogradsky rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green,
- 1832 - Évariste Galois presents a general condition for the solvability of algebraic equations, thereby essentially founding group theory and Galois theory,
- 1832 - Peter Dirichlet proves Fermat's last theorem for
*n*= 14, - 1835 - Peter Dirichlet proves Dirichlet's theorem about prime numbers in arithmetical progressions,
- 1837 - Pierre Wantsel proves that doubling the cube and trisecting the angle are impossible with only a compass and straightedge,
- 1841 - Karl Weierstrass discovers but does not publish the Laurent expansion theorem,
- 1843 - Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem,
- 1843 - William Hamilton discovers the calculus of quaternions and deduces that they are non-commutative,
- 1847 - George Boole formalizes symbolic logic in
*The Mathematical Analysis of Logic*, defining what are now called Boolean algebras, - 1849 - George Gabriel Stokes shows that solitary wavess can arise from a combination of periodic waves,
- 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,
- 1854 - Bernhard Riemann introduces Riemannian geometry,
- 1854 - Arthur Cayley shows that quaternions can be used to represent rotations in four-dimensional space,
- 1858 - August Ferdinand Möbius invents the Möbius strip,
- 1859 - Bernhard Riemann formulates the Riemann hypothesis which has strong implications about the distribution of prime numbers,
- 1870 - Felix Klein constructs an analytic geometry for Lobachevski's geometry thereby establishing its self-consistency and the logical independence of Euclid's fifth postulate,
- 1873 - Charles Hermite proves that e is transcendental,
- 1873 - Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points,
- 1874 - Georg Cantor shows that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his method was not his famous diagonal argument, which he published three years later. (Nor did he formulate set theory at this time.)
- 1878 - Charles Hermite solves the general quintic equation by means of elliptic and modular functions
- 1882 - Carl Louis Ferdinand von Lindemann proves that π is transcendental and that therefore the circle cannot be squared with a compass and straightedge,
- 1882 - Felix Klein invents the Klein bottle,
- 1895 - Diederik Korteweg and Gustav de Vries derive the KdV equation to describe the development of long solitary water waves in a canal of rectangular cross section,
- 1895 - Georg Cantor publishes a book about set theory containing the arithmetic of infinite cardinal numbers and the continuum hypothesis,
- 1896 - Jacques Hadamard and Charles de La Vallée-Poussin independently prove the prime number theorem,
- 1899 - Georg Cantor discovers a contradiction in his set theory,
- 1899 - David Hilbert presents a set of self-consistent geometric axioms in
*Foundations of Geometry*, - 1900 - David Hilbert states his list of 23 problems which show where some further mathematical work is needed,
- 1901 - Élie Cartan develops the exterior derivative,
- 1903 - Carle David Tolme Runge presents a fast Fourier Transform algorithm,
- 1903 - Edmund Georg Hermann Landau gives considerably simpler proof of the prime number theorem,
- 1908 - Ernst Zermelo axiomizes set theory, thus avoiding Cantor's contradictions,
- 1908 - Josip Plemelj solves the Riemman problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky - Plemelj formulae,
- 1912 - Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem,
- 1912 - Josip Plemelj publishes simplified proof for the Fermat's last theorem for exponent
*n*= 5, - 1914 - Srinivasa Aaiyangar Ramanujan publishes
*Modular Equations and Approximations to π*, - 1919 - Viggo Brun defines Brun's constant
*B*_{2}for twin primes, - 1928 - John von Neumann begins devising the principles of game theory and proves the minimax theorem,
- 1930 - Casimir Kuratowski shows that the three cottage problem has no solution,
- 1931 - Kurt Gödel proves his incompleteness theorem which shows that every axiomatic system for mathematics is either incomplete or inconsistent,
- 1931 - Georges De Rham develops theorem in cohomology and characteristic classes,
- 1933 - Karol Borsuk and Stanislaw Ulam present the Borsuk-Ulam antipodal-point theorem,
- 1933 - Andrey Nikolaevich Kolmogorov publishes his book
*Basic notions of the calculus of variations*(*Grundbegriffe der Wahrscheinlichkeitsrechnung*) which contains an axiomatization of probability based on measure theory, - 1940 - Kurt Gödel shows that neither the continuum hypothesis nor the axiom of choice can be disproven from the standard axioms of set theory,
- 1942 - G.C. Danielson and Cornelius Lanczos develop a Fast Fourier Transform algorithm,
- 1943 - Kenneth Levenberg proposes a method for nonlinear least squares fitting,
- 1948 - John von Neumann mathematically studies self-reproducing machines,
- 1949 - John von Neumann computes π to 2,037 decimal places using ENIAC,
- 1950 - Stanislaw Ulam and John von Neumann present cellular automata dynamical systems,
- 1953 - Nicholas Metropolis introduces the idea of thermodynamic simulated annealing algorithms,
- 1955 - Enrico Fermi, John Pasta, and Stanislaw Ulam numerically study a nonlinear spring model of heat conduction and discover solitary wave type behavior,
- 1960 - C. A. R. Hoare invents the quicksort algorithm,
- 1960 - Irving Reed and Gustave Solomon present the Reed-Solomon error-correcting code,
- 1961 - Daniel Shanks and John Wrench compute π to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer,
- 1962 - Donald Marquardt proposes the Levenberg-Marquardt nonlinear least squares fitting algorithm,
- 1963 - Paul Cohen uses his technique of forcing to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory,
- 1963 - Martin Kruskal and Norman Zabusky analytically study the Fermi-Pasta-Ulam heat conduction problem in the continuum limit and find that the KdV equation governs this system,
- 1965 - Martin Kruskal and Norman Zabusky numerically study colliding solitary waves in plasmas and find that they do not disperse after collisions,
- 1965 - James Cooley and John Tukey present an influential Fast Fourier Transform algorithm,
- 1966 - E.J. Putzer presents two methods for computing the exponential of a matrix in terms of a polynomial in that matrix,
- 1967 - Robert Langlands formulates the influential Langlands program of conjectures relating number theory and representation theory,
- 1968 - Michael Atiyah and Isadore Singer prove the Atiyah-Singer index theorem about the index of elliptic operators,
- 1976 - Kenneth Appel and Wolfgang Haken use a computer to prove the Four-Color Theorem,
- 1983 - Gerd Faltings proves the Mordell conjecture and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat's last theorem,
- 1983 - the classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed,
- 1985 - Louis de Branges de Bourcia proves the Bieberbach conjecture,
- 1987 - Yasumasa Kanada, David Bailey, Jonathan Borwein, and Peter Borwein use iterative modular equation approximations to elliptic integrals and a NEC SX-2 supercomputer to compute π to 134 million decimal places,
- 1991 - Alain Connes and John W. Lott develop non-commutative geometry,
- 1994 - Andrew Wiles proves part of the Taniyama-Shimura conjecture and thereby proves Fermat's last theorem,
- 1998 - Thomas Hales (almost certainly) proves the Kepler conjecture,
- 1999 - the full Taniyama-Shimura conjecture is proved.
- 2000 - the Clay Mathematics Institute establishes the seven Millennium Prize Problems of unsolved important classic mathematical questions,
- 2002 - Manindra Agrawal, Nitin Saxena, and Neeraj Kayal of Indian Institute of Technology (IIT), Kanpur, India, present a unconditional deterministic polynomial time algorithm to determine whether a given number is prime,
- 2002 - Yasumasa Kanada, Y. Ushiro, Hisayasu Kuroda, Makoto Kudoh and a team of nine more compute π to 1241 billion digits using a Hitachi 64-node supercomputer,

## Note

- This article is based on a timeline developed by Niel Brandt (1994) who has given permission for its use in Wikipedia. (See Talk:Timeline of mathematics.)