Chronology of Pure and Applied Mathematics

 -1700 Egyptian mathematicians employ primitive fractions. -530 Pythagoras studies propositional geometry and vibrating lyre strings. -370 Eudoxus states the method of exhaustion for area determination. -350 Aristotle discusses logical reasoning in Organon. -300 Euclid studies geometry as an axiomatic system in Elements and states the law of reflection in Catoptrics. -260 Archimedes computes pi to two decimal places using inscribed and cirumscribed polygons and computes the area under a parabolic segment. -200 Apollonius writes On Conic Sections and names the ellipse, parabola, and hyperbola. 250 Diophantus writes Arithmetica, the first systematic treatise on algebra. 450 Tsu Ch'ung-Chih and Tsu Keng-Chih compute pi to six decimal places. 550 Hindu mathematicians give zero a numeral representation in a positional notation system. 1202 Leonardo Fibonacci demonstrates the utility of Arabic numerals in his Book of the Abacus. 1424 Ghiyathal-Kashi computes pi to sixteen decimal places using inscribed and cirumscribed polygons. 1520 Scipione 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 pi to twenty decimal places using inscribed and cirumscribed polygons. 1614 John Napier discusses Napierian logarithms in Mirifici Logarithmorum Canonis Descriptio. 1617 Henry Briggs discusses decimal logarithms in Logarithmorum Chilias Prima. 1619 Rene Descartes discovers analytical geometry. 1629 Pierre de Fermat develops a rudimentary differential calculus. 1634 G.P. 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'Hopital states his rule for the examination of indeterminate forms. 1706 John Machin develops a quickly converging inverse-tangent series for pi and computes pi to 100 decimal places. 1712 Brook Taylor develops Taylor series'. 1722 Abraham De Moivre states De Moivre's theorem. 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 Geralamo Saccheri studies what geometry would be like if Euclid's fifth postulate were false. 1734 Leonhard Euler introduces the integrating factor technique for solving first order ordinary differential equations. 1736 Leonhard Euler solves the Koenigsberg bridge problem. 1739 Leonhard Euler solves the general homogeneous 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. 1744 Leonhard Euler shows the existence of transcendental numbers. 1748 Maria Agnesi discusses analysis in Instituzioni Analitiche ad Uso della Gioventu Italiana. 1761 Thomas Bayes proves Bayes' theorem. 1796 Karl Gauss presents a method for constructing a heptadecagon using only a compass and straight edge and also shows that only polygons with certain numbers of sides can be constructed. 1797 Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms. 1799 Karl Gauss proves that every polynomial equation has a solution among the complex numbers. 1806 Jean-Robert Argand associates vectors with complex numbers and studies complex number operations in geometrical terms. 1807 Joseph Fourier first announces his discoveries about the trigonometric decomposition of functions. 1811 Karl Gauss discusses the meaning of integrals with complex limits and briefly examines the dependence of such integrals on the chosen path of integration. 1815 Simeon Poisson carries out integrations along paths in the complex plane. 1817 Bernard Bolzano presents Bolzano's 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. 1824 Niels Abel partially proves that the general quintic or higher equations do not have algebraic solutions. 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. 1825 Peter Dirichlet and Adrien Legendre prove Fermat's Last Theorem for n=5. 1828 George Green proves Green's theorem. 1829 Nikolai Lobachevski publishes his work on hyperbolic non-Euclidean geometry. 1832 Evariste Galois presents a general condition for the solvability of algebraic equations. 1832 Peter Dirichlet proves Fermat's Last Theorem for n=14. 1837 Pierre Wantsel proves that doubling the cube and trisecting the angle are impossible with only a compass and straight edge. 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. 1849 George Stokes shows that solitary waves can arise from a combination of periodic waves. 1850 Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points. 1850 George Stokes 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 Mobius invents the Mobius strip. 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 is transcendental. 1873 Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points. 1878 Charles Hermite solves the general quintic equation by means of elliptic and modular functions. 1882 Ferdinand Lindeman proves that pi is transcendental and that the circle cannot be squared with a compass and straight edge. 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. 1896 Jacques Hadamard and Charles de La Vallee-Poussin independently prove the prime number theorem. 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 further mathematical work is needed. 1901 Elie Cartan develops the exterior derivative. 1903 C. Runge presents a fast Fourier transform algorithm. 1908 Ernst Zermelo axiomatizes set theory. 1912 L.E.J. Brouwer presents the Brouwer fixed-point theorem. 1914 Srinivasa Ramanujan publishes Modular Equations and Approximations to pi. 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 Godel shows that mathematical systems are not fully self-contained. 1933 Karol Borsuk and Stanislaw Ulam present the Borsuk-Ulam antipodal-point theorem. 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 pi 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 pi 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 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. 1976 Kenneth Appel and Wolfgang Haken use a computer to solve the four-color problem. 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. 1985 Louis de Branges 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 pi to 134 million decimal places. 1993 Andrew Wiles proves part of the Taniyama-Shimura Conjecture and thereby proves Fermat's Last Theorem.