**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. |