Symbolic stage Timeline of mathematics

1 symbolic stage

1.1 1000–1500

1.1.1 15th century


1.2 modern

1.2.1 16th century
1.2.2 17th century
1.2.3 18th century
1.2.4 19th century


1.3 contemporary

1.3.1 20th century
1.3.2 21st century

symbolic stage
1000–1500

c. 1000 – abū sahl al-qūhī (kuhi) solves equations higher second degree.
c. 1000 – abu-mahmud al-khujandi first states special case of fermat s last theorem.
c. 1000 – law of sines discovered muslim mathematicians, uncertain discovers first between abu-mahmud al-khujandi, abu nasr mansur, , abu al-wafa.
c. 1000 – pope sylvester ii introduces abacus using hindu-arabic numeral system europe.
1000 – al-karaji writes book containing first known proofs mathematical induction. used prove binomial theorem, pascal s triangle, , sum of integral cubes. first introduced theory of algebraic calculus .
c. 1000 – ibn tahir al-baghdadi studied slight variant of thabit ibn qurra s theorem on amicable numbers, , made improvements on decimal system.
1020 – abul wáfa gave formula: sin (α + β) = sin α cos β + sin β cos α. discussed quadrature of parabola , volume of paraboloid.
1021 – ibn al-haytham formulated , solved alhazen s problem geometrically.
1030 – ali ahmad nasawi writes treatise on decimal , sexagesimal number systems. arithmetic explains division of fractions , extraction of square , cubic roots (square root of 57,342; cubic root of 3, 652, 296) in modern manner.
1070 – omar khayyám begins write treatise on demonstration of problems of algebra , classifies cubic equations.
c. 1100 – omar khayyám gave complete classification of cubic equations geometric solutions found means of intersecting conic sections . became first find general geometric solutions of cubic equations , laid foundations development of analytic geometry , non-euclidean geometry. extracted roots using decimal system (hindu-arabic numeral system).
12th century – indian numerals have been modified arab mathematicians form modern hindu-arabic numeral system (used universally in modern world).
12th century – hindu-arabic numeral system reaches europe through arabs.
12th century – bhaskara acharya writes lilavati, covers topics of definitions, arithmetical terms, interest computation, arithmetical , geometrical progressions, plane geometry, solid geometry, shadow of gnomon, methods solve indeterminate equations, , combinations.
12th century – bhāskara ii (bhaskara acharya) writes bijaganita (algebra), first text recognize positive number has 2 square roots.
12th century – bhaskara acharya conceives differential calculus, , develops rolle s theorem, pell s equation, proof pythagorean theorem, proves division 0 infinity, computes π 5 decimal places, , calculates time taken earth orbit sun 9 decimal places.
1130 – al-samawal gave definition of algebra: [it concerned] operating on unknowns using arithmetical tools, in same way arithmetician operates on known.
1135 – sharafeddin tusi followed al-khayyam s application of algebra geometry, , wrote treatise on cubic equations represents essential contribution algebra aimed study curves means of equations, inaugurating beginning of algebraic geometry .
1202 – leonardo fibonacci demonstrates utility of hindu-arabic numerals in liber abaci (book of abacus).
1247 – qin jiushao publishes shùshū jiǔzhāng (mathematical treatise in 9 sections).
1248 – li ye writes ceyuan haijing, 12 volume mathematical treatise containing 170 formulas , 696 problems solved polynomial equations using method tian yuan shu.
1260 – al-farisi gave new proof of thabit ibn qurra s theorem, introducing important new ideas concerning factorization , combinatorial methods. gave pair of amicable numbers 17296 , 18416 have been joint attributed fermat thabit ibn qurra.
c. 1250 – nasir al-din al-tusi attempts develop form of non-euclidean geometry.
1303 – zhu shijie publishes precious mirror of 4 elements, contains ancient method of arranging binomial coefficients in triangle.
14th century – madhava considered father of mathematical analysis, worked on power series π , sine , cosine functions, , along other kerala school mathematicians, founded important concepts of calculus.
14th century – parameshvara, kerala school mathematician, presents series form of sine function equivalent taylor series expansion, states mean value theorem of differential calculus, , first mathematician give radius of circle inscribed cyclic quadrilateral.

15th century

1400 – madhava discovers series expansion inverse-tangent function, infinite series arctan , sin, , many methods calculating circumference of circle, , uses them compute π correct 11 decimal places.
c. 1400 – ghiyath al-kashi contributed development of decimal fractions not approximating algebraic numbers, real numbers such π. contribution decimal fractions major many years considered inventor. although not first so, al-kashi gave algorithm calculating nth roots, special case of methods given many centuries later [paolo] ruffini , [william george] horner. first use decimal point notation in arithmetic , arabic numerals. works include key of arithmetics, discoveries in mathematics, decimal point, , benefits of zero. contents of benefits of 0 introduction followed 5 essays: on whole number arithmetic , on fractional arithmetic , on astrology , on areas , , on finding unknowns [unknown variables] . wrote thesis on sine , chord , thesis on finding first degree sine.
15th century – ibn al-banna , al-qalasadi introduced symbolic notation algebra , mathematics in general.
15th century – nilakantha somayaji, kerala school mathematician, writes aryabhatiya bhasya, contains work on infinite-series expansions, problems of algebra, , spherical geometry.
1424 – ghiyath al-kashi computes π sixteen decimal places using inscribed , circumscribed polygons.
1427 – al-kashi completes key arithmetic containing work of great depth on decimal fractions. applies arithmetical , algebraic methods solution of various problems, including several geometric ones.
1464 – regiomontanus writes de triangulis omnimodus 1 of earliest texts treat trigonometry separate branch of mathematics.
1478 – anonymous author writes treviso arithmetic.
1494 – luca pacioli writes summa de arithmetica, geometria, proportioni et proportionalità; introduces primitive symbolic algebra using co (cosa) unknown.

modern
16th century

1501 – nilakantha somayaji writes tantrasamgraha.
1520 – scipione dal ferro develops method solving depressed cubic equations (cubic equations without x term), not publish.
1522 – adam ries explained use of arabic digits , advantages on roman numerals.
1535 – niccolò tartaglia independently develops method solving depressed cubic equations not publish.
1539 – gerolamo cardano learns tartaglia s method solving depressed cubics , discovers method depressing cubics, thereby creating method solving cubics.
1540 – lodovico ferrari solves quartic equation.
1544 – michael stifel publishes arithmetica integra.
1545 – gerolamo cardano conceives idea of complex numbers.
1550 – jyeshtadeva, kerala school mathematician, writes yuktibhāṣā, world s first calculus text, gives detailed derivations of many calculus theorems , formulae.
1572 – rafael bombelli writes algebra treatise , uses imaginary numbers solve cubic equations.
1584 – zhu zaiyu calculates equal temperament.
1596 – ludolf van ceulen computes π twenty decimal places using inscribed , circumscribed polygons.

17th century

1614 – john napier discusses napierian logarithms in mirifici logarithmorum canonis descriptio.
1617 – henry briggs discusses decimal logarithms in logarithmorum chilias prima.
1618 – john napier publishes first references e in work on logarithms.
1619 – rené descartes discovers analytic geometry (pierre de fermat claimed discovered independently).
1619 – johannes kepler discovers 2 of kepler-poinsot polyhedra.
1629 – pierre de fermat develops rudimentary differential calculus.
1634 – gilles de roberval shows area under cycloid 3 times area of generating circle.
1636 – muhammad baqir yazdi jointly discovered pair of amicable numbers 9,363,584 , 9,437,056 along descartes (1636).
1637 – pierre de fermat claims have proven fermat s last theorem in copy of diophantus arithmetica.
1637 – first use of term imaginary number rené descartes; meant derogatory.
1643 – rené descartes develops descartes theorem.
1654 – blaise pascal , pierre de fermat create theory of probability.
1655 – john wallis writes arithmetica infinitorum.
1658 – christopher wren shows length of cycloid 4 times diameter of generating circle.
1665 – isaac newton works on fundamental theorem of calculus , develops version of infinitesimal calculus.
1668 – nicholas mercator , william brouncker discover infinite series logarithm while attempting calculate area under hyperbolic segment.
1671 – james gregory develops series expansion inverse-tangent function (originally discovered madhava).
1671 – james gregory discovers taylor s theorem.
1673 – gottfried leibniz develops version of infinitesimal calculus.
1675 – isaac newton invents algorithm computation of functional roots.
1680s – gottfried leibniz works on symbolic logic.
1683 – seki takakazu discovers resultant , determinant.
1683 – seki takakazu develops elimination theory.
1691 – gottfried leibniz discovers technique of separation of variables ordinary differential equations.
1693 – edmund halley prepares first mortality tables statistically relating death rate age.
1696 – guillaume de l hôpital states rule computation of limits.
1696 – jakob bernoulli , johann bernoulli solve brachistochrone problem, first result in calculus of variations.
1699 – abraham sharp calculates π 72 digits 71 correct.

18th century

1706 – john machin develops converging inverse-tangent series π , computes π 100 decimal places.
1708 – seki takakazu discovers bernoulli numbers. jacob bernoulli whom numbers named after believed have independently discovered numbers shortly after takakazu.
1712 – brook taylor develops taylor series.
1722 – abraham de moivre states de moivre s formula connecting trigonometric functions , complex numbers.
1722 – takebe kenko introduces richardson extrapolation.
1724 – abraham de moivre studies mortality statistics , foundation of theory of annuities in annuities on lives.
1730 – james stirling publishes differential method.
1733 – giovanni gerolamo saccheri studies geometry if euclid s fifth postulate false.
1733 – abraham de moivre introduces normal distribution approximate binomial distribution in probability.
1734 – leonhard euler introduces integrating factor technique solving first-order ordinary differential equations.
1735 – leonhard euler solves basel problem, relating infinite series π.
1736 – leonhard euler solves problem of 7 bridges of königsberg, in effect creating graph theory.
1739 – leonhard euler solves general homogeneous linear ordinary differential equation constant coefficients.
1742 – christian goldbach conjectures every number greater 2 can expressed sum of 2 primes, known goldbach s conjecture.
1748 – maria gaetana agnesi discusses analysis in instituzioni analitiche ad uso della gioventu italiana.
1761 – thomas bayes proves bayes theorem.
1761 – johann heinrich lambert proves π irrational.
1762 – joseph louis lagrange discovers divergence theorem.
1789 – jurij vega improves machin s formula , computes π 140 decimal places, 136 of correct.
1794 – jurij vega publishes thesaurus logarithmorum completus.
1796 – carl friedrich gauss proves regular 17-gon can constructed using compass , straightedge.
1796 – adrien-marie legendre conjectures prime number theorem.
1797 – caspar wessel associates vectors complex numbers , studies complex number operations in geometrical terms.
1799 – carl friedrich gauss proves fundamental theorem of algebra (every polynomial equation has solution among complex numbers).
1799 – paolo ruffini partially proves abel–ruffini theorem quintic or higher equations cannot solved general formula.

19th century

1801 – disquisitiones arithmeticae, carl friedrich gauss s number theory treatise, published in latin.
1805 – adrien-marie legendre introduces method of least squares fitting curve given set of observations.
1806 – louis poinsot discovers 2 remaining kepler-poinsot polyhedra.
1806 – jean-robert argand publishes proof of fundamental theorem of algebra , argand diagram.
1807 – joseph fourier announces discoveries trigonometric decomposition of functions.
1811 – carl friedrich gauss discusses meaning of integrals complex limits , briefly examines dependence of such integrals on chosen path of integration.
1815 – siméon denis poisson carries out integrations along paths in complex plane.
1817 – bernard bolzano presents intermediate value theorem—a continuous function negative @ 1 point , positive @ point must 0 @ least 1 point in between.
1822 – augustin-louis cauchy presents cauchy integral theorem integration around boundary of rectangle in complex plane.
1822 – irisawa shintarō hiroatsu analyzes soddy s hexlet in sangaku.
1823 - sophie germain s theorem published in second edition of adrien-marie legendre s essai sur la théorie des nombres
1824 – niels henrik abel partially proves abel–ruffini theorem general quintic or higher equations cannot solved general formula involving arithmetical operations , roots.
1825 – augustin-louis cauchy presents cauchy integral theorem general integration paths—he assumes function being integrated has continuous derivative, , introduces theory of residues in complex analysis.
1825 – peter gustav lejeune dirichlet , adrien-marie legendre prove fermat s last theorem n = 5.
1825 – andré-marie ampère discovers stokes theorem.
1828 – george green proves green s theorem.
1829 – jános bolyai, gauss, , lobachevsky invent hyperbolic non-euclidean geometry.
1831 – mikhail vasilievich ostrogradsky rediscovers , gives first proof of divergence theorem earlier described lagrange, gauss , green.
1832 – Évariste galois presents general condition solvability of algebraic equations, thereby founding group theory , galois theory.
1832 – lejeune dirichlet proves fermat s last theorem n = 14.
1835 – lejeune dirichlet proves dirichlet s theorem prime numbers in arithmetical progressions.
1837 – pierre wantzel proves doubling cube , trisecting angle impossible compass , straightedge, full completion of problem of constructability of regular polygons.
1837 – peter gustav lejeune dirichlet develops analytic number theory.
1841 – karl weierstrass discovers not publish laurent expansion theorem.
1843 – pierre-alphonse laurent discovers , presents laurent expansion theorem.
1843 – william hamilton discovers calculus of quaternions , deduces non-commutative.
1847 – george boole formalizes symbolic logic in mathematical analysis of logic, defining called boolean algebra.
1849 – george gabriel stokes shows solitary waves can arise combination of periodic waves.
1850 – victor alexandre puiseux distinguishes between poles , branch points , introduces concept of essential singular points.
1850 – george gabriel stokes rediscovers , proves stokes theorem.
1854 – bernhard riemann introduces riemannian geometry.
1854 – arthur cayley shows quaternions can used represent rotations in four-dimensional space.
1858 – august ferdinand möbius invents möbius strip.
1858 – charles hermite solves general quintic equation means of elliptic , modular functions.
1859 – bernhard riemann formulates riemann hypothesis, has strong implications distribution of prime numbers.
1870 – felix klein constructs analytic geometry lobachevski s geometry thereby establishing self-consistency , logical independence of euclid s fifth postulate.
1872 – richard dedekind invents called dedekind cut defining irrational numbers, , used defining surreal numbers.
1873 – charles hermite proves e transcendental.
1873 – georg frobenius presents method finding series solutions linear differential equations regular singular points.
1874 – georg cantor proves set of real numbers uncountably infinite set of real algebraic numbers countably infinite. proof not use diagonal argument, published in 1891.
1882 – ferdinand von lindemann proves π transcendental , therefore circle cannot squared compass , straightedge.
1882 – felix klein invents klein bottle.
1895 – diederik korteweg , gustav de vries derive korteweg–de vries equation describe development of long solitary water waves in canal of rectangular cross section.
1895 – georg cantor publishes book set theory containing arithmetic of infinite cardinal numbers , continuum hypothesis.
1895 – henri poincare publishes paper analysis situs started modern topology.
1896 – jacques hadamard , charles jean de la vallée-poussin independently prove prime number theorem.
1896 – hermann minkowski presents geometry of numbers.
1899 – georg cantor discovers contradiction in set theory.
1899 – david hilbert presents set of self-consistent geometric axioms in foundations of geometry.
1900 – david hilbert states list of 23 problems, show further mathematical work needed.

contemporary
20th century

1901 – Élie cartan develops exterior derivative.
1901 – henri lebesgue publishes on lebesgue integration.
1903 – carle david tolmé runge presents fast fourier transform algorithm
1903 – edmund georg hermann landau gives considerably simpler proof of prime number theorem.
1908 – ernst zermelo axiomizes set theory, avoiding cantor s contradictions.
1908 – josip plemelj solves riemann problem existence of differential equation given monodromic group , uses sokhotsky – plemelj formulae.
1912 – luitzen egbertus jan brouwer presents brouwer fixed-point theorem.
1912 – josip plemelj publishes simplified proof fermat s last theorem exponent n = 5.
1915 – emmy noether proves symmetry theorem, shows every symmetry in physics has corresponding conservation law.
1916 – srinivasa ramanujan introduces ramanujan conjecture. conjecture later generalized hans petersson.
1919 – viggo brun defines brun s constant b2 twin primes.
1921 – emmy noether introduces first general definition of commutative ring.
1928 – john von neumann begins devising principles of game theory , proves minimax theorem.
1929 – emmy noether introduces first general representation theory of groups , algebras.
1930 – casimir kuratowski shows three-cottage problem has no solution.
1930 – alonzo church introduces lambda calculus.
1931 – kurt gödel proves incompleteness theorem, shows every axiomatic system mathematics either incomplete or inconsistent.
1931 – georges de rham develops theorems in cohomology , characteristic classes.
1933 – karol borsuk , stanislaw ulam present borsuk–ulam antipodal-point theorem.
1933 – andrey nikolaevich kolmogorov publishes book basic notions of calculus of probability (grundbegriffe der wahrscheinlichkeitsrechnung), contains axiomatization of probability based on measure theory.
1940 – kurt gödel shows neither continuum hypothesis nor axiom of choice can disproven standard axioms of set theory.
1942 – g.c. danielson , cornelius lanczos develop fast fourier transform algorithm.
1943 – kenneth levenberg proposes method nonlinear least squares fitting.
1945 – stephen cole kleene introduces realizability.
1945 – saunders mac lane , samuel eilenberg start category theory.
1945 – norman steenrod , samuel eilenberg give eilenberg–steenrod axioms (co-)homology.
1946 – jean leray introduces spectral sequence.
1948 – john von neumann mathematically studies self-reproducing machines.
1948 – alan turing introduces lu decomposition.
1949 – john wrench , l.r. smith compute π 2,037 decimal places using eniac.
1949 – claude shannon develops notion of information theory.
1950 – stanisław ulam , john von neumann present cellular automata dynamical systems.
1953 – nicholas metropolis introduces idea of thermodynamic simulated annealing algorithms.
1955 – h. s. m. coxeter et al. publish complete list of uniform polyhedron.
1955 – enrico fermi, john pasta, stanisław ulam, , mary tsingou numerically study nonlinear spring model of heat conduction , discover solitary wave type behavior.
1956 – noam chomsky describes hierarchy of formal languages.
1957 – kiyosi itô develops itô calculus.
1957 – stephen smale provides existence proof crease-free sphere eversion.
1958 – alexander grothendieck s proof of grothendieck–riemann–roch theorem published.
1959 – kenkichi iwasawa creates iwasawa theory.
1960 – c. a. r. hoare invents quicksort algorithm.
1960 – irving s. reed , gustave solomon present reed–solomon error-correcting code.
1961 – daniel shanks , john wrench compute π 100,000 decimal places using inverse-tangent identity , ibm-7090 computer.
1961 – john g. f. francis , vera kublanovskaya independently develop qr algorithm calculate eigenvalues , eigenvectors of matrix.
1961 – stephen smale proves poincaré conjecture dimensions greater or equal 5.
1962 – donald marquardt proposes levenberg–marquardt nonlinear least squares fitting algorithm.
1962 – gloria conyers hewitt becomes third african american woman receive phd in mathematics.
1963 – paul cohen uses technique of forcing show neither continuum hypothesis nor axiom of choice can proven standard axioms of set theory.
1963 – martin kruskal , norman zabusky analytically study fermi–pasta–ulam–tsingou heat conduction problem in continuum limit , find kdv equation governs system.
1963 – meteorologist , mathematician edward norton lorenz published solutions simplified mathematical model of atmospheric turbulence – known chaotic behaviour , strange attractors or lorenz attractor – butterfly effect.
1965 – iranian mathematician lotfi asker zadeh founded fuzzy set theory extension of classical notion of set , founded field of fuzzy mathematics.
1965 – martin kruskal , norman zabusky numerically study colliding solitary waves in plasmas , find not disperse after collisions.
1965 – james cooley , john tukey present influential fast fourier transform algorithm.
1966 – e. j. putzer presents 2 methods computing exponential of matrix in terms of polynomial in matrix.
1966 – abraham robinson presents non-standard analysis.
1967 – robert langlands formulates influential langlands program of conjectures relating number theory , representation theory.
1968 – michael atiyah , isadore singer prove atiyah–singer index theorem index of elliptic operators.
1973 – lotfi zadeh founded field of fuzzy logic.
1975 – benoît mandelbrot publishes les objets fractals, forme, hasard et dimension.
1976 – kenneth appel , wolfgang haken use computer prove 4 color theorem.
1978 – olga taussky-todd awarded austrian cross of honour science , art, 1st class, highest scientific award of government of austria.
1981 – richard feynman gives influential talk simulating physics computers (in 1980 yuri manin proposed same idea quantum computations in computable , uncomputable (in russian)).
1983 – gerd faltings proves mordell conjecture , thereby shows there finitely many whole number solutions each exponent of fermat s last theorem.
1983 – classification of finite simple groups, collaborative work involving hundred mathematicians , spanning thirty years, completed.
1985 – louis de branges de bourcia proves bieberbach conjecture.
1986 – ken ribet proves ribet s theorem.
1987 – yasumasa kanada, david bailey, jonathan borwein, , peter borwein use iterative modular equation approximations elliptic integrals , nec sx-2 supercomputer compute π 134 million decimal places.
1991 – alain connes , john w. lott develop non-commutative geometry.
1992 – david deutsch , richard jozsa develop deutsch–jozsa algorithm, 1 of first examples of quantum algorithm exponentially faster possible deterministic classical algorithm.
1994 – andrew wiles proves part of taniyama–shimura conjecture , thereby proves fermat s last theorem.
1994 – peter shor formulates shor s algorithm, quantum algorithm integer factorization.
1995 – simon plouffe discovers bailey–borwein–plouffe formula capable of finding nth binary digit of π.
1998 – thomas callister hales (almost certainly) proves kepler conjecture.
1999 – full taniyama–shimura conjecture proven.
2000 – clay mathematics institute proposes 7 millennium prize problems of unsolved important classic mathematical questions.

21st century

2002 – manindra agrawal, nitin saxena, , neeraj kayal of iit kanpur present unconditional deterministic polynomial time algorithm determine whether given number prime (the aks primality test).
2002 – yasumasa kanada, y. ushiro, hisayasu kuroda, makoto kudoh , team of 9 more compute π 1241.1 billion digits using hitachi 64-node supercomputer.
2002 – preda mihăilescu proves catalan s conjecture.
2003 – grigori perelman proves poincaré conjecture.
2004 – ben green , terence tao prove green-tao theorem.
2007 – team of researchers throughout north america , europe used networks of computers map e8.
2009 – fundamental lemma (langlands program) had been proved ngô bảo châu.
2010 – larry guth , nets hawk katz solve erdős distinct distances problem.
2013 – yitang zhang proves first finite bound on gaps between prime numbers.
2014 – project flyspeck announces completed proof of kepler s conjecture.
2014 – using alexander yee s y-cruncher houkouonchi calculated π 13.3 trillion digits.
2015 – terence tao solved erdös discrepancy problem
2015 – lászló babai found quasipolynomial complexity algorithm solve graph isomorphism problem
2016 – using alexander yee s y-cruncher peter trueb calculated π 22.4 trillion digits




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^ f. woepcke (1853). extrait du fakhri, traité d algèbre par abou bekr mohammed ben alhacan alkarkhi. paris.
^ o connor, john j.; robertson, edmund f., abu l hasan ali ibn ahmad al-nasawi , mactutor history of mathematics archive, university of st andrews .
^ arabic mathematics, mactutor history of mathematics archive, university of st andrews, scotland
^ various ap lists , statistics
^ https://www.agnesscott.edu/lriddle/women/germain-flt/sgandflt.htm
^ paul benacerraf , hilary putnam, cambridge university press, philosophy of mathematics: selected readings, isbn 0-521-29648-x
^ elizabeth a. thompson, mit news office, math research team maps e8 mathematicians map e8, harminka, 2007-03-20
^ laumon, g.; ngô, b. c. (2004), le lemme fondamental pour les groupes unitaires, arxiv:math/0404454  
^ unh mathematician s proof breakthrough toward centuries-old problem . university of new hampshire. may 1, 2013. retrieved may 20, 2013. 
^ announcement of completion. project flyspeck, google code.
^ team announces construction of formal computer-verified proof of kepler conjecture. august 13, 2014 bob yirk.
^ proof confirmed of 400-year-old fruit-stacking problem, 12 august 2014; new scientist.
^ formal proof of kepler conjecture, arxiv.
^ solved: 400-year-old maths theory proven. sky news, 16:39, uk, tuesday 12 august 2014.
^ y-cruncher - multi-threaded pi program . numberworld.org. retrieved 29 august 2015. 
^ y-cruncher - multi-threaded pi program . numberworld.org. retrieved 15 december 2016.