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{{Use mdy dates|date=February 2015}}
This is a [[timeline]] of [[pure mathematics|pure]] and [[applied mathematics]] [[History of mathematics|history]]. It is divided here into three stages, corresponding to stages in the [[History of mathematical notation|development 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.

==Rhetorical stage==
===Before 1000 BC===
* ca. [[Middle Paleolithic|70,000 BC]]&nbsp;– South Africa, ochre rocks adorned with scratched [[Geometry|geometric]] patterns (see [[Blombos Cave]]).<ref>[http://www.accessexcellence.org/WN/SU/caveart.html Art Prehistory], Sean Henahan, January 10, 2002. {{webarchive |url=https://web.archive.org/web/20080719073344/http://www.accessexcellence.org/WN/SU/caveart.html |date=July 19, 2008 }}</ref>
* ca. [[35,000 BC]] to [[Upper Paleolithic|20,000 BC]]&nbsp;– Africa and France, earliest known [[prehistory|prehistoric]] attempts to quantify time (see [[Lebombo bone]]).<ref>[http://www.tacomacc.edu/home/jkellerm/Papers/Menses/Menses.htm How Menstruation Created Mathematics], [[Tacoma Community College]], [https://web.archive.org/web/20051223112514/http://www.tacomacc.edu/home/jkellerm/Papers/Menses/Menses.htm (archive link).]</ref><ref>{{cite web|url=http://www.math.buffalo.edu/mad/Ancient-Africa/lebombo.html|title=OLDEST Mathematical Object is in Swaziland|publisher=|accessdate=March 15, 2015}}</ref><ref>{{cite web|url=http://www.math.buffalo.edu/mad/Ancient-Africa/ishango.html|title=an old Mathematical Object|publisher=|accessdate=March 15, 2015}}</ref>
* c. 20,000 BC&nbsp;– [[Nile Valley]], [[Ishango bone]]: possibly the earliest reference to [[prime number]]s and [[Egyptian multiplication]].
* c. 3400 BC&nbsp;– [[Mesopotamia]], the [[Sumer]]ians invent the first [[Babylonian cuneiform numerals|numeral system]], and a system of [[Ancient Mesopotamian units of measurement|weights and measures]].
* c. 3100 BC&nbsp;– [[Egypt]], earliest known [[decimal|decimal system]] allows indefinite counting by way of introducing new symbols.<ref name="buffalo1">{{cite web|url=http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin.|title=Egyptian Mathematical Papyri - Mathematicians of the African Diaspora|publisher=|accessdate=March 15, 2015}}</ref>
* c. 2800 BC&nbsp;– [[Indus Valley Civilisation]] on the [[Indian subcontinent]], earliest use of decimal ratios in a uniform system of [[Ancient Indus Valley units of measurement|ancient weights and measures]], the smallest unit of measurement used is 1.704 millimetres and the smallest unit of mass used is 28&nbsp;grams.
* 2700 BC&nbsp;– Egypt, precision [[surveying]].
* 2400 BC&nbsp;– Egypt, precise [[Egyptian calendar|astronomical calendar]], used even in the [[Middle Ages]] for its mathematical regularity.
* c. 2000 BC&nbsp;– Mesopotamia, the [[Babylonians]] use a base-60 positional numeral system, and compute the first known approximate value of [[pi|π]] at 3.125.
* c. 2000 BC&nbsp;– Scotland, [[carved stone balls]] exhibit a variety of symmetries including all of the symmetries of [[Platonic solid]]s, though it is not known if this was deliberate.
* c. 1800 BC&nbsp;– The [[Plimpton 322]] Babylonian tablet records the oldest known examples of [[Pythagorean triple]]s.<ref>{{citation |last=Joyce |first=David E. |year=1995 |title=Plimpton 322 |url=http://aleph0.clarku.edu/~djoyce/mathhist/plimpnote.html}} and {{citation |last=Maor |first=Eli |year=1993 |title=Trigonometric Delights |publisher=Princeton University Press |isbn=978-0-691-09541-7 |chapter=Plimpton 322: The Earliest Trigonometric Table? |chapter-url=http://press.princeton.edu/titles/6287.html |accessdate=November 28, 2010 |url-status=dead |pages=30–34 |archiveurl=https://web.archive.org/web/20100805230810/http://press.princeton.edu/titles/6287.html |archivedate=5 August 2010}}</ref>
* 1800 BC&nbsp;– Egypt, [[Moscow Mathematical Papyrus]], finding the volume of a [[frustum]].
* c. 1800 BC&nbsp;– [[Berlin Papyrus 6619]] (Egypt, 19th dynasty) contains a quadratic equation and its solution.<ref name="buffalo1"/>
* 1650 BC&nbsp;– [[Rhind Mathematical Papyrus]], copy of a lost scroll from around 1850 BC, the scribe [[Ahmes]] presents one of the first known approximate values of π at 3.16, the first attempt at [[squaring the circle]], earliest known use of a sort of [[cotangent]], and knowledge of solving first order linear equations.
*The earliest recorded use of [[combinatorial]] techniques comes from problem 79 of the [[Rhind Mathematical Papyrus|Rhind papyrus]] which dates to the 16th century BCE.<ref name="Biggs">{{cite book
 | last = Biggs
 | first = Norman
 |author2=Keith Lloyd |author3=Robin Wilson
 | editor = Ronald Graham |editor2=Martin Grötschel | editor2-link = Martin Grötschel |editor3=László Lovász
 | title = Handbook of Combinatorics
 | year = 1995
 | url = https://books.google.com/books?id=kfiv_-l2KyQC
 | format = Google book
 | access-date = 2008-03-08
 | publisher = MIT Press
 | isbn = 0-262-57172-2
 | pages = 2163–2188
 | chapter = 44
 }}</ref>

==Syncopated stage==

===1st millennium BC===
* c. 1000 BC&nbsp;– [[Simple fraction]]s used by the [[Egyptians]]. However, only unit fractions are used (i.e., those with 1 as the numerator) and [[interpolation]] tables are used to approximate the values of the other fractions.<ref>Carl B. Boyer, ''A History of Mathematics'', 2nd Ed.</ref>
* first half of 1st millennium BC&nbsp;– [[Vedic civilization|Vedic India]]&nbsp;– [[Yajnavalkya]], in his [[Shatapatha Brahmana]], describes the motions of the Sun and the Moon, and advances a [[Yajnavalkya 95 Years Cycle|95-year cycle]] to synchronize the motions of the Sun and the Moon.
* c. 800 BC&nbsp;– [[Baudhayana]], author of the Baudhayana [[Shulba Sutras|Shulba Sutra]], a [[Vedic Sanskrit]] geometric text, contains [[quadratic equation]]s, calculates the [[Square root of 2|square 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."<ref>*{{cite book| last1=Hayashi| first1=Takao| year=1995| title=The Bakhshali Manuscript, An ancient Indian mathematical treatise| publisher=Groningen: Egbert Forsten, 596 pages| isbn=90-6980-087-X|page=363}}</ref>
* c. 8th century BC&nbsp;– the [[Yajurveda]], one of the four [[Hindu]] [[Vedas]], contains the earliest concept of [[infinity]], 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&nbsp;– China, ''[[Zhoubi Suanjing]]'', arithmetic, geometric algorithms, and proofs.
* 624 BC – 546 BC&nbsp;– Greece, [[Thales of Miletus]] has various theorems attributed to him.
* c. 600 BC&nbsp;– Greece, the other Vedic "Sulba Sutras" ("rule of chords" in [[Sanskrit]]) use [[Pythagorean triple]]s, contain of a number of geometrical proofs, and approximate [[pi|π]] at 3.16.
* second half of 1st millennium BC&nbsp;– The [[Luoshu Square]], the unique normal [[magic square]] of order three, was discovered in China.
* 530 BC&nbsp;– Greece, [[Pythagoras]] studies propositional [[geometry]] and vibrating lyre strings; his group also discovers the [[irrational number|irrationality]] of the [[square root of two]].
* c. 510 BC&nbsp;– Greece, [[Anaxagoras]]
* c. 500 BC&nbsp;– [[History of India|Indian]] grammarian [[Pāṇini|Pānini]] writes the ''[[Astadhyayi]]'', which contains the use of metarules, [[transformation (mathematics)|transformations]] and [[recursion]]s, originally for the purpose of systematizing the grammar of Sanskrit.
* c. 500 BC&nbsp;– Greece, [[Oenopides of Chios]]
* 470 BC – 410 BC&nbsp;– Greece, [[Hippocrates of Chios]] utilizes [[Lune (mathematics)|lunes]] in an attempt to [[squaring the circle|square the circle]].
* 490 BC – 430 BC – Greece, [[Zeno of Elea]] ''[[Zeno's paradoxes]]''
* 5th century BC&nbsp;– India, [[Apastamba]], author of the Apastamba Sulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the [[square root of 2]] correct to five decimal places.
* 5th c. BC – Greece, [[Theodorus of Cyrene]]
* 5th century&nbsp;– Greece, [[Antiphon the Sophist]]
* 460 BC – 370 BC&nbsp;– Greece, [[Democritus]]
* 460 BC – 399 BC&nbsp;– Greece, [[Hippias]]
* 5th century (late)&nbsp;– Greece, [[Bryson of Heraclea]]
* 428 BC – 347 BC&nbsp;– Greece, [[Archytas]]
* 423 BC – 347 BC&nbsp;– Greece, [[Plato]]
* 417 BC – 317 BC&nbsp;– Greece, [[Theaetetus (mathematician)|Theaetetus]]
* c. 400 BC&nbsp;– India, write the ''Surya Prajinapti'', a mathematical text classifying all numbers into three sets: enumerable, innumerable and [[Infinite set|infinite]]. 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&nbsp;– Greece, [[Eudoxus of Cnidus]]
* 400 BC – 350 BC&nbsp;– Greece, [[Thymaridas]]
* 395 BC – 313 BC&nbsp;– Greece, [[Xenocrates]]
* 390 BC – 320 BC&nbsp;– Greece, [[Dinostratus]]
* 380–290&nbsp;– Greece, [[Autolycus of Pitane]]
* 370 BC&nbsp;– Greece, [[Eudoxus of Cnidus|Eudoxus]] states the [[method of exhaustion]] for [[area]] determination.
* 370 BC – 300 BC&nbsp;– Greece, [[Aristaeus the Elder]]
* 370 BC – 300 BC&nbsp;– Greece, [[Callippus]]
* 350 BC&nbsp;– Greece, [[Aristotle]] discusses [[logic]]al reasoning in ''[[Organon]]''.
* 4th century BC&nbsp;– [[Indian mathematics|Indian]] texts use the Sanskrit word "Shunya" to refer to the concept of "void" ([[0 (number)|zero]]).
*4th century BC&nbsp;– China, [[Counting rods]]
* 330 BC&nbsp;– China, the earliest known work on [[History of geometry#Chinese geometry|Chinese geometry]], the ''Mo Jing'', is compiled.
* 310 BC – 230 BC&nbsp;– Greece, [[Aristarchus of Samos]]
* 390 BC – 310 BC&nbsp;– Greece, [[Heraclides Ponticus]]
* 380 BC – 320 BC&nbsp;– Greece, [[Menaechmus]]
* 300 BC&nbsp;– India, ''Bhagabati Sutra'', which contains the earliest information on [[combinations]].
* 300 BC&nbsp;&nbsp;– Greece, [[Euclid]] in his ''[[Euclid's Elements|Elements]]'' studies geometry as an [[axiomatic system]], proves the infinitude of [[prime number]]s and presents the [[Euclidean algorithm]]; he states the law of reflection in ''Catoptrics'', and he proves the [[fundamental theorem of arithmetic]].
* c. 300 BC&nbsp;– India, [[Brahmi numerals]] (ancestor of the common modern [[base 10]] [[numeral system]])
* 370 BC – 300 BC&nbsp;– Greece, [[Eudemus of Rhodes]] works on histories of arithmetic, geometry and astronomy now lost.<ref name="CorsiWeindling1983">{{cite book|last1=Corsi|first1=Pietro|last2=Weindling|first2=Paul|title=Information sources in the history of science and medicine|url=https://books.google.com/books?id=sV0ZAAAAMAAJ|accessdate=July 6, 2014|year=1983|publisher=Butterworth Scientific|isbn=9780408107648}}</ref>
* 300 BC&nbsp;– [[Mesopotamia]], the [[Babylonians]] invent the earliest calculator, the [[abacus]].
* c. 300 BC&nbsp;– [[Indian mathematicians|Indian mathematician]] [[Pingala]] writes the ''Chhandah-shastra'', which contains the first Indian use of zero as a digit (indicated by a dot) and also presents a description of a [[binary numeral system]], along with the first use of [[Fibonacci numbers]] and [[Pascal's triangle]].
* 280 BC – 210 BC&nbsp;– Greece, [[Nicomedes (mathematician)]]
* 280 BC – 220BC&nbsp;– Greece, [[Philo of Byzantium]]
* 280 BC – 220 BC&nbsp;– Greece, [[Conon of Samos]]
* 279 BC – 206 BC&nbsp;– Greece, [[Chrysippus]]
* c. 3rd century BC – India, [[Kātyāyana]]
* 250 BC – 190 BC&nbsp;– Greece, [[Dionysodorus]]
* 262 -198 BC&nbsp;– Greece, [[Apollonius of Perga]]
* 260 BC&nbsp;– 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&nbsp;– late [[Olmec]]s had already begun to use a true zero (a shell glyph) several centuries before [[Ptolemy]] in the New World. See [[0 (number)]].
* 240 BC – Greece, [[Eratosthenes]] uses [[Sieve of Eratosthenes|his sieve algorithm]] to quickly isolate prime numbers.
* 240 BC 190 BC– Greece, [[Diocles (mathematician)]]
* 225 BC&nbsp;– Greece, [[Apollonius of Perga]] writes ''On [[Conic section|Conic Sections]]'' and names the [[ellipse]], [[parabola]], and [[hyperbola]].
* 202 BC to 186 BC&nbsp;–China, ''[[Book on Numbers and Computation]]'', a mathematical treatise, is written in [[Han dynasty]].
* 200 BC – 140 BC – Greece, [[Zenodorus (mathematician)]]
* 150 BC&nbsp;– India, [[Jainism|Jain]] mathematicians in India write the ''Sthananga Sutra'', which contains work on the theory of numbers, arithmetical operations, geometry, operations with [[fractions]], simple equations, [[cubic equations]], quartic equations, and [[permutations]] and combinations.
* c. 150 BC&nbsp;– Greece, [[Perseus (geometer)]]
* 150 BC&nbsp;– China, A method of [[Gaussian elimination]] appears in the Chinese text ''[[The Nine Chapters on the Mathematical Art]]''.
* 150 BC&nbsp;– China, [[Horner's method]] appears in the Chinese text ''[[The Nine Chapters on the Mathematical Art]]''.
* 150 BC&nbsp;– China, [[Negative numbers]] appear in the Chinese text ''[[The Nine Chapters on the Mathematical Art]]''.
* 150 BC – 75 BC – Phoenician, [[Zeno of Sidon]]
* 190 BC – 120 BC – Greece, [[Hipparchus]] develops the bases of [[trigonometry]].
* 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&nbsp;– [[Indian numerals]], a descendant of the [[Brahmi numerals]] (the first [[positional notation]] [[base-10]] [[numeral system]]), begins development in [[History of India|India]].
* mid 1st century [[Cleomedes]] (as late as 400 AD)
* final centuries BC&nbsp;– Indian astronomer [[Lagadha]] writes the ''Vedanga Jyotisha'', a Vedic text on [[astronomy]] 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 (scholar)|Liu Xin]]

===1st millennium AD===
* 1st century&nbsp;– 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&nbsp;– Greece, [[Ptolemy]] of [[Alexandria]] wrote the ''[[Almagest]]''.
* 132 – 192 – China, [[Cai Yong]]
* 240 – 300 – Greece, [[Sporus of Nicaea]]
* 250&nbsp;– Greece, [[Diophantus]] uses symbols for unknown numbers in terms of syncopated [[algebra]], and writes ''[[Arithmetica]]'', one of the earliest treatises on algebra.
* 263&nbsp;– China, [[Liu Hui]] computes [[pi|π]] using [[Liu Hui's π algorithm]].
* 300&nbsp;– the earliest known use of [[0 (number)|zero]] as a decimal digit is introduced by [[Indian mathematicians]].
* 234 – 305 – Greece, [[Porphyry (philosopher)]]
* 300 – 360 – Greece, [[Serenus of Antinoöpolis]]
* 335 – 405– Greece, [[Theon of Alexandria]]
* c. 340&nbsp;– Greece, [[Pappus of Alexandria]] states his [[Pappus's hexagon theorem|hexagon theorem]] and his [[Pappus's centroid theorem|centroid theorem]].
* 350 – 415 – Eastern Roman Empire, [[Hypatia]]
* c. 400&nbsp;– India, the [[Bakhshali manuscript]], which describes a theory of the infinite containing different levels of [[Infinite set|infinity]], shows an understanding of [[Indexed family|indices]], as well as [[logarithms]] to [[base 2]], and computes [[square roots]] of numbers as large as a million correct to at least 11 decimal places.
* 300 to 500&nbsp;– the [[Chinese remainder theorem]] is developed by [[Sun Tzu (mathematician)|Sun Tzu]].
* 300 to 500&nbsp;– China, a description of [[rod calculus]] is written by [[Sun Tzu (mathematician)|Sun Tzu]].
* 412 – 485 – Greece, [[Proclus]]
* 420 – 480 – Greece, [[Domninus of Larissa]]
* b 440 – Greece, [[Marinus of Neapolis]] "I wish everything was mathematics."
* 450&nbsp;– China, [[Zu Chongzhi]] computes [[pi|π]] 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&nbsp;– India, [[Aryabhata]] writes the ''Aryabhata-Siddhanta'', which first introduces the trigonometric functions and methods of calculating their approximate numerical values. It defines the concepts of [[sine]] and [[cosine]], and also contains the [[Aryabhata's sine table|earliest 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&nbsp;– Aryabhata gives accurate calculations for astronomical constants, such as the [[solar eclipse]] and [[lunar eclipse]], computes π to four decimal places, and obtains whole number solutions to [[linear equations]] by a method equivalent to the modern method.
* 505 – 587 – India, [[Varāhamihira]]
* 6th century – India, [[Yativṛṣabha]]
* 535 – 566 – China, [[Zhen Luan]]
* 550&nbsp;– [[Hindu]] mathematicians give zero a numeral representation in the [[positional notation]] [[Indian numerals|Indian numeral]] system.
* 600 – China, Liu Zhuo uses quadratic interpolation.
* 602 – 670 – China, [[Li Chunfeng]] 
* 625 China, Wang Xiaotong writes the ''Jigu Suanjing'', where cubic and quartic equations are solved.
* 7th century&nbsp;– India, [[Bhāskara I]] gives a rational approximation of the sine function.
* 7th century&nbsp;– 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&nbsp;– Brahmagupta writes the ''[[Brāhmasphuṭasiddhānta|Brahma-sphuta-siddhanta]]'', where zero is clearly explained, and where the modern [[place-value]] Indian numeral system is fully developed. It also gives rules for manipulating both [[negative and positive numbers]], methods for computing square roots, methods of solving [[linear equation|linear]] and [[quadratic equation]]s, and rules for summing [[series (mathematics)|series]], [[Brahmagupta's identity]], and the [[Brahmagupta theorem]].
* 721 – China, Zhang Sui (Yi Xing) computes the first tangent table.
* 8th century&nbsp;– India, [[Virasena]] gives explicit rules for the [[Fibonacci sequence]], gives the derivation of the [[volume]] of a [[frustum]] using an [[Infinity|infinite]] procedure, and also deals with the [[logarithm]] to base 2 and knows its laws.
* 8th century&nbsp;– India, [[Sridhara]] gives the rule for finding the volume of a sphere and also the formula for solving quadratic equations.
* 773&nbsp;– Iraq, Kanka brings Brahmagupta's Brahma-sphuta-siddhanta to [[Baghdad]] to explain the Indian system of arithmetic [[astronomy]] and the Indian numeral system.
* 773&nbsp;– [[Muḥammad ibn Ibrāhīm al-Fazārī]] translates the Brahma-sphuta-siddhanta into Arabic upon the request of King Khalif Abbasid Al Mansoor.
* 9th century&nbsp;– India, [[Govindasvāmi]] discovers the Newton-Gauss interpolation formula, and gives the fractional parts of Aryabhata's tabular [[Sine (trigonometric function)|sines]].
* 810&nbsp;– The [[House of Wisdom]] is built in Baghdad for the translation of Greek and [[Sanskrit]] mathematical works into Arabic.
* 820&nbsp;– [[Al-Khwarizmi]]&nbsp;– [[Persian people|Persian]] mathematician, father of algebra, writes the ''[[The Compendious Book on Calculation by Completion and Balancing|Al-Jabr]]'', later transliterated as ''[[Algebra]]'', which introduces systematic algebraic techniques for solving linear and quadratic equations. Translations of his book on [[arithmetic]] will introduce the [[Arabic numerals|Hindu–Arabic]] [[decimal]] number system to the Western world in the 12th century. The term ''[[algorithm]]'' is also named after him.
* 820&nbsp;– Iran, [[Al-Mahani]] conceived the idea of reducing [[Geometry|geometrical]] problems such as [[doubling the cube]] to problems in algebra.
* c. 850&nbsp;– Iraq, [[al-Kindi]] pioneers [[cryptanalysis]] and [[frequency analysis]] in his book on [[cryptography]].
* c. 850&nbsp;– India, [[Mahāvīra (mathematician)|Mahāvīra]] writes the Gaṇitasārasan̄graha otherwise known as the Ganita Sara Samgraha which gives systematic rules for expressing a fraction as the [[Egyptian fraction|sum of unit fractions]].
* 895&nbsp;– Syria, [[Thābit ibn Qurra]]: the only surviving fragment of his original work contains a chapter on the solution and properties of [[cubic equation]]s. He also generalized the [[Pythagorean theorem]], and discovered the [[Thabit number|theorem]] by which pairs of [[amicable number]]s can be found, (i.e., two numbers such that each is the sum of the proper divisors of the other).
* c. 900&nbsp;– Egypt, [[Abu Kamil]] had begun to understand what we would write in symbols as <math>x^n \cdot x^m = x^{m+n}</math>
* 940&nbsp;– Iran, [[Abu al-Wafa' al-Buzjani]] extracts [[root of a function|roots]] using the Indian numeral system.
* 953&nbsp;– The arithmetic of the [[Hindu–Arabic numeral system]] at first required the use of a dust board (a sort of handheld [[blackboard]]) because "the methods required moving the numbers around in the calculation and rubbing some out as the calculation proceeded." [[Abu'l-Hasan al-Uqlidisi|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&nbsp;– 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 the [[monomial]]s <math>x</math>, <math>x^2</math>, <math>x^3</math>, ... and <math>1/x</math>, <math>1/x^2</math>, <math>1/x^3</math>, ... and to give rules for [[product (mathematics)|products]] of any two of these. He started a school of algebra which flourished for several hundreds of years". He also discovered the [[binomial theorem]] for [[integer]] [[exponent]]s, which "was a major factor in the development of [[numerical analysis]] based on the decimal system".
* 975&nbsp;– 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: <math> \sin \alpha = \tan \alpha / \sqrt{1+\tan^2 \alpha} </math> and <math> \cos \alpha = 1 / \sqrt{1 + \tan^2 \alpha}</math>.

==Symbolic stage==

===1000–1500===
* c. 1000&nbsp;– [[Abu Sahl al-Quhi]] (Kuhi) solves [[equation]]s higher than the [[Quadratic equation|second degree]].
* c. 1000&nbsp;– [[Abu-Mahmud Khujandi]] first states a special case of [[Fermat's Last Theorem]].
* c. 1000&nbsp;– [[Law of sines]] is discovered by [[Islamic mathematics|Muslim mathematicians]], but it is uncertain who discovers it first between [[Abu-Mahmud al-Khujandi]], [[Abu Nasr Mansur]], and [[Abu al-Wafa' al-Buzjani]].
* c. 1000&nbsp;– [[Pope Sylvester II]] introduces the [[abacus]] using the [[Hindu–Arabic numeral system]] to Europe.
* 1000&nbsp;– [[Al-Karaji]] writes a book containing the first known [[Mathematical proof|proofs]] by [[mathematical induction]]. He used it to prove the [[binomial theorem]], [[Pascal's triangle]], and the sum of [[Integer|integral]] [[Cube (algebra)|cubes]].<ref>Victor J. Katz (1998). ''History of Mathematics: An Introduction'', p. 255–259. [[Addison-Wesley]]. {{isbn|0-321-01618-1}}.</ref> He was "the first who introduced the theory of [[algebra]]ic [[calculus]]".<ref>F. Woepcke (1853). ''Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi''. Paris.</ref>
* c. 1000&nbsp;– [[Abu Mansur al-Baghdadi]] studied a slight variant of [[Thābit ibn Qurra]]'s theorem on [[amicable number]]s, and he also made improvements on the decimal system.
* 1020&nbsp;– [[Abu al-Wafa' al-Buzjani]] gave the 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]].
* 1021&nbsp;– [[Ibn al-Haytham]] formulated and solved [[Alhazen's problem]] geometrically.
* 1030&nbsp;– [[Alī ibn Ahmad al-Nasawī]] writes a treatise on the [[decimal]] and [[sexagesimal]] 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.<ref>{{MacTutor|id=Al-Nasawi|title=Abu l'Hasan Ali ibn Ahmad Al-Nasawi}}</ref>
* 1070&nbsp;– [[Omar Khayyam]] begins to write ''Treatise on Demonstration of Problems of Algebra'' and classifies cubic equations.
* c. 1100&nbsp;– Omar Khayyám "gave a complete classification of [[cubic equation]]s with geometric solutions found by means of intersecting [[conic section]]s". He became the first to find general [[geometry|geometric]] solutions of cubic equations and laid the foundations for the development of [[analytic geometry]] and [[non-Euclidean geometry]]. He also extracted [[root of a function|roots]] using the decimal system (Hindu–Arabic numeral system).
* 12th century&nbsp;– [[Indian numerals]] have been modified by Arab mathematicians to form the modern [[Arabic numeral]] system .
* 12th century&nbsp;– the Arabic numeral system reaches Europe through the [[Arabs]].
* 12th century&nbsp;– [[Bhāskara II|Bhaskara Acharya]] writes the [[Lilavati]], which covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, [[solid geometry]], the shadow of the [[gnomon]], methods to solve indeterminate equations, and [[combinations]].
* 12th century&nbsp;– [[Bhāskara II]] (Bhaskara Acharya) writes the ''[[Bijaganita]]'' (''[[Algebra]]''), which is the first text to recognize that a positive number has two square roots. Furthermore, it also gives the ''[[Chakravala method]]'' which was the first generalized solution of so-called ''[[Pell's equation]]''
* 12th century&nbsp;– Bhaskara Acharya develops preliminary concepts of [[Derivative|differentiation]], and also develops [[Rolle's theorem]], [[Pell's equation]], a proof for the [[Pythagorean theorem]], proves that division by zero is infinity, computes [[pi|π]] to 5 decimal places, and calculates the time taken for the Earth to orbit the Sun to 9 decimal places.
* 1130&nbsp;– [[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."<ref name=MacTutor/>
* 1135&nbsp;– [[Sharafeddin 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".<ref name=MacTutor>[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_mathematics.html Arabic mathematics], ''[[MacTutor History of Mathematics archive]]'', [[University of St Andrews]], Scotland</ref>
* 1202&nbsp;– [[Leonardo of Pisa|Leonardo Fibonacci]] demonstrates the utility of [[Hindu–Arabic numerals]] in his [[Liber Abaci]] (''Book of the Abacus'').
* 1247&nbsp;– [[Qin Jiushao]] publishes ''Shùshū Jiǔzhāng'' (''[[Mathematical Treatise in Nine Sections]]'').
* 1248&nbsp;– [[Li Ye (mathematician)|Li Ye]] writes ''[[Ceyuan haijing]]'', a 12 volume mathematical treatise containing 170 formulas and 696 problems mostly solved by polynomial equations using the method [[tian yuan shu]].
* 1260&nbsp;– [[Al-Farisi]] gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning [[factorization]] and [[Combinatorics|combinatorial]] methods. He also gave the pair of amicable numbers 17296 and 18416 that have also been jointly attributed to [[Fermat]] as well as Thabit ibn Qurra.<ref name="Various AP Lists and Statistics">[http://amicable.homepage.dk/apstat.htm#discoverer Various AP Lists and Statistics] {{Webarchive|url=https://web.archive.org/web/20120728163824/http://amicable.homepage.dk/apstat.htm#discoverer |date=July 28, 2012 }}</ref>
* c. 1250&nbsp;– [[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&nbsp;– [[Zhu Shijie]] publishes ''Precious Mirror of the Four Elements'', which contains an ancient method of arranging [[binomial coefficient]]s in a triangle.
*1356- [[Narayana Pandita (mathematician)|Narayana Pandita]] completes his treatise [[Ganita Kaumudi]], generalized fibonacci sequence, and the first ever algorithm to systematically generate all permutations as well as many new magic figure techniques.
* 14th century&nbsp;– [[Madhava of Sangamagrama|Madhava]] discovers the [[power series]] expansion for <math>\sin x</math>, <math>\cos x</math>, <math>\arctan x</math> and <math>\pi/4</math> <ref>{{Cite web |last=Weisstein |first=Eric W. |title=Taylor Series |url=https://mathworld.wolfram.com/ |access-date=2022-11-03 |website=mathworld.wolfram.com |language=en}}</ref><ref>{{Cite journal |date=August 1932 |title=The Taylor Series: an Introduction to the Theory of Functions of a Complex Variable |journal=Nature |language=en |volume=130 |issue=3275 |pages=188 |doi=10.1038/130188b0 |bibcode=1932Natur.130R.188. |s2cid=4088442 |issn=1476-4687|doi-access=free }}</ref> This theory is now well known in the Western world as the [[Taylor series]] or infinite series.<ref>{{Cite web |last=Saeed |first=Mehreen |date=2021-08-19 |title=A Gentle Introduction to Taylor Series |url=https://machinelearningmastery.com/a-gentle-introduction-to-taylor-series/ |access-date=2022-11-03 |website=Machine Learning Mastery |language=en-US}}</ref>
* 14th century&nbsp;– [[Parameshvara Nambudiri]], a Kerala school mathematician, presents a series form of the [[sine function]] that is equivalent to its [[Taylor series]] expansion, states the [[mean value theorem]] of differential calculus, and is also the first mathematician to give the radius of circle with inscribed [[cyclic quadrilateral]].

====15th century====
* 1400&nbsp;– 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&nbsp;– [[Jamshid al-Kashi]]『contributed to the development of [[decimal fraction]]s not only for approximating [[algebraic number]]s, but also for [[real number]]s 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 the [[decimal point]] notation in [[arithmetic]] and [[Arabic numerals]]. His works include ''The Key of arithmetics, Discoveries in mathematics, The Decimal point'', and ''The benefits of the zero''. The contents of the ''Benefits 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 the ''Thesis on the sine and the chord'' and ''Thesis on finding the first degree sine''.
* 15th century&nbsp;– [[Ibn al-Banna' al-Marrakushi]] and [[Abu'l-Hasan ibn Ali al-Qalasadi]] introduced [[Mathematical notation|symbolic notation]] for algebra and for mathematics in general.<ref name=MacTutor/>
* 15th century&nbsp;– [[Nilakantha Somayaji]], a Kerala school mathematician, writes the ''Aryabhatiya Bhasya'', which contains work on infinite-series expansions, problems of algebra, and spherical geometry.
* 1424&nbsp;– Ghiyath al-Kashi computes π to sixteen decimal places using inscribed and circumscribed polygons.
* 1427&nbsp;– [[Jamshid al-Kashi]] completes ''The 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&nbsp;– [[Regiomontanus]] writes ''De Triangulis omnimodus'' which is one of the earliest texts to treat trigonometry as a separate branch of mathematics.
* 1478&nbsp;– An anonymous author writes the ''[[Treviso Arithmetic]]''.
* 1494&nbsp;– [[Luca Pacioli]] writes ''[[Summa de arithmetica|Summa de arithmetica, geometria, proportioni et proportionalità]]''; introduces primitive symbolic algebra using "co" (cosa) for the unknown.

===Modern===

====16th century====
* 1501&nbsp;– [[Nilakantha Somayaji]] writes the [[Tantrasamgraha]] which is the first treatment of all 10 cases in spherical trigonometry.
* 1520&nbsp;– [[Scipione del Ferro]] develops a method for solving "depressed" cubic equations (cubic equations without an x2 term), but does not publish.
* 1522&nbsp;– [[Adam Ries]] explained the use of Arabic digits and their advantages over Roman numerals.
* 1535&nbsp;– [[Nicolo Tartaglia]] independently develops a method for solving depressed cubic equations but also does not publish.
* 1539&nbsp;– [[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&nbsp;– [[Lodovico Ferrari]] solves the [[quartic equation]].
* 1544&nbsp;– [[Michael Stifel]] publishes ''Arithmetica integra''.
* 1545&nbsp;– [[Gerolamo Cardano]] conceives the idea of [[complex number]]s.
* 1550&nbsp;– [[Jyeṣṭhadeva]], a [[Kerala school of astronomy and mathematics|Kerala school]] mathematician, writes the ''[[Yuktibhāṣā]]'' which gives proofs of power series expansion of some trigonometry functions.
* 1572&nbsp;– [[Rafael Bombelli]] writes ''Algebra'' treatise and uses imaginary numbers to solve cubic equations.
* 1584&nbsp;– [[Zhu Zaiyu]] calculates [[equal temperament]].
* 1596&nbsp;– [[Ludolph van Ceulen]] computes π to twenty decimal places using inscribed and circumscribed polygons.

====17th century====
* 1614&nbsp;– [[John Napier]] publishes a table of Napierian [[logarithm]]s in ''[[Mirifici Logarithmorum Canonis Descriptio]]''.
* 1617&nbsp;– [[Henry Briggs (mathematician)|Henry Briggs]] discusses decimal logarithms in ''Logarithmorum Chilias Prima''.
* 1618&nbsp;– John Napier publishes the first references to [[E (mathematical constant)|''e'']] in a work on [[logarithms]].
* 1619&nbsp;– [[René Descartes]] discovers [[analytic geometry]] ([[Pierre de Fermat]] claimed that he also discovered it independently).
* 1619&nbsp;– [[Johannes Kepler]] discovers two of the [[Kepler-Poinsot polyhedra]].
* 1629&nbsp;– Pierre de Fermat develops a rudimentary [[differential calculus]].
* 1634&nbsp;– [[Gilles de Roberval]] shows that the area under a [[cycloid]] is three times the area of its generating circle.
* 1636&nbsp;– [[Muhammad Baqir Yazdi]] jointly discovered the pair of [[amicable number]]s 9,363,584 and 9,437,056 along with [[René Descartes|Descartes]] (1636).<ref name="Various AP Lists and Statistics"/>
* 1637&nbsp;– Pierre de Fermat claims to have proven [[Fermat's Last Theorem]] in his copy of [[Diophantus]]' ''Arithmetica''.
* 1637&nbsp;– First use of the term [[imaginary number]] by René Descartes; it was meant to be derogatory.
* 1643&nbsp;– René Descartes develops [[Descartes' theorem]].
* 1654&nbsp;– [[Blaise Pascal]] and Pierre de Fermat create the theory of [[probability]].
* 1655&nbsp;– [[John Wallis]] writes ''Arithmetica Infinitorum''.
* 1658&nbsp;– [[Christopher Wren]] shows that the length of a cycloid is four times the diameter of its generating circle.
* 1665&nbsp;– [[Isaac Newton]] works on the [[fundamental theorem of calculus]] and develops his version of [[infinitesimal calculus]].
* 1668&nbsp;– [[Nicholas Mercator]] and [[William Brouncker, 2nd Viscount Brouncker|William Brouncker]] discover an [[infinite series]] for the logarithm while attempting to calculate the area under a [[hyperbolic segment]].
* 1671&nbsp;– [[James Gregory (astronomer and mathematician)|James Gregory]] develops a series expansion for the inverse-[[tangent (trigonometric function)|tangent]] function (originally discovered by [[Madhava of Sangamagrama|Madhava]]).
* 1671&nbsp;– James Gregory discovers [[Taylor's theorem]].
* 1673&nbsp;– [[Gottfried Leibniz]] also develops his version of infinitesimal calculus.
* 1675&nbsp;– Isaac Newton invents an algorithm for the [[Newton's method|computation of functional roots]].
* 1680s – Gottfried Leibniz works on symbolic logic.
* 1683&nbsp;– [[Seki Takakazu]] discovers the [[resultant]] and [[determinant]].
* 1683&nbsp;– Seki Takakazu develops [[elimination theory]].
* 1691&nbsp;– Gottfried Leibniz discovers the technique of separation of variables for ordinary [[differential equation]]s.
* 1693&nbsp;– [[Edmund Halley]] prepares the first mortality tables statistically relating death rate to age.
* 1696&nbsp;– [[Guillaume de l'Hôpital]] states [[L'Hôpital's rule|his rule]] for the computation of certain [[limit (mathematics)|limits]].
* 1696&nbsp;– [[Jakob Bernoulli]] and [[Johann Bernoulli]] solve [[brachistochrone curve|brachistochrone problem]], the first result in the [[calculus of variations]].
* 1699&nbsp;– [[Abraham Sharp]] calculates π to 72 digits but only 71 are correct.

====18th century====
* 1706&nbsp;– [[John Machin]] develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places.
* 1708&nbsp;– [[Seki Takakazu]] discovers [[Bernoulli numbers]]. [[Jacob Bernoulli]] whom the numbers are named after is believed to have independently discovered the numbers shortly after Takakazu.
* 1712&nbsp;– [[Brook Taylor]] develops [[Taylor series]].
* 1722&nbsp;– [[Abraham de Moivre]] states [[de Moivre's formula]] connecting [[trigonometric function]]s and [[complex number]]s.
* 1722&nbsp;– [[Takebe Kenko]] introduces [[Richardson extrapolation]].
* 1724&nbsp;– Abraham De Moivre studies mortality statistics and the foundation of the theory of annuities in ''Annuities on Lives''.
* 1730&nbsp;– [[James Stirling (mathematician)|James Stirling]] publishes ''The Differential Method''.
* 1733&nbsp;– [[Giovanni Gerolamo Saccheri]] studies what geometry would be like if [[parallel postulate|Euclid's fifth postulate]] were false.
* 1733&nbsp;– Abraham de Moivre introduces the [[normal distribution]] to approximate the [[binomial distribution]] in probability.
* 1734&nbsp;– [[Leonhard Euler]] introduces the [[integrating factor technique]] for solving first-order ordinary [[differential equation]]s.
* 1735&nbsp;– Leonhard Euler solves the [[Basel problem]], relating an infinite series to π.
* 1736&nbsp;– Leonhard Euler solves the problem of the [[Seven bridges of Königsberg]], in effect creating [[graph theory]].
* 1739&nbsp;– Leonhard Euler solves the general [[homogeneous linear ordinary differential equation]] with [[constant coefficients]].
* 1742&nbsp;– [[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]].
* 1747&nbsp;– [[Jean le Rond d'Alembert]] [[d'Alembert's formula|solves]] the [[String vibration|vibrating string]] problem (one-dimensional [[wave equation]]).<ref>D'Alembert (1747) [https://archive.org/details/bub_gb_lJQDAAAAMAAJ/page/n244  "Recherches sur la courbe que forme une corde tenduë mise en vibration"] (Researches on the curve that a tense cord [string] forms [when] set into vibration), ''Histoire de l'académie royale des sciences et belles lettres de Berlin'', vol. 3, pages 214-219.</ref>
* 1748&nbsp;– [[Maria Gaetana Agnesi]] discusses analysis in ''Instituzioni Analitiche ad Uso della Gioventu Italiana''.
* 1761&nbsp;– [[Thomas Bayes]] proves [[Bayes' theorem]].
* 1761&nbsp;– [[Johann Heinrich Lambert]] proves that π is irrational.
* 1762&nbsp;– [[Joseph-Louis Lagrange]] discovers the [[divergence theorem]].
* 1789&nbsp;– [[Jurij Vega]] improves Machin's formula and computes π to 140 decimal places, 136 of which were correct.
* 1794&nbsp;– Jurij Vega publishes ''[[Thesaurus Logarithmorum Completus]]''.
* 1796&nbsp;– [[Carl Friedrich Gauss]] proves that the [[heptadecagon|regular 17-gon]] can be constructed using only a [[compass and straightedge]].
* 1796&nbsp;– [[Adrien-Marie Legendre]] conjectures the [[prime number theorem]].
* 1797&nbsp;– [[Caspar Wessel]] associates vectors with complex numbers and studies complex number operations in geometrical terms.
* 1799&nbsp;– Carl Friedrich Gauss proves the [[fundamental theorem of algebra]] (every polynomial equation has a solution among the complex numbers).
* 1799&nbsp;– [[Paolo Ruffini (mathematician)|Paolo Ruffini]] partially proves the [[Abel–Ruffini theorem]] that [[Quintic equation|quintic]] or higher equations cannot be solved by a general formula.

====19th century====
* 1801&nbsp;– ''[[Disquisitiones Arithmeticae]]'', Carl Friedrich Gauss's [[number theory]] treatise, is published in Latin.
* 1805&nbsp;– Adrien-Marie Legendre introduces the [[method of least squares]] for fitting a curve to a given set of observations.
* 1806&nbsp;– [[Louis Poinsot]] discovers the two remaining [[Kepler-Poinsot polyhedra]].
* 1806&nbsp;– [[Jean-Robert Argand]] publishes proof of the [[Fundamental theorem of algebra]] and the [[Argand diagram]].
* 1807&nbsp;– [[Joseph Fourier]] announces his discoveries about the [[Fourier series|trigonometric decomposition of functions]].
* 1811&nbsp;– 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&nbsp;– [[Siméon Denis Poisson]] carries out integrations along paths in the complex plane.
* 1817&nbsp;– [[Bernard Bolzano]] presents the [[intermediate value theorem]]—a [[continuous 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&nbsp;– [[Augustin-Louis Cauchy]] publishes [[Cours d'Analyse]] which purportedly contains an erroneous “proof” that the [[Pointwise convergence|pointwise limit]] of continuous functions is continuous.
* 1822&nbsp;– [[Augustin-Louis Cauchy]] presents the [[Cauchy's integral theorem]] for integration around the boundary of a rectangle in the [[complex plane]].
* 1822&nbsp;– Irisawa Shintarō Hiroatsu analyzes [[Soddy's hexlet]] in a [[Sangaku]].
* 1823&nbsp;– [[Sophie Germain's theorem|Sophie Germain's Theorem]] is published in the second edition of [[Adrien-Marie Legendre|Adrien-Marie Legendre's]] Essai sur la théorie des nombres<ref>{{Cite web|url=https://www.agnesscott.edu/lriddle/women/germain-FLT/SGandFLT.htm|title = Sophie Germain and FLT}}</ref>
* 1824&nbsp;– [[Niels Henrik Abel]] partially proves the [[Abel–Ruffini theorem]] that the general [[Quintic equation|quintic]] or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.
* 1825&nbsp;– Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths—he assumes the function being integrated has a continuous derivative, and he introduces the theory of [[residue (complex analysis)|residue]]s in [[complex analysis]].
* 1825&nbsp;– [[Peter Gustav Lejeune Dirichlet]] and Adrien-Marie Legendre prove Fermat's Last Theorem for ''n'' = 5.
* 1825&nbsp;– [[André-Marie Ampère]] discovers [[Stokes' theorem]].
* 1826&nbsp;– [[Niels Henrik Abel]] gives counterexamples to [[Augustin-Louis Cauchy]]’s purported “proof” that the [[pointwise limit]] of continuous functions is continuous.
* 1828&nbsp;– George Green proves [[Green's theorem]].
* 1829&nbsp;– [[János Bolyai]], [[Carl Friedrich Gauss|Gauss]], and [[Nikolai Ivanovich Lobachevsky|Lobachevsky]] invent hyperbolic [[non-Euclidean geometry]].
* 1831&nbsp;– [[Mikhail Vasilievich Ostrogradsky]] rediscovers and gives the first proof of the divergence theorem earlier described by Lagrange, Gauss and Green.
* 1832&nbsp;– [[Évariste Galois]] presents a general condition for the solvability of [[algebraic equation]]s, thereby essentially founding [[group theory]] and [[Galois theory]].
* 1832&nbsp;– Lejeune Dirichlet proves Fermat's Last Theorem for ''n'' = 14.
* 1835&nbsp;– Lejeune Dirichlet proves [[Dirichlet's theorem on arithmetic progressions|Dirichlet's theorem]] about prime numbers in arithmetical progressions.
* 1837&nbsp;– [[Pierre Wantzel]] proves that doubling the cube and [[trisecting 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&nbsp;– [[Peter Gustav Lejeune Dirichlet]] develops [[Analytic number theory]].
* 1838&nbsp;– First mention of [[uniform convergence]] in a paper by [[Christoph Gudermann]]; later formalized by [[Karl Weierstrass]]. Uniform convergence is required to fix [[Augustin-Louis Cauchy]] erroneous “proof” that the [[Pointwise convergence|pointwise limit]] of continuous functions is continuous from Cauchy's 1821 [[Cours d'Analyse]].
* 1841&nbsp;– [[Karl Weierstrass]] discovers but does not publish the [[Laurent expansion theorem]].
* 1843&nbsp;– [[Pierre-Alphonse Laurent]] discovers and presents the Laurent expansion theorem.
* 1843&nbsp;– [[William Rowan Hamilton|William Hamilton]] discovers the calculus of [[quaternion]]s and deduces that they are non-commutative.
* 1844&nbsp;- [[Hermann Grassmann]] publishes his ''Ausdehnungslehre'', from which [[linear algebra]] is later developed.
* 1847&nbsp;– [[George Boole]] formalizes [[symbolic logic]] in ''The Mathematical Analysis of Logic'', defining what is now called [[Boolean algebra (logic)|Boolean algebra]].
* 1849&nbsp;– [[George Gabriel Stokes]] shows that [[soliton|solitary wave]]s can arise from a combination of periodic waves.
* 1850&nbsp;– [[Victor Alexandre Puiseux]] distinguishes between poles and branch points and introduces the concept of [[mathematical singularity|essential singular points]].
* 1850&nbsp;– George Gabriel Stokes rediscovers and proves Stokes' theorem.
* 1854&nbsp;– [[Bernhard Riemann]] introduces [[Riemannian geometry]].
* 1854&nbsp;– [[Arthur Cayley]] shows that quaternions can be used to represent rotations in four-dimensional [[space]].
* 1858&nbsp;– [[August Ferdinand Möbius]] invents the [[Möbius strip]].
* 1858&nbsp;– [[Charles Hermite]] solves the general quintic equation by means of elliptic and modular functions.
* 1859&nbsp;– Bernhard Riemann formulates the [[Riemann hypothesis]], which has strong implications about the distribution of [[prime number]]s.
* 1868&nbsp;– [[Eugenio Beltrami]] demonstrates [[Independence (mathematical logic)|independence]] of [[Euclid]]’s [[parallel postulate]] from the other axioms of [[Euclidean geometry]].
* 1870&nbsp;– [[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&nbsp;– [[Richard Dedekind]] invents what is now called the Dedekind Cut for defining irrational numbers, and now used for defining surreal numbers.
* 1873&nbsp;– [[Charles Hermite]] proves that [[e (mathematical constant)|e]] is [[Transcendental number|transcendental]].
* 1873&nbsp;– [[Georg Frobenius]] presents his method for finding series solutions to linear differential equations with [[regular singular point]]s.
* 1874&nbsp;– [[Georg Cantor]] proves that the set of all [[real number]]s is [[uncountable|uncountably infinite]] but the set of all real [[algebraic number]]s is [[countable|countably infinite]]. [[Cantor's first uncountability proof|His proof]] does not use his [[Cantor's diagonal argument|diagonal argument]], which he published in 1891.
* 1882&nbsp;– [[Ferdinand von Lindemann]] proves that π is transcendental and that therefore the circle cannot be squared with a compass and straightedge.
* 1882&nbsp;– Felix Klein invents the [[Klein bottle]].
* 1895&nbsp;– [[Diederik Korteweg]] and [[Gustav de Vries]] derive the [[Korteweg–de Vries equation]] to describe the development of long solitary water waves in a canal of rectangular cross section.
* 1895&nbsp;– Georg Cantor publishes a book about set theory containing the arithmetic of infinite [[cardinal number]]s and the [[continuum hypothesis]].
* 1895&nbsp;– [[Henri Poincaré]] publishes paper "[[Analysis Situs (paper)|Analysis Situs]]" which started modern topology.
* 1896&nbsp;– [[Jacques Hadamard]] and [[Charles Jean de la Vallée-Poussin]] independently prove the [[prime number theorem]].
* 1896&nbsp;– [[Hermann Minkowski]] presents ''Geometry of numbers''.
* 1899&nbsp;– Georg Cantor discovers a contradiction in his set theory.
* 1899&nbsp;– [[David Hilbert]] presents a set of self-consistent geometric axioms in ''Foundations of Geometry''.
* 1900&nbsp;– David Hilbert states his [[Hilbert's problems|list of 23 problems]], which show where some further mathematical work is needed.

===Contemporary===

====20th century====
<ref>Paul Benacerraf and Hilary Putnam, Cambridge University Press, ''Philosophy of Mathematics: Selected Readings, {{isbn|0-521-29648-X}}''</ref>
* 1901&nbsp;– [[Élie Cartan]] develops the [[exterior derivative]].
* 1901&nbsp;– [[Henri Lebesgue]] publishes on [[Lebesgue integration]].
* 1903&nbsp;– [[Edmund Georg Hermann Landau]] gives considerably simpler proof of the prime number theorem.
* 1908&nbsp;– [[Ernst Zermelo]] axiomizes [[set theory]], thus avoiding Cantor's contradictions.
* 1908&nbsp;– [[Josip Plemelj]] solves the Riemann problem about the existence of a differential equation with a given [[monodromic group]] and uses Sokhotsky – Plemelj formulae.
* 1912&nbsp;– [[Luitzen Egbertus Jan Brouwer]] presents the [[Brouwer fixed-point theorem]].
* 1912&nbsp;– Josip Plemelj publishes simplified proof for the Fermat's Last Theorem for exponent ''n'' = 5.
* 1915&nbsp;– [[Emmy Noether]] proves [[Noether's theorem|her symmetry theorem]], which shows that every [[symmetry in physics]] has a corresponding [[conservation law]].
* 1916&nbsp;– [[Srinivasa Ramanujan]] introduces [[Ramanujan conjecture]]. This conjecture is later generalized by [[Hans Petersson]].
* 1919&nbsp;– [[Viggo Brun]] defines [[Brun's constant]] ''B''2 for [[twin prime]]s.
* 1921&nbsp;– Emmy Noether introduces the first general definition of a [[commutative ring]]. 
* 1928&nbsp;– [[John von Neumann]] begins devising the principles of [[game theory]] and proves the [[minimax theorem]].
* 1929&nbsp;– Emmy Noether introduces the first general representation theory of groups and algebras.
* 1930&nbsp;– [[Casimir Kuratowski]] shows that the [[three-cottage problem]] has no solution.
* 1931&nbsp;– [[Kurt Gödel]] proves [[Gödel's incompleteness theorem|his incompleteness theorem]], which shows that every axiomatic system for mathematics is either incomplete or inconsistent.
* 1931&nbsp;– [[Georges de Rham]] develops theorems in [[cohomology]] and [[characteristic class]]es.
* 1933&nbsp;– [[Karol Borsuk]] and [[Stanislaw Ulam]] present the [[Borsuk–Ulam theorem|Borsuk–Ulam antipodal-point theorem]].
* 1933&nbsp;– [[Andrey Nikolaevich Kolmogorov]] publishes his book ''Basic notions of the calculus of probability'' (''Grundbegriffe der Wahrscheinlichkeitsrechnung''), which contains an [[probability axiom|axiomatization of probability]] based on [[measure theory]].
* 1936&nbsp;– [[Alonzo Church]] and [[Alan Turing]] create, respectively, the [[Lambda calculus|λ-calculus]] and the [[Turing machine]], formalizing the notion of computation and computability.
* 1938&nbsp;– [[Tadeusz Banachiewicz]] introduces [[LU decomposition]].
* 1940&nbsp;– Kurt Gödel shows that neither the [[continuum hypothesis]] nor the [[axiom of choice]] can be disproven from the standard axioms of set theory.
* 1942&nbsp;– [[G.C. Danielson]] and [[Cornelius Lanczos]] develop a [[fast Fourier transform]] algorithm.
* 1943&nbsp;– [[Kenneth Levenberg]] proposes a method for nonlinear least squares fitting.
* 1945&nbsp;– [[Stephen Cole Kleene]] introduces [[realizability]].
* 1945&nbsp;– [[Saunders Mac Lane]] and [[Samuel Eilenberg]] start [[category theory]].
* 1945&nbsp;– [[Norman Steenrod]] and [[Samuel Eilenberg]] give the [[Eilenberg–Steenrod axioms]] for (co-)homology.
* 1946&nbsp;– [[Jean Leray]] introduces the [[Spectral sequence]].
* 1947&nbsp;– [[George Dantzig]] publishes the [[simplex method]] for linear programming.
* 1948&nbsp;– John von Neumann mathematically studies [[Self-replicating machine|self-reproducing machines]].
* 1948&nbsp;– [[Atle Selberg]] and [[Paul Erdős]] prove independently in an elementary way the [[prime number theorem]].
* 1949&nbsp;– [[John Wrench]] and L.R. Smith compute π to 2,037 decimal places using [[ENIAC]].
* 1949&nbsp;– [[Claude Shannon]] develops notion of [[Information Theory]].
* 1950&nbsp;– [[Stanisław Ulam]] and John von Neumann present [[cellular automata]] dynamical systems.
* 1953&nbsp;– [[Nicholas Metropolis]] introduces the idea of thermodynamic [[simulated annealing]] algorithms.
* 1955&nbsp;– [[H. S. M. Coxeter]] et al. publish the complete list of [[uniform polyhedron]].
* 1955&nbsp;– [[Enrico Fermi]], [[John Pasta]], Stanisław Ulam, and [[Mary Tsingou]] numerically study a nonlinear spring model of heat conduction and discover solitary wave type behavior.
* 1956&nbsp;– [[Noam Chomsky]] describes a [[Chomsky hierarchy|hierarchy]] of [[formal language]]s.
* 1956&nbsp;– [[John Milnor]] discovers the existence of an [[Exotic sphere]] in seven dimensions, inaugurating the field of [[differential topology]]. 
* 1957&nbsp;– [[Kiyosi Itô]] develops [[Itô calculus]].
* 1957&nbsp;– [[Stephen Smale]] provides the [[existence proof]] for crease-free [[sphere eversion]].
* 1958&nbsp;– [[Alexander Grothendieck]]'s proof of the [[Grothendieck–Riemann–Roch theorem]] is published.
* 1959&nbsp;– [[Kenkichi Iwasawa]] creates [[Iwasawa theory]].
* 1960&nbsp;– [[Tony Hoare]] invents the [[quicksort]] algorithm.
* 1960&nbsp;– [[Irving S. Reed]] and [[Gustave Solomon]] present the [[Reed–Solomon code|Reed–Solomon error-correcting code]].
* 1961&nbsp;– [[Daniel Shanks]] and [[John Wrench]] compute π to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer.
* 1961&nbsp;– [[John G. F. Francis]] and [[Vera Kublanovskaya]] independently develop the [[QR algorithm]] to calculate the [[eigenvalues]] and [[eigenvectors]] of a matrix.
* 1961&nbsp;– Stephen Smale proves the [[Poincaré conjecture]] for all dimensions greater than or equal to 5.
* 1962&nbsp;– [[Donald Marquardt]] proposes the [[Levenberg–Marquardt nonlinear least squares fitting algorithm]].
* 1963&nbsp;– [[Paul Cohen (mathematician)|Paul Cohen]] uses his technique of [[forcing (mathematics)|forcing]] to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory.
* 1963&nbsp;– [[Martin Kruskal]] and [[Norman Zabusky]] analytically study the [[Fermi–Pasta–Ulam–Tsingou problem|Fermi–Pasta–Ulam–Tsingou heat conduction problem]] in the [[continuum limit]] and find that the [[KdV equation]] governs this system.
* 1963&nbsp;– meteorologist and mathematician [[Edward Norton Lorenz]] published solutions for a simplified mathematical model of atmospheric turbulence – generally known as chaotic behaviour and [[strange attractor]]s or [[Lorenz Attractor]] – also the [[Butterfly Effect]].
* 1965&nbsp;– Iranian mathematician [[Lotfi Asker Zadeh]] founded [[fuzzy set]] theory as an extension of the classical notion of [[Set (mathematics)|set]] and he founded the field of [[Fuzzy mathematics]].
* 1965&nbsp;– Martin Kruskal and Norman Zabusky numerically study colliding [[Soliton|solitary waves]] in [[Plasma (physics)|plasmas]] and find that they do not disperse after collisions.
* 1965&nbsp;– [[James Cooley]] and [[John Tukey]] present an influential fast Fourier transform algorithm.
* 1966&nbsp;– [[E. J. Putzer]] presents two methods for computing the [[Matrix exponential|exponential of a matrix]] in terms of a polynomial in that matrix.
* 1966&nbsp;– [[Abraham Robinson]] presents [[non-standard analysis]].
* 1967&nbsp;– [[Robert Langlands]] formulates the influential [[Langlands program]] of conjectures relating number theory and representation theory.
* 1968&nbsp;– [[Michael Atiyah]] and [[Isadore Singer]] prove the [[Atiyah–Singer index theorem]] about the index of [[elliptic operator]]s.
* 1973&nbsp;– [[Lotfi Zadeh]] founded the field of [[fuzzy logic]].
* 1974&nbsp;– [[Pierre Deligne]] solves the last and deepest of the [[Weil conjectures]], completing the program of Grothendieck.
* 1975&nbsp;– [[Benoit Mandelbrot]] publishes ''Les objets fractals, forme, hasard et dimension''.
* 1976&nbsp;– [[Kenneth Appel]] and [[Wolfgang Haken]] use a computer to prove the [[Four color theorem]].
* 1981&nbsp;– [[Richard Feynman]] gives an influential talk "Simulating Physics with Computers" (in 1980 [[Yuri Manin]] proposed the same idea about quantum computations in "Computable and Uncomputable" (in Russian)).
* 1983&nbsp;– [[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.
* 1984&nbsp;– [[Vaughan Jones]] discovers the [[Jones polynomial]] in knot theory, which leads to other new knot polynomials as well as connections between knot theory and other fields.
* 1985&nbsp;– [[Louis de Branges de Bourcia]] proves the [[Bieberbach conjecture]].
* 1986&nbsp;– [[Ken Ribet]] proves [[Ribet's theorem]].
* 1987&nbsp;– [[Yasumasa Kanada]], [[David H. Bailey (mathematician)|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&nbsp;– [[Alain Connes]] and [[John Lott (mathematician)|John W. Lott]] develop [[non-commutative geometry]].
* 1992&nbsp;– [[David Deutsch]] and [[Richard Jozsa]] develop the [[Deutsch–Jozsa algorithm]], one of the first examples of a [[quantum algorithm]] that is exponentially faster than any possible deterministic classical algorithm.
* 1994&nbsp;– [[Andrew Wiles]] proves part of the [[Taniyama–Shimura conjecture]] and thereby proves [[Fermat's Last Theorem]].
* 1994&nbsp;– [[Peter Shor]] formulates [[Shor's algorithm]], a [[quantum algorithm]] for [[integer factorization]].
* 1995&nbsp;– [[Simon Plouffe]] discovers [[Bailey–Borwein–Plouffe formula]] capable of finding the ''n''th binary digit of π.
* 1998&nbsp;– [[Thomas Callister Hales]] (almost certainly) proves the [[Kepler conjecture]].
* 1999&nbsp;– the full [[Taniyama–Shimura conjecture]] is proven.
* 2000&nbsp;– the [[Clay Mathematics Institute]] proposes the seven [[Millennium Prize Problems]] of unsolved important classic mathematical questions.

====21st century====
* 2002&nbsp;– [[Manindra Agrawal]], [[Nitin Saxena]], and [[Neeraj Kayal]] of [[IIT Kanpur]] present an unconditional deterministic [[polynomial time]] algorithm to determine whether a given number is [[prime number|prime]] (the [[AKS primality test]]).
* 2002&nbsp;– [[Preda Mihăilescu]] proves [[Catalan's conjecture]].
* 2003&nbsp;– [[Grigori Perelman]] proves the [[Poincaré conjecture]].
* 2004&nbsp;– the [[classification of finite simple groups]], a collaborative work involving some hundred mathematicians and spanning fifty years, is completed.
* 2004&nbsp;– [[Ben Green (mathematician)|Ben Green]] and [[Terence Tao]] prove the [[Green–Tao theorem]].
* 2009&nbsp;– [[Fundamental lemma (Langlands program)]] is [[Mathematical proof|proved]] by [[Ngô Bảo Châu]].<ref>{{citation|first1=G.|last1=Laumon|first2=B. C.|last2=Ngô|arxiv=math/0404454|year=2004|title=Le lemme fondamental pour les groupes unitaires|bibcode=2004math......4454L}}</ref>
* 2010&nbsp;– [[Larry Guth]] and [[Nets Hawk Katz]] solve the [[Erdős distinct distances problem]].
* 2013&nbsp;– [[Yitang Zhang]] proves the first finite bound on gaps between prime numbers.<ref>{{cite web|title=UNH Mathematician's Proof Is Breakthrough Toward Centuries-Old Problem|url=http://www.unh.edu/news/releases/2013/may/bp16zhang.cfm|publisher=[[University of New Hampshire]]|accessdate=May 20, 2013|date=May 1, 2013}}</ref>
* 2014&nbsp;– Project Flyspeck<ref>[https://code.google.com/p/flyspeck/wiki/AnnouncingCompletion Announcement of Completion.] Project Flyspeck, [[Google Code]].</ref> announces that it completed a proof of [[Kepler conjecture|Kepler's conjecture]].<ref>[http://phys.org/news/2014-08-team-formal-computer-verified-proof-kepler.html Team announces construction of a formal computer-verified proof of the Kepler conjecture.] August 13, 2014 by Bob Yirk.</ref><ref>[https://www.newscientist.com/article/dn26041-proof-confirmed-of-400-year-old-fruit-stacking-problem/ Proof confirmed of 400-year-old fruit-stacking problem], 12 August 2014; [[New Scientist]].
</ref><ref>[https://arxiv.org/abs/1501.02155 A formal proof of the Kepler conjecture], [[arXiv]].</ref><ref>[http://pnews.sky.com/story/1317501/solved-400-year-old-maths-theory-finally-proven Solved: 400-Year-Old Maths Theory Finally Proven.] [[Sky News]], 16:39, UK, Tuesday 12 August 2014.</ref>
* 2015&nbsp;– [[Terence Tao]] solves the [[Paul Erdős|Erdős]] [[Sign sequence#Erdős discrepancy problem|discrepancy problem]].
* 2015&nbsp;– [[László Babai]] finds that a quasipolynomial complexity algorithm would solve the [[Graph isomorphism problem]].
* 2016&nbsp;– [[Maryna Viazovska]] solves the [[sphere packing]] problem in dimension 8. Subsequent work building on this leads to a solution for dimension 24.

==See also==
{{portal|Mathematics}}
* [[History of mathematical notation]] explains Rhetorical, Syncopated and Symbolic
* {{annotated link|Timeline of ancient Greek mathematicians}}
* [[Timeline of mathematical innovation in South and West Asia]]
* {{annotated link|Timeline of mathematical logic}}
* [[Timeline of women in mathematics]]
* [[Timeline of women in mathematics in the United States]]

==References==
{{reflist}}
* David Eugene Smith, 1929 and 1959, ''A Source Book in Mathematics'', [[Dover Publications]]. {{isbn|0-486-64690-4}}.

==External links==
* {{MacTutor Biography|class=Chronology|id=full|title=A Mathematical Chronology}}

[[Category:Mathematics timelines| ]]
[[Category:History of mathematics| ]]

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Greek: Ά ά Έ έ Ή ή Ί ί Ό ό Ύ ύ Ώ ώ Α α Β β Γ γ Δ δ Ε ε Ζ ζ Η η Θ θ Ι ι Κ κ Λ λ Μ μ Ν ν Ξ ξ Ο ο Π π Ρ ρ Σ σ ς Τ τ Υ υ Φ φ Χ χ Ψ ψ Ω ω {{Polytonic|}}
Cyrillic: А а Б б В в Г г Ґ ґ Ѓ ѓ Д д Ђ ђ Е е Ё ё Є є Ж ж З з Ѕ ѕ И и І і Ї ї Й й Ј ј К к Ќ ќ Л л Љ љ М м Н н Њ њ О о П п Р р С с Т т Ћ ћ У у Ў ў Ф ф Х х Ц ц Ч ч Џ џ Ш ш Щ щ Ъ ъ Ы ы Ь ь Э э Ю ю Я я ́
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