- 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 Mathematical Papyrus, copy of a lost scroll from around
1850 BC, a great and still widely misunderstood synopsis of early
geometry and mathematics. - 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 π 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 Kng-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-Khawarzimi – 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 tanÐgent, secant and their
reciprocals. Derived the formula: sin α = tan α / … (1+tan2
α) and cos α = 1 / …(1 + tan2 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 – RenŽ 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’H™pital 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
Kšnigsberg, 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 – SimŽon-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 – AndrŽ-Marie Ampre 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 of earlier described by Lagrange, Gauss
and Green, - 1832 – ƒvariste 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 waves 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 Mšbius invents the Mšbius 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 formulates set theory and uses his diagonal
argument to show that the set of all real numbers is uncountably
infinite but the set of all algebraic numbers is countably infinite, - 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 VallŽe-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 – ƒlie 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 B2 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 Gšdel 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 Gšdel 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, - 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,
Timeline of Mathematics
