This is atimeline ofpure andapplied mathematicshistory. It is divided here into three stages, corresponding to stages in thedevelopment of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

• ca.70,000 BC – South Africa, ochre rocks adorned with scratchedgeometric patterns (seeBlombos Cave).^[1]
• ca.35,000 BC to20,000 BC – Africa and France, earliest knownprehistoric attempts to quantify time (seeLebombo bone).^[2][3][4]
• c. 20,000 BC –Nile Valley,Ishango bone: possibly the earliest reference toprime numbers andEgyptian multiplication.
• c. 3400 BC –Mesopotamia, theSumerians invent the firstnumeral system, and a system ofweights and measures.
• c. 3100 BC –Egypt, earliest knowndecimal system allows indefinite counting by way of introducing new symbols.^[5]
• c. 2800 BC –Indus Valley Civilisation on theIndian subcontinent, earliest use of decimal ratios in a uniform system ofancient weights and measures, the smallest unit of measurement used is 1.704 millimetres and the smallest unit of mass used is 28 grams.
• 2700 BC – Egypt, precisionsurveying.
• 2400 BC – Egypt, preciseastronomical calendar, used even in theMiddle Ages for its mathematical regularity.
• c. 2000 BC – Mesopotamia, theBabylonians use a base-60 positional numeral system, and compute the first known approximate value ofπ at 3.125.
• c. 2000 BC – Scotland,carved stone balls exhibit a variety of symmetries including all of the symmetries ofPlatonic solids, though it is not known if this was deliberate.
• c. 1800 BC – ThePlimpton 322 Babylonian tablet records the oldest known examples ofPythagorean triples.^[6]
• 1800 BC – Egypt,Moscow Mathematical Papyrus, finding the volume of afrustum.
• c. 1800 BC –Berlin Papyrus 6619 (Egypt, 19th dynasty) contains a quadratic equation and its solution.^[5]
• 1650 BC –Rhind Mathematical Papyrus, copy of a lost scroll from around 1850 BC, the scribeAhmes presents one of the first known approximate values of π at 3.16, the first attempt atsquaring the circle, earliest known use of a sort ofcotangent, and knowledge of solving first order linear equations.
• The earliest recorded use ofcombinatorial techniques comes from problem 79 of theRhind papyrus which dates to the 16th century BCE.^[7]

• c. 1000 BC –Simple fractions used by theEgyptians. However, only unit fractions are used (i.e., those with 1 as the numerator) andinterpolation tables are used to approximate the values of the other fractions.^[8]
• first half of 1st millennium BC –Vedic India –Yajnavalkya, in hisShatapatha Brahmana, describes the motions of the Sun and the Moon, and advances a95-year cycle to synchronize the motions of the Sun and the Moon.
• c. 800 BC –Baudhayana, author of the BaudhayanaShulba Sutra, aVedic Sanskrit geometric text, containsquadratic equations, calculates thesquare root of two correctly to five decimal places, and contains "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."^[9]
• c. 8th century BC – theYajurveda, one of the fourHinduVedas, contains the earliest concept ofinfinity, and states "if you remove a part from infinity or add a part to infinity, still what remains is infinity."
• 1046 BC to 256 BC – China,Zhoubi Suanjing, arithmetic, geometric algorithms, and proofs.
• 624 BC – 546 BC – Greece,Thales of Miletus has various theorems attributed to him.
• c. 600 BC – India, the other Vedic "Shulba Sutras" ("rule of chords" inSanskrit) usePythagorean triples, contain a number of geometrical proofs, and approximateπ at 3.16.
• second half of 1st millennium BC – TheLuoshu Square, the unique normalmagic square of order three, was discovered in China.
• 530 BC – Greece,Pythagoras studies propositionalgeometry and vibrating lyre strings; his group also discovers theirrationality of thesquare root of two.
• c. 510 BC – Greece,Anaxagoras
• c. 500 BC –Indian grammarianPāṇini writes theAṣṭādhyāyī, which contains the use of metarules,transformations andrecursions, originally for the purpose of systematizing the grammar of Sanskrit.
• c. 500 BC – Greece,Oenopides of Chios
• 470 BC – 410 BC – Greece,Hippocrates of Chios utilizeslunes in an attempt tosquare the circle.
• 490 BC – 430 BC – Greece,Zeno of Elea createsZeno's paradoxes.
• 5th century BC – India,Apastamba, author of the Apastamba Shulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates thesquare root of 2 correct to five decimal places.
• 5th c. BC – Greece,Theodorus of Cyrene
• 5th century – Greece,Antiphon the Sophist
• 460 BC – 370 BC – Greece,Democritus
• 460 BC – 399 BC – Greece,Hippias
• 5th century (late) – Greece,Bryson of Heraclea
• 428 BC – 347 BC – Greece,Archytas
• 423 BC – 347 BC – Greece,Plato
• 417 BC – 317 BC – Greece,Theaetetus
• c. 400 BC – India, write theSurya Prajnapti, a mathematical text classifying all numbers into three sets: enumerable, innumerable andinfinite. It also recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
• 408 BC – 355 BC – Greece,Eudoxus of Cnidus
• 400 BC – 350 BC – Greece,Thymaridas
• 395 BC – 313 BC – Greece,Xenocrates
• 390 BC – 320 BC – Greece,Dinostratus
• 380–290 – Greece,Autolycus of Pitane
• 370 BC – Greece,Eudoxus states themethod of exhaustion forarea determination.
• 370 BC – 300 BC – Greece,Aristaeus the Elder
• 370 BC – 300 BC – Greece,Callippus
• 350 BC – Greece,Aristotle discusseslogical reasoning inOrganon.
• 4th century BC – China,Counting rods
• 330 BC – China, the earliest known work onChinese geometry, theMo Jing, is compiled.
• 310 BC – 230 BC – Greece,Aristarchus of Samos
• 390 BC – 310 BC – Greece,Heraclides Ponticus
• 380 BC – 320 BC – Greece,Menaechmus
• 300 BC – India,Bhagavati Sutra, which contains the earliest information oncombinations.
• 300 BC  – Greece,Euclid in hisElements studies geometry as anaxiomatic system, proves the infinitude ofprime numbers and presents theEuclidean algorithm; he states the law of reflection inCatoptrics, and he proves thefundamental theorem of arithmetic.
• c. 300 BC – India,Brahmi numerals (ancestor of the common modernbase 10numeral system)
• 370 BC – 300 BC – Greece,Eudemus of Rhodes works on histories of arithmetic, geometry and astronomy now lost.^[10]
• 300 BC –Mesopotamia, theBabylonians invent the earliest calculator, theabacus.
• c. 300 BC –Indian mathematicianPingala writes theChandaḥśāstra, which contains the first Indian use of zero as a digit (indicated by a dot) and also presents a description of abinary numeral system, along with the first use ofFibonacci numbers andPascal's triangle.
• 3rd century BC –Indian texts use the Sanskrit word "Shunya" to refer to the concept of "void" (zero).
• 280 BC – 210 BC – Greece,Nicomedes (mathematician)
• 280 BC – 220BC – Greece,Philo of Byzantium
• 280 BC – 220 BC – Greece,Conon of Samos
• 279 BC – 206 BC – Greece,Chrysippus
• c. 3rd century BC – India,Kātyāyana
• 250 BC – 190 BC – Greece,Dionysodorus
• 262 -198 BC – Greece,Apollonius of Perga
• 260 BC – Greece,Archimedes proved that the value of π lies between 3 + 1/7 (approx. 3.1429) and 3 + 10/71 (approx. 3.1408), that the area of a circle was equal to π multiplied by the square of the radius of the circle and that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He also gave a very accurate estimate of the value of the square root of 3.
• c. 250 BC – lateOlmecs had already begun to use a true zero (a shell glyph) several centuries beforePtolemy in the New World. See0 (number).
• 240 BC – Greece,Eratosthenes useshis sieve algorithm to quickly isolate prime numbers.
• 240 BC 190 BC– Greece,Diocles (mathematician)
• 225 BC – Greece,Apollonius of Perga writesOnConic Sections and names theellipse,parabola, andhyperbola.
• 202 BC to 186 BC –China,Book on Numbers and Computation, a mathematical treatise, is written inHan dynasty.
• 200 BC – 140 BC – Greece,Zenodorus (mathematician)
• 150 BC – India,Jain mathematicians in India write theSthananga Sutra, which contains work on the theory of numbers, arithmetical operations, geometry, operations withfractions, simple equations,cubic equations, quartic equations, andpermutations and combinations.
• c. 150 BC – Greece,Perseus (geometer)
• 150 BC – China, A method ofGaussian elimination appears in the Chinese textThe Nine Chapters on the Mathematical Art.
• 150 BC – China,Horner's method appears in the Chinese textThe Nine Chapters on the Mathematical Art.
• 150 BC – China,Negative numbers appear in the Chinese textThe Nine Chapters on the Mathematical Art.
• 150 BC – 75 BC – Phoenician,Zeno of Sidon
• 190 BC – 120 BC – Greece,Hipparchus develops the bases oftrigonometry.
• 190 BC – 120 BC – Greece,Hypsicles
• 160 BC – 100 BC – Greece,Theodosius of Bithynia
• 135 BC – 51 BC – Greece,Posidonius
• 78 BC – 37 BC – China,Jing Fang
• 50 BC –Indian numerals, a descendant of theBrahmi numerals (the firstpositional notationbase-10numeral system), begins development inIndia.
• mid 1st centuryCleomedes (as late as 400 AD)
• final centuries BC – Indian astronomerLagadha writes theVedanga Jyotisha, a Vedic text onastronomy that describes rules for tracking the motions of the Sun and the Moon, and uses geometry and trigonometry for astronomy.
• 1st C. BC – Greece,Geminus
• 50 BC – 23 AD – China,Liu Xin

• 1st century – Greece,Heron of Alexandria, Hero, the earliest, fleeting reference to square roots of negative numbers.
• c 100 – Greece,Theon of Smyrna
• 60 – 120 – Greece,Nicomachus
• 70 – 140 – Greece,Menelaus of Alexandria,Spherical trigonometry
• 78 – 139 – China,Zhang Heng
• c. 2nd century – Greece,Ptolemy ofAlexandria wrote theAlmagest.
• 132 – 192 – China,Cai Yong
• 240 – 300 – Greece,Sporus of Nicaea
• 250 – Greece,Diophantus uses symbols for unknown numbers in terms of syncopatedalgebra, and writesArithmetica, one of the earliest treatises on algebra.
• 263 – China,Liu Hui computesπ usingLiu Hui's π algorithm.
• 300 – the earliest known use ofzero as a decimal digit is introduced byIndian mathematicians.
• 234 – 305 – Greece,Porphyry (philosopher)
• 300 – 360 – Greece,Serenus of Antinoöpolis
• 335 – 405– Greece,Theon of Alexandria
• c. 340 – Greece,Pappus of Alexandria states hishexagon theorem and hiscentroid theorem.
• 350 – 415 – Eastern Roman Empire,Hypatia
• c. 400 – India, theBakhshali manuscript, which describes a theory of the infinite containing different levels ofinfinity, shows an understanding ofindices, as well aslogarithms tobase 2, and computessquare roots of numbers as large as a million correct to at least 11 decimal places.
• 300 to 500 – theChinese remainder theorem is developed bySun Tzu.
• 300 to 500 – China, a description ofrod calculus is written bySun Tzu.
• 412 – 485 – Greece,Proclus
• 420 – 480 – Greece,Domninus of Larissa
• b 440 – Greece,Marinus of Neapolis "I wish everything was mathematics."
• 450 – China,Zu Chongzhi computesπ to seven decimal places. This calculation remains the most accurate calculation for π for close to a thousand years.
• c. 474 – 558 – Greece,Anthemius of Tralles
• 500 – India,Aryabhata writes theAryabhata-Siddhanta, which first introduces the trigonometric functions and methods of calculating their approximate numerical values. It defines the concepts ofsine andcosine, and also contains theearliest tables of sine and cosine values (in 3.75-degree intervals from 0 to 90 degrees).
• 480 – 540 – Greece,Eutocius of Ascalon
• 490 – 560 – Greece,Simplicius of Cilicia
• 6th century – Aryabhata gives accurate calculations for astronomical constants, such as thesolar eclipse andlunar eclipse, computes π to four decimal places, and obtains whole number solutions tolinear equations by a method equivalent to the modern method.
• 505 – 587 – India,Varāhamihira
• 6th century – India,Yativṛṣabha
• 535 – 566 – China,Zhen Luan
• 550 –Hindu mathematicians give zero a numeral representation in thepositional notationIndian numeral system.
• 600 – China, Liu Zhuo uses quadratic interpolation.
• 602 – 670 – China,Li Chunfeng
• 625 China,Wang Xiaotong writes theJigu Suanjing, where cubic and quartic equations are solved.
• 7th century – India,Bhāskara I gives a rational approximation of the sine function.
• 7th century – India,Brahmagupta invents the method of solving indeterminate equations of the second degree and is the first to use algebra to solve astronomical problems. He also develops methods for calculations of the motions and places of various planets, their rising and setting, conjunctions, and the calculation of eclipses of the sun and the moon.
• 628 – Brahmagupta writes theBrāhmasphuṭasiddhānta, where zero is clearly explained, and where the modernplace-value Indian numeral system is fully developed. It also gives rules for manipulating bothnegative and positive numbers, methods for computing square roots, methods of solvinglinear andquadratic equations, and rules for summingseries,Brahmagupta's identity, and theBrahmagupta theorem.
• 721 – China, Zhang Sui (Yi Xing) computes the first tangent table.
• 8th century – India,Virasena gives explicit rules for theFibonacci sequence, gives the derivation of thevolume of afrustum using aninfinite procedure, and also deals with thelogarithm to base 2 and knows its laws.
• 8th century – India,Sridhara gives the rule for finding the volume of a sphere and also the formula for solving quadratic equations.
• 773 – Iraq, Kanka brings Brahmagupta's Brāhmasphuṭasiddhānta toBaghdad to explain the Indian system of arithmeticastronomy and the Indian numeral system.
• 773 –Muḥammad ibn Ibrāhīm al-Fazārī translates the Brāhmasphuṭasiddhānta into Arabic upon the request of King Khalif Abbasid Al Mansoor.
• 9th century – India,Govindasvāmi discovers the Newton-Gauss interpolation formula, and gives the fractional parts of Aryabhata's tabularsines.
• 810 – TheHouse of Wisdom is built in Baghdad for the translation of Greek andSanskrit mathematical works into Arabic.
• 820 –Al-Khwarizmi –Persian mathematician, father of algebra, writes theAl-Jabr, later transliterated asAlgebra, which introduces systematic algebraic techniques for solving linear and quadratic equations. Translations of his book onarithmetic will introduce theHindu–Arabicdecimal number system to the Western world in the 12th century. The termalgorithm is also named after him.
• 820 – Iran,Al-Mahani conceived the idea of reducinggeometrical problems such asdoubling the cube to problems in algebra.
• c. 850 – Iraq,al-Kindi pioneerscryptanalysis andfrequency analysis in his book oncryptography.
• c. 850 – India,Mahāvīra writes theGaṇitasārasan̄graha otherwise known as the Ganita Sara Samgraha which gives systematic rules for expressing a fraction as thesum of unit fractions.
• 895 – Syria,Thābit ibn Qurra: the only surviving fragment of his original work contains a chapter on the solution and properties ofcubic equations. He also generalized thePythagorean theorem, and discovered thetheorem by which pairs ofamicable numbers can be found, (i.e., two numbers such that each is the sum of the proper divisors of the other).
• c. 900 – Egypt,Abu Kamil had begun to understand what we would write in symbols as${\displaystyle x^n\cdot x^m=x^{m+n}}$
• c. 900 – Mesopotamia,al-Battani extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tangent, secant and their inverse functions. Derived the formulae:${\displaystyle \sin \alpha =\tan \alpha /\sqrt{1+{\tan }^2\alpha }}$ and${\displaystyle \cos \alpha =1/\sqrt{1+{\tan }^2\alpha }}$.
• 940 – Iran,Abu al-Wafa' al-Buzjani extractsroots using the Indian numeral system.
• 953 – The arithmetic of theHindu–Arabic numeral system at first required the use of a dust board (a sort of handheldblackboard) because "the methods required moving the numbers around in the calculation and rubbing some out as the calculation proceeded."Al-Uqlidisi modified these methods for pen and paper use. Eventually the advances enabled by the decimal system led to its standard use throughout the region and the world.
• 953 – Persia,Al-Karaji is the "first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today. He was first to define themonomials${\displaystyle x}$,${\displaystyle x^2}$,${\displaystyle x^3}$, ... and${\displaystyle 1/x}$,${\displaystyle 1/x^2}$,${\displaystyle 1/x^3}$, ... and to give rules forproducts of any two of these. He started a school of algebra which flourished for several hundreds of years". He also discovered thebinomial theorem forintegerexponents, which "was a major factor in the development ofnumerical analysis based on the decimal system".

• c. 1000 –Abu Sahl al-Quhi (Kuhi) solves polynomialequations higher than thesecond degree.
• c. 1000 –Abu-Mahmud Khujandi first states a special case ofFermat's Last Theorem.
• c. 1000 –Law of sines is discovered byMuslim mathematicians, but it is uncertain who discovers it first betweenAbu-Mahmud al-Khujandi,Abu Nasr Mansur, andAbu al-Wafa' al-Buzjani.
• c. 1000 –Pope Sylvester II introduces theabacus using theHindu–Arabic numeral system to Europe.
• 1000 –Al-Karaji writes a book containing the first knownproofs bymathematical induction. He used it to prove thebinomial theorem,Pascal's triangle, and the sum ofintegralcubes.^[11] He was "the first who introduced the theory ofalgebraiccalculus".^[12]
• c. 1000 –Abu Mansur al-Baghdadi studied a slight variant ofThābit ibn Qurra's theorem onamicable numbers, and he also made improvements on the decimal system.
• 1020 –Abu al-Wafa' al-Buzjani gave the formula: sin (α + β) = sin α cos β + sin β cos α. Also discussed the quadrature of theparabola and the volume of theparaboloid.
• 1021 –Ibn al-Haytham formulated and solvedAlhazen's problem geometrically.
• 1030 –Alī ibn Ahmad al-Nasawī writes a treatise on thedecimal andsexagesimal number systems. His arithmetic explains the division of fractions and the extraction of square and cubic roots (square root of 57,342; cubic root of 3,652,296) in an almost modern manner.^[13]
• 1070 –Omar Khayyam begins to writeTreatise on Demonstration of Problems of Algebra and classifies cubic equations.
• c. 1100 – Omar Khayyám "gave a complete classification ofcubic equations with geometric solutions found by means of intersectingconic sections". He became the first to find generalgeometric solutions of cubic equations and laid the foundations for the development ofanalytic geometry andnon-Euclidean geometry. He also extractedroots using the decimal system (Hindu–Arabic numeral system).
• 12th century –Indian numerals have been modified by Arab mathematicians to form the modernArabic numeral system.
• 12th century – the Arabic numeral system reaches Europe through theArabs.
• 12th century –Bhaskara Acharya writes theLilavati, which covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry,solid geometry, the shadow of thegnomon, methods to solve indeterminate equations, andcombinations.
• 12th century –Bhāskara II (Bhaskara Acharya) writes theBijaganita (Algebra), which is the first text to recognize that a positive number has two square roots. Furthermore, it also gives theChakravala method which was the first generalized solution of so-calledPell's equation.
• 12th century – Bhaskara Acharya develops preliminary concepts ofdifferentiation, and also developsRolle's theorem,Pell's equation, a proof for thePythagorean theorem, proves that division by zero is infinity, computesπ to 5 decimal places, and calculates the time taken for the Earth to orbit the Sun to 9 decimal places.
• 1130 –Al-Samawal al-Maghribi gave a definition of algebra: "[it is concerned] with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known."^[14]
• 1135 –Sharaf al-Din al-Tusi followed al-Khayyam's application of algebra to geometry, and wrote a treatise on cubic equations that "represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry."^[14]
• 1202 –Leonardo Fibonacci demonstrates the utility ofHindu–Arabic numerals in hisLiber Abaci (Book of the Abacus).
• 1247 –Qin Jiushao publishesShùshū Jiǔzhāng (Mathematical Treatise in Nine Sections).
• 1248 –Li Ye writesCeyuan haijing, a 12 volume mathematical treatise containing 170 formulas and 696 problems mostly solved by polynomial equations using the methodtian yuan shu.
• 1260 –Al-Farisi gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerningfactorization andcombinatorial methods. He also gave the pair of amicable numbers 17296 and 18416 that have also been jointly attributed toFermat as well as Thabit ibn Qurra.^[15]
• c. 1250 –Nasir al-Din al-Tusi attempts to develop a form of non-Euclidean geometry.
• 1280 –Guo Shoujing and Wang Xun use cubic interpolation for generating sine.
• 1303 –Zhu Shijie publishesJade Mirror of the Four Unknowns, which contains an ancient method of arrangingbinomial coefficients in a triangle.
• 1356-Narayana Pandita completes his treatiseGanita Kaumudi, generalized Fibonacci sequence, and the first ever algorithm to systematically generate all permutations as well as many new magic figure techniques.
• 14th century –Madhava discovers thepower series expansion for${\displaystyle \sin x}$,${\displaystyle \cos x}$,${\displaystyle \arctan x}$ and${\displaystyle \pi /4}$^[16][17] This theory is now well known in the Western world as theTaylor series or infinite series.^[18]
• 14th century –Parameshvara Nambudiri, a Kerala school mathematician, presents a series form of thesine function that is equivalent to itsTaylor series expansion, states themean value theorem of differential calculus, and is also the first mathematician to give the radius of circle with inscribedcyclic quadrilateral.

• 1400 – Madhava discovers the series expansion for the inverse-tangent function, the infinite series for arctan and sin, and many methods for calculating the circumference of the circle, and uses them to compute π correct to 11 decimal places.
• c. 1400 –Jamshid al-Kashi "contributed to the development ofdecimal fractions not only for approximatingalgebraic numbers, but also forreal numbers such as π. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots, which is a special case of the methods given many centuries later by [Paolo] Ruffini and [William George] Horner." He is also the first to use thedecimal point notation inarithmetic andArabic numerals. His works includeThe Key of arithmetics, Discoveries in mathematics, The Decimal point, andThe benefits of the zero. The contents of theBenefits of the Zero are an introduction followed by five essays: "On whole number arithmetic", "On fractional arithmetic", "On astrology", "On areas", and "On finding the unknowns [unknown variables]". He also wrote theThesis on the sine and the chord andThesis on finding the first degree sine.
• 15th century –Ibn al-Banna' al-Marrakushi andAbu'l-Hasan ibn Ali al-Qalasadi introducedsymbolic notation for algebra and for mathematics in general.^[14]
• 15th century –Nilakantha Somayaji, a Kerala school mathematician, writes theAryabhatiya Bhasya, which contains work on infinite-series expansions, problems of algebra, and spherical geometry.
• 1424 – Jamshid al-Kashi computes π to sixteen decimal places using inscribed and circumscribed polygons.
• 1427 – Jamshid al-Kashi completesThe Key to Arithmetic containing work of great depth on decimal fractions. It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones.
• 1464 –Regiomontanus writesDe Triangulis omnimodus which is one of the earliest texts to treat trigonometry as a separate branch of mathematics.
• 1478 – An anonymous author writes theTreviso Arithmetic.
• 1494 –Luca Pacioli writesSumma de arithmetica, geometria, proportioni et proportionalità; introduces primitive symbolic algebra using "co" (cosa) for the unknown.

• 1501 –Nilakantha Somayaji writes theTantrasamgraha which is the first treatment of all 10 cases in spherical trigonometry.
• 1520 –Scipione del Ferro develops a method for solving "depressed" cubic equations (cubic equations without an x² term), but does not publish.
• 1522 –Adam Ries explained the use of Arabic digits and their advantages over Roman numerals.
• 1535 –Nicolo Tartaglia independently develops a method for solving depressed cubic equations but also does not publish.
• 1539 –Gerolamo Cardano learns Tartaglia's method for solving depressed cubics and discovers a method for depressing cubics, thereby creating a method for solving all cubics.
• 1540 –Lodovico Ferrari solves thequartic equation.
• 1544 –Michael Stifel publishesArithmetica integra.
• 1545 –Gerolamo Cardano conceives the idea ofcomplex numbers.
• 1550 –Jyeṣṭhadeva, aKerala school mathematician, writes theYuktibhāṣā which gives proofs of power series expansion of some trigonometry functions.
• 1572 –Rafael Bombelli writesAlgebra treatise and uses imaginary numbers to solve cubic equations.
• 1584 –Zhu Zaiyu calculatesequal temperament.
• 1596 –Ludolph van Ceulen computes π to twenty decimal places using inscribed and circumscribed polygons.

• 1614 –John Napier publishes a table of Napierianlogarithms inMirifici Logarithmorum Canonis Descriptio.
• 1617 –Henry Briggs discusses decimal logarithms inLogarithmorum Chilias Prima.
• 1618 – John Napier publishes the first references toe in a work onlogarithms.
• 1619 –Johannes Kepler discovers two of theKepler-Poinsot polyhedra.
• 1629 – Pierre de Fermat develops a rudimentarydifferential calculus.
• 1634 –Gilles de Roberval shows that the area under acycloid is three times the area of its generating circle.
• 1636 –Pierre de Fermat circulates manuscriptMethodus ad disquirendam maximam et minimam et de tangentibus linearum curvarum a rudimentarydifferential calculus containing his version ofanalytic geometry
• 1636 –Muhammad Baqir Yazdi jointly discovered the pair ofamicable numbers 9,363,584 and 9,437,056 along withDescartes (1636).^[15]
• 1637 –René Descartes publishesLa Géométrie containing his version ofAnalytic geometry
• 1637 – Pierre de Fermat claims to have provenFermat's Last Theorem in his copy ofDiophantus'Arithmetica.
• 1637 – First use of the termimaginary number by René Descartes; it was meant to be derogatory.
• 1643 – René Descartes developsDescartes' theorem.
• 1654 –Blaise Pascal and Pierre de Fermat create the theory ofprobability.
• 1655 –John Wallis writesArithmetica Infinitorum.
• 1658 –Christopher Wren shows that the length of a cycloid is four times the diameter of its generating circle.
• 1665 –Isaac Newton works on thefundamental theorem of calculus and develops his version ofinfinitesimal calculus.
• 1668 –Nicholas Mercator andWilliam Brouncker discover aninfinite series for the logarithm while attempting to calculate the area under a hyperbolic segment.
• 1671 –James Gregory develops a series expansion for the inverse-tangent function (originally discovered byMadhava).
• 1671 – James Gregory discoversTaylor's theorem.
• 1673 –Gottfried Leibniz also develops his version of infinitesimal calculus.
• 1675 – Isaac Newton invents an algorithm for thecomputation of functional roots.
• 1680s – Gottfried Leibniz works on symbolic logic.
• 1683 –Seki Takakazu discovers theresultant anddeterminant.
• 1683 – Seki Takakazu developselimination theory.
• 1691 – Gottfried Leibniz discovers the technique of separation of variables for ordinarydifferential equations.
• 1693 –Edmund Halley prepares the first mortality tables statistically relating death rate to age.
• 1696 –Guillaume de l'Hôpital stateshis rule for the computation of certainlimits.
• 1696 –Jakob Bernoulli andJohann Bernoulli solvebrachistochrone problem, the first result in thecalculus of variations.
• 1699 –Abraham Sharp calculates π to 72 digits but only 71 are correct.

• 1706 –John Machin develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places.
• 1708 –Seki Takakazu discoversBernoulli numbers.Jacob Bernoulli whom the numbers are named after is believed to have independently discovered the numbers shortly after Takakazu.
• 1712 –Brook Taylor developsTaylor series.
• 1722 –Abraham de Moivre statesde Moivre's formula connectingtrigonometric functions andcomplex numbers.
• 1722 –Takebe Kenko introducesRichardson extrapolation.
• 1724 – Abraham De Moivre studies mortality statistics and the foundation of the theory of annuities inAnnuities on Lives.
• 1730 –James Stirling publishesThe Differential Method.
• 1733 –Giovanni Gerolamo Saccheri studies what geometry would be like ifEuclid's fifth postulate were false.
• 1733 – Abraham de Moivre introduces thenormal distribution to approximate thebinomial distribution in probability.
• 1734 –Leonhard Euler introduces theintegrating factor technique for solving first-order ordinarydifferential equations.
• 1735 – Leonhard Euler solves theBasel problem, relating an infinite series to π.
• 1736 – Leonhard Euler solves the problem of theSeven bridges of Königsberg, in effect creatinggraph theory.
• 1739 – Leonhard Euler solves the generalhomogeneous linear ordinary differential equation withconstant coefficients.
• 1742 –Christian Goldbach conjectures that every even number greater than two can be expressed as the sum of two primes, now known asGoldbach's conjecture.
• 1747 –Jean le Rond d'Alembertsolves thevibrating string problem (one-dimensionalwave equation).^[19]
• 1748 –Maria Gaetana Agnesi discusses analysis inInstituzioni Analitiche ad Uso della Gioventu Italiana.
• 1761 –Thomas Bayes provesBayes' theorem.
• 1761 –Johann Heinrich Lambert proves that π is irrational.
• 1762 –Joseph-Louis Lagrange discovers thedivergence theorem.
• 1789 –Jurij Vega improves Machin's formula and computes π to 140 decimal places, 136 of which were correct.
• 1794 – Jurij Vega publishesThesaurus Logarithmorum Completus.
• 1796 –Carl Friedrich Gauss proves that theregular 17-gon can be constructed using only acompass and straightedge.
• 1796 –Adrien-Marie Legendre conjectures theprime number theorem.
• 1797 –Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms.
• 1799 – Carl Friedrich Gauss proves thefundamental theorem of algebra (every polynomial equation has a solution among the complex numbers).
• 1799 –Paolo Ruffini partially proves theAbel–Ruffini theorem thatquintic or higher equations cannot be solved by a general formula.

• 1801 –Disquisitiones Arithmeticae, Carl Friedrich Gauss'snumber theory treatise, is published in Latin.
• 1805 – Adrien-Marie Legendre introduces themethod of least squares for fitting a curve to a given set of observations.
• 1806 –Louis Poinsot discovers the two remainingKepler-Poinsot polyhedra.
• 1806 –Jean-Robert Argand publishes proof of theFundamental theorem of algebra and theArgand diagram.
• 1807 –Joseph Fourier announces his discoveries about thetrigonometric 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 theintermediate value theorem—acontinuous function that is negative at one point and positive at another point must be zero for at least one point in between. Bolzano gives a first formal(ε, δ)-definition of limit.
• 1821 –Augustin-Louis Cauchy publishesCours d'Analyse which purportedly contains an erroneous “proof” that thepointwise limit of continuous functions is continuous.
• 1822 –Augustin-Louis Cauchy presents theCauchy's integral theorem for integration around the boundary of a rectangle in thecomplex plane.
• 1822 – Irisawa Shintarō Hiroatsu analyzesSoddy's hexlet in aSangaku.
• 1823 –Sophie Germain's Theorem is published in the second edition ofAdrien-Marie Legendre's Essai sur la théorie des nombres^[20]
• 1824 –Niels Henrik Abel partially proves theAbel–Ruffini theorem that the generalquintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.
• 1825 – Augustin-Louis Cauchy presents theCauchy integral theorem for general integration paths—he assumes the function being integrated has a continuous derivative, and he introduces the theory ofresidues incomplex analysis.
• 1825 –Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre prove Fermat's Last Theorem forn = 5.
• 1825 –André-Marie Ampère discoversStokes' theorem.
• 1826 –Niels Henrik Abel gives counterexamples toAugustin-Louis Cauchy’s purported “proof” that thepointwise limit of continuous functions is continuous.
• 1828 – George Green provesGreen's theorem.
• 1829 –János Bolyai,Gauss, andLobachevsky invent hyperbolicnon-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 ofalgebraic equations, thereby essentially foundinggroup theory andGalois theory.
• 1832 – Lejeune Dirichlet proves Fermat's Last Theorem forn = 14.
• 1835 – Lejeune Dirichlet provesDirichlet's theorem about prime numbers in arithmetical progressions.
• 1837 –Pierre Wantzel proves that doubling the cube andtrisecting the angle are impossible with only a compass and straightedge, as well as the full completion of the problem of constructability of regular polygons.
• 1837 –Peter Gustav Lejeune Dirichlet developsAnalytic number theory.
• 1838 – First mention ofuniform convergence in a paper byChristoph Gudermann; later formalized byKarl Weierstrass. Uniform convergence is required to fixAugustin-Louis Cauchy erroneous “proof” that thepointwise limit of continuous functions is continuous from Cauchy's 1821Cours d'Analyse.
• 1841 –Karl Weierstrass discovers but does not publish theLaurent expansion theorem.
• 1843 –Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem.
• 1843 –William Hamilton discovers the calculus ofquaternions and deduces that they are non-commutative.
• 1844 -Hermann Grassmann publishes hisAusdehnungslehre, from whichlinear algebra is later developed.
• 1847 –George Boole formalizessymbolic logic inThe Mathematical Analysis of Logic, defining what is now calledBoolean algebra.
• 1849 –George Gabriel Stokes shows thatsolitary waves can arise from a combination of periodic waves.
• 1850 –Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept ofessential singular points.
• 1850 – George Gabriel Stokes rediscovers and proves Stokes' theorem.
• 1854 –Bernhard Riemann introducesRiemannian geometry.
• 1854 –Arthur Cayley shows that quaternions can be used to represent rotations in four-dimensionalspace.
• 1858 –August Ferdinand Möbius invents theMöbius strip.
• 1858 –Charles Hermite solves the general quintic equation by means of elliptic and modular functions.
• 1859 – Bernhard Riemann formulates theRiemann hypothesis, which has strong implications about the distribution ofprime numbers.
• 1868 –Eugenio Beltrami demonstratesindependence ofEuclid’sparallel postulate from the other axioms ofEuclidean geometry.
• 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.
• 1872 –Richard Dedekind invents what is now called theDedekind Cut for defining irrational numbers, and now used for defining surreal numbers.
• 1873 –Charles Hermite proves thate istranscendental.
• 1873 –Georg Frobenius presents his method for finding series solutions to linear differential equations withregular singular points.
• 1874 –Georg Cantor proves that the set of allreal numbers isuncountably infinite but the set of all realalgebraic numbers iscountably infinite.His proof does not use hisdiagonal argument, which he published in 1891.
• 1882 –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 theKlein bottle.
• 1888 -Sophus Lie publishes work on transformation groups, serving as the foundation for the modern theory of Lie groups.
• 1895 –Diederik Korteweg andGustav de Vries derive theKorteweg–de Vries 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 infinitecardinal numbers and thecontinuum hypothesis.
• 1895 –Henri Poincaré publishes paper "Analysis Situs" which started modern topology.
• 1896 –Jacques Hadamard andCharles Jean de la Vallée-Poussin independently prove theprime number theorem.
• 1896 –Hermann Minkowski presentsGeometry of numbers.
• 1899 – Georg Cantor discovers a contradiction in his set theory.
• 1899 –David Hilbert presents a set of self-consistent geometric axioms inFoundations of Geometry.
• 1900 – David Hilbert states hislist of 23 problems, which show where some further mathematical work is needed.

^[21]

• 1901 –Élie Cartan develops theexterior derivative.
• 1901 –Henri Lebesgue publishes onLebesgue integration.
• 1903 –Edmund Georg Hermann Landau gives considerably simpler proof of the prime number theorem.
• 1908 –Ernst Zermelo axiomizesset theory, thus avoiding Cantor's contradictions.
• 1908 –Josip Plemelj solves the Riemann problem about the existence of a differential equation with a givenmonodromic group and uses Sokhotsky – Plemelj formulae.
• 1912 –Luitzen Egbertus Jan Brouwer presents theBrouwer fixed-point theorem.
• 1912 – Josip Plemelj publishes simplified proof for the Fermat's Last Theorem for exponentn = 5.
• 1915 –Emmy Noether provesher symmetry theorem, which shows that everysymmetry in physics has a correspondingconservation law.
• 1916 –Srinivasa Ramanujan introducesRamanujan conjecture. This conjecture is later generalized byHans Petersson.
• 1919 –Viggo Brun definesBrun's constantB₂ fortwin primes.
• 1921 – Emmy Noether introduces the first general definition of acommutative ring.
• 1928 –John von Neumann begins devising the principles ofgame theory and proves theminimax theorem.
• 1929 – Emmy Noether introduces the first general representation theory of groups and algebras.
• 1930 –Casimir Kuratowski shows that thethree-cottage problem has no solution.
• 1931 –Kurt Gödel proveshis incompleteness theorem, which shows that every axiomatic system for mathematics is either incomplete or inconsistent.
• 1931 –Georges de Rham develops theorems incohomology andcharacteristic classes.
• 1932 -Stefan Banach brought the abstract study of functional analysis to the broader mathematical community.
• 1933 –Karol Borsuk andStanislaw Ulam present theBorsuk–Ulam antipodal-point theorem.
• 1933 –Andrey Nikolaevich Kolmogorov publishes his bookBasic notions of the calculus of probability (Grundbegriffe der Wahrscheinlichkeitsrechnung), which contains anaxiomatization of probability based onmeasure theory.
• 1936 –Alonzo Church andAlan Turing create, respectively, theλ-calculus and theTuring machine, formalizing the notion of computation and computability.
• 1938 –Tadeusz Banachiewicz introducesLU decomposition.
• 1940 – Kurt Gödel shows that neither thecontinuum hypothesis nor theaxiom of choice can be disproven from the standard axioms of set theory.
• 1941 –Cahit Arf defines theArf invariant.
• 1942 –G.C. Danielson andCornelius Lanczos develop afast Fourier transform algorithm.
• 1943 –Kenneth Levenberg proposes a method for nonlinear least squares fitting.
• 1945 –Stephen Cole Kleene introducesrealizability.
• 1945 –Saunders Mac Lane andSamuel Eilenberg startcategory theory.
• 1945 –Norman Steenrod andSamuel Eilenberg give theEilenberg–Steenrod axioms for (co-)homology.
• 1946 –Jean Leray introduces theSpectral sequence.
• 1947 –George Dantzig publishes thesimplex method for linear programming.
• 1948 – John von Neumann mathematically studiesself-reproducing machines.
• 1948 -Norbert Wiener begins the study of cybernetics, the science of communication as it relates to living things and machines.
• 1948 –Atle Selberg andPaul Erdős prove independently in an elementary way theprime number theorem.
• 1949 -André Weil proposed his famous conjectures.
• 1949 –John Wrench and L. R. Smith compute π to 2,037 decimal places usingENIAC.
• 1949 –Claude Shannon develops notion ofinformation theory.
• 1950 –Stanisław Ulam and John von Neumann presentcellular automata dynamical systems.
• 1953 –Nicholas Metropolis introduces the idea of thermodynamicsimulated annealing algorithms.
• 1955 –H. S. M. Coxeter et al. publish the complete list ofuniform polyhedron.
• 1955 –Enrico Fermi,John Pasta, Stanisław Ulam, andMary Tsingou numerically study a nonlinear spring model of heat conduction and discover solitary wave type behavior.
• 1956 –Noam Chomsky describes ahierarchy offormal languages.
• 1956 –John Milnor discovers the existence of anExotic sphere in seven dimensions, inaugurating the field ofdifferential topology.
• 1957 –Kiyosi Itô developsItô calculus.
• 1957 –Stephen Smale provides theexistence proof for crease-freesphere eversion.
• 1958 –Alexander Grothendieck's proof of theGrothendieck–Riemann–Roch theorem is published.
• 1959 –Kenkichi Iwasawa createsIwasawa theory.
• 1960 –Tony Hoare invents thequicksort algorithm.
• 1960 -Rudolf Kalman introduced theKalman filter in his "A New Approach to Linear Filtering and Prediction Problems".
• 1960 –Irving S. Reed andGustave Solomon present theReed–Solomon error-correcting code.
• 1961 –Daniel Shanks andJohn Wrench compute π to 100,000 decimal places using an inverse-tangent identity and anIBM 7090 computer.
• 1961 –John G. F. Francis andVera Kublanovskaya independently develop theQR algorithm to calculate theeigenvalues andeigenvectors of a matrix.
• 1961 – Stephen Smale proves thePoincaré conjecture for all dimensions greater than or equal to 5.
• 1962 –Donald Marquardt proposes theLevenberg–Marquardt nonlinear least squares fitting algorithm.
• 1963 –Paul Cohen uses his technique offorcing 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 andNorman Zabusky analytically study theFermi–Pasta–Ulam–Tsingou heat conduction problem in thecontinuum limit and find that theKdV equation governs this system.
• 1963 – meteorologist and mathematicianEdward Norton Lorenz published solutions for a simplified mathematical model of atmospheric turbulence – generally known as chaotic behaviour andstrange attractors orLorenz attractor – also thebutterfly effect.
• 1965 – Iranian mathematicianLotfi Asker Zadeh foundedfuzzy set theory as an extension of the classical notion ofsets and he founded the field offuzzy mathematics.
• 1965 – Martin Kruskal and Norman Zabusky numerically study collidingsolitary waves inplasmas and find that they do not disperse after collisions.
• 1965 –James Cooley andJohn Tukey present an influential fast Fourier transform algorithm.
• 1966 –E. J. Putzer presents two methods for computing theexponential of a matrix in terms of a polynomial in that matrix.
• 1966 –Abraham Robinson presentsnon-standard analysis.
• 1967 –Robert Langlands formulates the influentialLanglands program of conjectures relating number theory and representation theory.
• 1968 –Michael Atiyah andIsadore Singer prove theAtiyah–Singer index theorem about the index ofelliptic operators.
• 1970 –Yuri Matiyasevich proves that there exists no general algorithm to solve allDiophantine equations, thus giving a negative answer toHilbert's 10th problem.
• 1973 –Lotfi Zadeh founded the field offuzzy logic.
• 1974 –Pierre Deligne solves the last and deepest of theWeil conjectures, completing the program of Grothendieck.
• 1975 –Benoit Mandelbrot publishesLes objets fractals, forme, hasard et dimension.
• 1976 –Kenneth Appel andWolfgang Haken use a computer to prove theFour color theorem.
• 1981 –Richard Feynman gives an influential talk "Simulating Physics with Computers" (in 1980Yuri Manin proposed the same idea about quantum computations in "Computable and Uncomputable" (in Russian)).
• 1983 –Gerd Faltings proves theMordell conjecture and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat's Last Theorem.
• 1984 –Vaughan Jones discovers theJones polynomial in knot theory, which leads to other new knot polynomials as well as connections between knot theory and other fields.
• 1985 –Louis de Branges de Bourcia proves theBieberbach conjecture.
• 1986 –Ken Ribet provesRibet's theorem.
• 1987 –Yasumasa Kanada,David Bailey,Jonathan Borwein, andPeter Borwein use iterative modular equation approximations to elliptic integrals and aNEC SX-2supercomputer to compute π to 134 million decimal places.
• 1991 –Alain Connes andJohn W. Lott developnon-commutative geometry.
• 1992 –David Deutsch andRichard Jozsa develop theDeutsch–Jozsa algorithm, one of the first examples of aquantum algorithm that is exponentially faster than any possible deterministic classical algorithm.
• 1994 –Andrew Wiles proves part of theTaniyama–Shimura conjecture and thereby provesFermat's Last Theorem.
• 1994 –Peter Shor formulatesShor's algorithm, aquantum algorithm forinteger factorization.
• 1995 –Simon Plouffe discoversBailey–Borwein–Plouffe formula capable of finding thenth binary digit of π.
• 1998 –Thomas Callister Hales (almost certainly) proves theKepler conjecture.
• 1999 – the fullTaniyama–Shimura conjecture is proven.
• 2000 – theClay Mathematics Institute proposes the sevenMillennium Prize Problems of unsolved important classic mathematical questions.

• 2002 –Manindra Agrawal,Nitin Saxena, andNeeraj Kayal ofIIT Kanpur present an unconditional deterministicpolynomial time algorithm to determine whether a given number isprime (theAKS primality test).
• 2002 –Preda Mihăilescu provesCatalan's conjecture.
• 2003 –Grigori Perelman proves thePoincaré conjecture.
• 2004 – theclassification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning fifty years, is completed.
• 2004 –Ben Green andTerence Tao prove theGreen–Tao theorem.
• 2009 –Fundamental lemma (Langlands program) isproved byNgô Bảo Châu.^[22]
• 2010 –Larry Guth andNets Hawk Katz solve theErdős distinct distances problem.
• 2013 –Yitang Zhang proves the first finite bound on gaps between prime numbers.^[23]
• 2014 – Project Flyspeck^[24] announces that it completed a proof ofKepler's conjecture.^{[25][26][27][28]}
• 2015 –Terence Tao solves theErdősdiscrepancy problem.
• 2015 –László Babai finds that a quasipolynomial complexity algorithm would solve theGraph isomorphism problem.
• 2016 –Maryna Viazovska solves thesphere packing problem in dimension 8. Subsequent work building on this leads to a solution for dimension 24.
• 2023 – Elia Bruè,Aaron Naber, and Daniele Semola disprove theMilnor conjecture for six or more dimensions.^[29][30]

• History of mathematical notation explains Rhetorical, Syncopated and Symbolic
• Timeline of ancient Greek mathematicians
• Timeline of mathematical innovation in South and West Asia
• Timeline of mathematical logic
• Timeline of women in mathematics
• Timeline of women in mathematics in the United States

• David Eugene Smith, 1929 and 1959,A Source Book in Mathematics,Dover Publications.ISBN 0-486-64690-4.

• O'Connor, John J.;Robertson, Edmund F.,"A Mathematical Chronology",MacTutor History of Mathematics Archive,University of St Andrews

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