# Mathematics timeline

This timeline is an adaptation of one presented on Wikipedia several years ago. For a more up-to-date version (without the historical commentary) see the current **Wikipedia mathematics timeline**.

The first records of mathematical activity date back to early Egypt and Mesopotamia also India and China. But maths underwent a major transition in ancient Greece where proofs were first demonstrated and geometry took precedence over arithmetic. During the European Dark Age further mathematical progress was made by Arab mathematicians during an Islamic Golden Age when translations of Greeks mathematics became available. Only with the Renaissance was there a revival of mathematics in Europe that culminated in the 17th century as a Heroic Age of mathematics. Being a symbolic ‘language’ mathematics has always involved the development of the most efficient systems of notation.

#### Mesopotamia, Egypt, India, and China

Evidence for mathematical principles and procedures appear in the early civilizations of Mesopotamia, Egypt, the Indus Valley and China. All had arithmetic and geometry, the Egyptians and Babylonians using fractions by about 2000 BCE. China had decimal fractions about 1500 years before Europe. Babylonians were aware of Pythagoras’s theorem before Pythagoras while Babylonians used a sexagesimal (base 60) for numbers and fractions which persisted into the 17th century CE. Indo-Arabic numerals began in India around 3rd century BCE. But none used formal proofs or demonstrations of logical underpinnings. Related to commerce, surveying and building but also recreational puzzles.

#### Ancient Greece

Deductive proofs began with the ancient Greeks, the first attributed to Thales of Miletus beginning with agreed axioms and definitions connected by logic. Greeks worked mostly with geometry with little number theory and a little trigonometry as an extension of geometry used in astronomy – even how the Earth moved round the Sun. This was done by Euclid (4th BCE) studied by all subsequent major mathematicians and the Pythagoreans (6th BCE) who discovered irrational numbers. Archimedes (3rd BCE) excelled at geometry and number theory and the preliminary ideas relating to future calculus as one of the greatest mathematicians of all time. Appolonius (2nd BCE) carried out foundational studies in conic sections (ellipse, parabola etc.) so important later in astronomy.

#### Islamic Golden Age

400-1200 CE was a European Dark Age with mathematical work being done by Islamic scholars after the establishment of Islam by Mohammed in the 7th century CE building on ancient texts and used Hindu/Arabic (Indian) numerals and decimal fractions c. 800-1200 CE. The word ‘Algebra’ is derived from the Arabic ‘Al-Jabr’ in geometric form c. 825 cubic equations by Omar Khayam who was not surpassed in Europe until 16th century.

#### European Renaissance & Scientific Revolution

Around 1000 CE Hindu/Arabic numerals are first used in Europe, the Fibonnacci series produced the Liber Abaci in 1200s ittook300years to get these numerals accepted. In 1450 universities could cope with addition and subtraction but multiplication and division was only available in Italy (Europeans were still using cumbersome Roman numerals).

Only in 1585 did Belgian Simon Stevin advocate base 10 (decimal) fractions although his notation was clumsy and ignored until the 1620s. Scotsman John Napier invented logarithms in 1614 noting that you could convert multiplication into addition (products to sums), quotients into differences etc. but his base was clumsy so Briggs in 1624 developed base 10 logs. Today’s symbolic algebra (that of today) was launched in Europe in 16th century, moving from rhetorical to syncopated algebra as mathematics in general was gathering momentum. Geometry and algebra now become united with the modern-day coordinate system established by Frenchman Fermat and Descartes (method) in the 1630s. Could find the solutions to algebraic equations by graphing them.

#### The 17th century European Heroic Century for mathematics

The 17th century in Europe is known as the Heroic Century for mathematics as in 1600 it passes from an emphasis on geometry to calculus, as formal analysis in 1700 marking the transition from ancient to modern mathematics – a major transition. In 1650 there was a foundation of Greek geometry, symbolic algebra, Hindu/Arabic numerals, logarithms and the combination of algebra and geometry to create coordinate systems and graphs. This set the scene for the birth of calculus via people like Kepler, Fermat, Borrow at Cambridge and others. The two towering figures of Newton and Liebniz (the notation we use today) developed their own versions, Liebniz publishing in 1684. From now on advances were given momentum by the application of calculus to physics, astronomy, and engineering using the calculus tools of functions, curves, series, derivatives, and integrals. The work of Bernoulli was passed on to people like Leonard Euler (1707-1783) the greatest mathematician of the 18th century.

#### The 19th century challenge to foundations

By the 19th century the foundations of mathematics were under challenge. Euclidian geometry’s parallel postulate led to three angles adding up to 180. Gauss when 15 was convinced it couldn’t be done and in hyperbolic space it was less than 180 used by Einstein and Riemann’s elliptic geometry ‘fat’ triangles. By 1870 the hyperbolic geometry was clearly accepted and Euclid equalled. Gauss (differential geometry and number theory) was one of the greatest mathematicians of the 19th century with contemporaries Lobachevsky (1829) and Bolyai (1832) adding to the work of the day. Fourier worked with trigonometric series and now the foundations of calculus began to look shaky. They depended on limits which depended on real numbers (as yet undefined), reals were based on rationals, rationals on integers, integers on natural numbers. By 1900 the arithmetization of analysis was in place (Gauss, Cauchy, Riemann, Weierstrass, Dedekind, Peano) establishing the foundations of analysis. Cantor developed the idea of ‘the set of all the natural numbers’ that some infinities are bigger than others.

#### 20th century global mathematics

In the last 100 years much more mathematics has been created than in the last 5000, much of it extremely difficult. Of note are the incompleteness theorems of Kurt Gödel (1906-1978) challenged the foundations of mathematics in 1931. That the formal axiomatic system of natural numbers is impossible to prove within that system, that the system is complete. It is also impossible to prove (within the system) the statement that ‘this system is consistent’. American Claude Shannon (1916-2001) in 1948 founded information theory, transmission of information (words, images, sounds) using 1s and 0s.

### BCE

#### Mesopotamia, Egypt, India, and China

**c. 3400** – first numeral system, and a system of weights and measures. Sumeria

**c. 3100** – earliest decimal system allows indefinite counting by way of introducing new symbols Egypt

**c. 2800** – earliest use of decimal ratios in a uniform system of ancient weights and measures, the smallest unit of measurement used is 1.704 millimetres and the smallest unit of mass used is 28 gramsIndus Valley Civilization

**2700 **– precision surveying Egypt

**2400** – precise astronomical calendar used into the Middle Ages Egypt

**c. 2000** – Sexagesimal 60-base numeral system. First known computation of π to 3.125.Babylon

**c. 2000** – carved stone balls with all of the symmetries of Platonic solids Scotland

**1800** – findings volume of a frustum Egypt, Moscow Mathematical Papyrus

**c. 1800** – Berlin Papyrus 6619 (Egypt, 19th dynasty) contains a quadratic equation and its solution

**1650** – 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

**1046 to 256** – arithmetic and geometric algorithms and proofs China, attributed to Zhoubi Suanjing

**c. 1000** – simple fractions but 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

**c.500** – Yajnavalkya, in *Shatapatha Brahmana* describes the motions of the sun and the moon, and advances a 95-year cycle to synchronize the motions of the sun and the moon Vedic India

**c. 8th century** – Yajur Veda (one of four Hindu Vedas) has mathematical reference to infinity ‘if you remove a part from infinity or add a part to infinity, still what remains is infinity’

**800** – Baudhayana, author of the Baudhayana Sulba Sutra, a Vedic Sanskrit geometric text, contains quadratic equations, and calculates the square root of two correctly to five decimal places

**624 – 546** – Thales of Miletus has various theorems attributed to him

**c. 600** – the other Vedic “Sulba Sutras” (“rule of chords” in Sanskrit) use Pythagorean triples, contain of a number of geometrical proofs, and approximate π at 3.16.

**second half of 1st millennium**– The Lo Shu Square, the unique normal magic square of order three, was discovered in China

**530**– Pythagoras studies propositional geometry and vibrating lyre strings; his group also discovers the irrationality of the square root of two.

**c. 500** – Indian grammarian Pānini writes the Astadhyayi, which contains the use of metarules, transformations and recursions, originally for the purpose of systematizing the grammar of Sanskrit.

**470 BC – 410** – Hippocrates of Chios utilizes lunes in an attempt to square the circle.

**5th century** – 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.

**490 BC – 430** Zeno of Elea Zeno’s paradoxes

**5th c. ** Theodorus of Cyrene, Democritus, Hippias, Archytas, Plato, Theaetetus (mathematician)

**c. 400** – Jaina mathematicians in India write the Surya Prajinapti, a mathematical text classifying all numbers into three sets: enumerable, innumerable and 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** Eudoxus of Cnidus, Antiphon, Bryson of Heraclea, Xenocrates

**4th century** –Dinostratus, Autolycus of Pitane, Aristaeus the Elder, Callippus

**370**– Eudoxus states the method of exhaustion for area determination

**350** – Aristotle discusses logical reasoning in Organon

**330**– the earliest known work on Chinese geometry, the Mo Jing, is compiled

**3rd century** – Aristarchus of Samos, Heraclides of Pontus, Menaechmus

**300 BC – Jain mathematicians in India write the Bhagabati Sutra, which contains the earliest information on combinations. **

**300 BC – Euclid in his Elements studies geometry as an axiomatic system, proves the infinitude of prime numbers and presents the Euclidean algorithm; he states the law of reflection in Catoptrics, and he proves the fundamental theorem of arithmetic. **

**c. 300 BC – Brahmi numerals (ancestor of the common modern base 10 numeral system) are conceived in India. **

**370 BC – 300 BC – Eudemus of Rhodes works on histories of arithmetic, geometry and astronomy now lost **

**300 BC – Mesopotamia, the Babylonians invent the earliest calculator, the abacus. **

**c. 300 BC – 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. **

**c. 3rd century Nicomedes (mathematician), Philon of Byzantium, Chrysippus, Conon of Samos, Dionysodorus, Apollonius of Perga, Diocles (mathematician) **

**202 BC to 186 BC – Book on Numbers and Computation, a mathematical treatise, is written in Han Dynasty China. **

**260 BC – 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 – late Olmecs had already begun to use a true zero (a shell glyph) several centuries before Ptolemy in the New World. See 0 (number). **

**240 BC – Eratosthenes uses his sieve algorithm to quickly isolate prime numbers. **

**225 BC – Apollonius of Perga writes On Conic Sections and names the ellipse, parabola, and hyperbola. **

**206 BC to 8 AD – Counting rods are invented in China. **

**c. 2nd century Zenodorus (mathematician), Perseus (geometer), Zeno of Sidon, Hypsicles, Theodosius of Bithynia, Posidonius**

**150 BC – 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. **

**150 BC – A method of Gaussian elimination appears in the Chinese text The Nine Chapters on the Mathematical Art. **

**150 BC – Horner’s method appears in the Chinese text The Nine Chapters on the Mathematical Art. **

**150 BC – Negative numbers appear in the Chinese text The Nine Chapters on the Mathematical Art. **

**190 BC – 120 BC – Hipparchus develops the bases of trigonometry. **

**50 BC – Indian numerals, a descendant of the Brahmi numerals (the first positional notation base-10 numeral system), begins development in India. **

**final centuries BC – 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 Century Geminus, Cleomedes, Heron of Alexandria, (Hero) the earliest fleeting reference to square roots of negative numbers, Theon of Smyrna, Nicomachus, Menelaus of Alexandria Spherical trigonometry**

**50 BC – 23 AD Liu Xin**

**CE**

**1st to 2nd century Zhang Heng, Cai Yong**

**c. 2nd century – Ptolemy of Alexandria wrote the Almagest. **

**3rd century – Sporus of Nicaea, Diophantus uses symbols for unknown numbers in terms of syncopated algebra, and writes Arithmetica, one of the earliest treatises on algebra. **

**263 – Liu Hui computes π using Liu Hui’s π algorithm. **

**300 – the earliest known use of zero as a decimal digit is introduced by Indian mathematicians. **

**3rd to 4th century – Porphyry (philosopher), Serenus of Antinouplis. **

**300 to 500 – the Chinese remainder theorem is developed by Sun Tzu. **

**300 to 500 – a description of rod calculus is written by Sun Tzu. **

**4th to 5th century – Theon of Alexandria, Pappus of Alexandria states his hexagon theorem and his centroid theorem, Hypatia, Proclus, Domninus of Larissa, Marinus of Neapolis, Anthemius of Tralles**

**c. 400 – the Bakhshali manuscript is written by Jaina mathematicians, which describes a theory of the infinite containing different levels of infinity, shows an understanding of 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. **

**450 – Zu Chongzhi computes π to seven decimal places. This calculation remains the most accurate calculation for π for close to a thousand years. **

**500 – 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 earliest tables of sine and cosine values (in 3.75-degree intervals from 0 to 90 degrees). **

**5th to 6th century – Eutocius of Ascalon, 490 – 560 Simplicius of Cilicia.**

**6th century – 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. **

**6th century – Yativṛṣabha, Varāhamihira.**

**535 – 566 Zhen Luan.**

**550 – Hindu mathematicians give zero a numeral representation in the positional notation Indian numeral system. **

**7th century – Bhaskara I gives a rational approximation of the sine function.**

**7th century – 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 the 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 and quadratic equations, and rules for summing series, Brahmagupta’s identity, and the Brahmagupta theorem.**

**602 – 670 Li Chunfeng**

**Islamic Golden Age**

**8th century – Virasena gives explicit rules for the Fibonacci sequence, gives the derivation of the volume of a frustum using an infinite procedure, and also deals with the logarithm to base 2 and knows its laws. **

**8th century – Shridhara gives the rule for finding the volume of a sphere and the formula for solving quadratic equations. **

**773 – Kanka brings Brahmagupta’s Brahma-sphuta-siddhanta to Baghdad to explain the Indian system of arithmetic astronomy and the Indian numeral system. **

**773 – Al-Fazari translates the Brahma-sphuta-siddhanta into Arabic upon the request of King Khalif Abbasid Al Mansoor. **

**9th century – Govindsvamin discovers the Newton-Gauss interpolation formula, and gives the fractional parts of Aryabhata’s tabular sines. **

**810 – The House of Wisdom is built in Baghdad for the translation of Greek and Sanskrit mathematical works into Arabic. **

**820 – Al-Khwarizmi – Persian mathematician, father of algebra, writes the 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 Hindu-Arabic decimal number system to the Western world in the 12th century. The term algorithm is also named after him. **

**820 – Al-Mahani conceived the idea of reducing geometrical problems such as doubling the cube to problems in algebra. **

**c. 850 – Al-Kindi pioneers cryptanalysis and frequency analysis in his book on cryptography. **

**c. 850 – Mahāvīra writes the Gaṇitasārasan̄graha otherwise known as the Ganita Sara Samgraha which gives systematic rules for expressing a fraction as the sum of unit fractions. **

**895 – Thabit ibn Qurra: the only surviving fragment of his original work contains a chapter on the solution and properties of cubic equations. He also generalized the Pythagorean theorem, and discovered the theorem by which pairs of amicable numbers can be found, (i.e., two numbers such that each is the sum of the proper divisors of the other). **

**c. 900 – Abu Kamil of Egypt devises new notation**

**940 – Abu’l-Wafa al-Buzjani extracts roots using the Indian numeral system. **

**953 – 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.” 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 – 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 started a school of algebra which flourished for several hundreds of years”. He also discovered the binomial theorem for integer exponents, which “was a major factor in the development of numerical analysis based on the decimal system”.**

**975 – Al-Batani extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tangent, secant and their inverse functions.**

**c. 1000 – Abū Sahl al-Qūhī (Kuhi) solves equations higher than the second degree. **

**c. 1000 – Abu-Mahmud al-Khujandi first states a special case of Fermat’s Last Theorem. **

**c. 1000 – Law of sines is discovered by Muslim mathematicians, but it is uncertain who discovers it first between Abu-Mahmud al-Khujandi, Abu Nasr Mansur, and Abu al-Wafa. **

**c. 1000 – Pope Sylvester II introduces the abacus using the Hindu-Arabic numeral system to Europe. **

**1000 – . He used it to prove the binomial theorem, Pascal’s triangle, and the sum of integral cubes. He was “the first who introduced the theory of algebraic calculus”.**

**c. 1000 – Ibn Tahir al-Baghdadi studied a slight variant of Thabit ibn Qurra’s theorem on amicable numbers, and he also made improvements on the decimal system. **

**1020 – Abul Wáfa 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 – Ibn al-Haytham formulated and solved Alhazen’s problem geometrically. **

**1030 – Ali Ahmad Nasawi 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. **

**1070 – Omar Khayyám begins to write Treatise on Demonstration of Problems of Algebra and classifies cubic equations. **

**c. 1100 – Omar Khayyám “gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections”. He became the first to find general geometric solutions of cubic equations and laid the foundations for the development of analytic geometry and non-Euclidean geometry. He also extracted roots using the decimal system (Hindu-Arabic numeral system). **

**12th century – Indian numerals have been modified by Arab mathematicians to form the modern Hindu-Arabic numeral system (used universally in the modern world). **

**12th century – the Hindu-Arabic numeral system reaches Europe through the Arabs. **

**12th century – 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 – Bhāskara II (Bhaskara Acharya) writes the Bijaganita (Algebra), which is the first text to recognize that a positive number has two square roots.**

**12th century – Bhaskara Acharya conceives differential calculus, and also develops Rolle’s theorem, Pell’s equation, a proof for the Pythagorean Theorem, proves that division by zero is infinity, computes π to 5 decimal places, and calculates the time taken for the earth to orbit the sun to 9 decimal places. **

**1130 – Al-Samawal 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.” **

**1135 – 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”.**

**1202 – Leonardo Fibonacci demonstrates the utility of Hindu-Arabic numerals in his Liber Abaci (Book of the Abacus). **

**1247 – Qin Jiushao publishes Shùshū Jiǔzhāng (Mathematical Treatise in Nine Sections). **

**1248 – 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 – Al-Farisi gave a new proof of Thabit ibn Qurra’s theorem, introducing important new ideas concerning factorization and combinatorial methods. He also gave the pair of amicable numbers 17296 and 18416 that have also been joint attributed to Fermat as well as Thabit ibn Qurra. **

**c. 1250 – Nasir Al-Din Al-Tusi attempts to develop a form of non-Euclidean geometry. **

**1303 – Zhu Shijie publishes Precious Mirror of the Four Elements, which contains an ancient method of arranging binomial coefficients in a triangle. **

**14th century – Madhava is considered the father of mathematical analysis, who also worked on the power series for π and for sine and cosine functions, and along with other Kerala school mathematicians, founded the important concepts of calculus. **

**14th century – Parameshvara, 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. **

**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 – Ghiyath al-Kashi “contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real 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 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 – Ibn al-Banna and al-Qalasadi introduced symbolic notation for algebra and for mathematics in general**

**15th century – Nilakantha Somayaji, a Kerala school mathematician, writes the Aryabhatiya Bhasya, which contains work on infinite-series expansions, problems of algebra, and spherical geometry. **

**1424 – Ghiyath al-Kashi computes π to sixteen decimal places using inscribed and circumscribed polygons. **

**1427 – 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 – Regiomontanus writes De Triangulis omnimodus which is one of the earliest texts to treat trigonometry as a separate branch of mathematics. **

**1478 – An anonymous author writes the Treviso Arithmetic. **

**1494 – Luca Pacioli writes Summa de arithmetica, geometria, proportioni et proportionalità; introduces primitive symbolic algebra using “co” (cosa) for the unknown. **

**1501 – Nilakantha Somayaji writes the Tantrasamgraha. **

**1520 – Scipione dal 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 – Niccolò 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 the quartic equation. **

**1544 – Michael Stifel publishes Arithmetica integra. **

**1545 – Gerolamo Cardano conceives the idea of complex numbers. **

**1550 – Jyeshtadeva, a Kerala school mathematician, writes the Yuktibhāṣā, the world’s first calculus text, which gives detailed derivations of many calculus theorems and formulae. **

**1572 – Rafael Bombelli writes Algebra treatise and uses imaginary numbers to solve cubic equations. **

**1584 – Zhu Zaiyu calculates equal temperament. **

**1596 – Ludolf van Ceulen computes π to twenty decimal places using inscribed and circumscribed polygons. **

**17th century**

**1614 – John Napier discusses Napierian logarithms in Mirifici Logarithmorum Canonis Descriptio. **

**1617 – Henry Briggs discusses decimal logarithms in Logarithmorum Chilias Prima. **

**1618 – John Napier publishes the first references to e in a work on logarithms. **

**1619 – René Descartes discovers analytic geometry (Pierre de Fermat claimed that he also discovered it independently). **

**1619 – Johannes Kepler discovers two of the Kepler-Poinsot polyhedra. **

**1629 – Pierre de Fermat develops a rudimentary differential calculus. **

**1634 – Gilles de Roberval shows that the area under a cycloid is three times the area of its generating circle. **

**1636 – Muhammad Baqir Yazdi jointly discovered the pair of amicable numbers 9,363,584 and 9,437,056 along with Descartes (1636). **

**1637 – Pierre de Fermat claims to have proven Fermat’s Last Theorem in his copy of Diophantus’ Arithmetica. **

**1637 – First use of the term imaginary number by René Descartes; it was meant to be derogatory. **

**1643 – René Descartes develops Descartes’ theorem. **

**1654 – Blaise Pascal and Pierre de Fermat create the theory of probability. **

**1655 – John Wallis writes Arithmetica Infinitorum. **

**1658 – Christopher Wren shows that the length of a cycloid is four times the diameter of its generating circle. **

**1665 – Isaac Newton works on the fundamental theorem of calculus and develops his version of infinitesimal calculus. **

**1668 – Nicholas Mercator and William Brouncker discover an infinite series for the logarithm while attempting to calculate the area under a hyperbolic segment. **

**1671 – James Gregory develops a series expansion for the inverse-tangent function (originally discovered by Madhava). **

**1671 – James Gregory discovers Taylor’s Theorem. **

**1673 – Gottfried Leibniz also develops his version of infinitesimal calculus. **

**1675 – Isaac Newton invents an algorithm for the computation of functional roots. **

**1680s – Gottfried Leibniz works on symbolic logic. **

**1683 – Seki Takakazu discovers the resultant and determinant. **

**1683 – Seki Takakazu develops elimination theory. **

**1691 – Gottfried Leibniz discovers the technique of separation of variables for ordinary differential equations. **

**1693 – Edmund Halley prepares the first mortality tables statistically relating death rate to age. **

**1696 – Guillaume de L’Hôpital states his rule for the computation of certain limits. **

**1696 – Jakob Bernoulli and Johann Bernoulli solve brachistochrone problem, the first result in the calculus of variations. **

**1699 – Abraham Sharp calculates π to 72 digits but only 71 are correct. **

**18th century**

**1706 – John Machin develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places.
1708 – 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 – Brook Taylor develops Taylor series.
1722 – Abraham de Moivre states de Moivre’s formula connecting trigonometric functions and complex numbers.
1722 – Takebe Kenko introduces Richardson extrapolation.
1724 – Abraham De Moivre studies mortality statistics and the foundation of the theory of annuities in Annuities on Lives.
1730 – James Stirling publishes The Differential Method.
1733 – Giovanni Gerolamo Saccheri studies what geometry would be like if Euclid’s fifth postulate were false.
1733 – Abraham de Moivre introduces the normal distribution to approximate the binomial distribution in probability.
1734 – Leonhard Euler introduces the integrating factor technique for solving first-order ordinary differential equations.
1735 – Leonhard Euler solves the Basel problem, relating an infinite series to π.
1736 – Leonhard Euler solves the problem of the Seven bridges of Königsberg, in effect creating graph theory.
1739 – Leonhard Euler solves the general homogeneous linear ordinary differential equation with constant coefficients.
1742 – Christian Goldbach conjectures that every even number greater than two can be expressed as the sum of two primes, now known as Goldbach’s conjecture.
1748 – Maria Gaetana Agnesi discusses analysis in Instituzioni Analitiche ad Uso della Gioventu Italiana.
1761 – Thomas Bayes proves Bayes’ theorem.
1761 – Johann Heinrich Lambert proves that π is irrational.
1762 – Joseph Louis Lagrange discovers the divergence theorem.
1789 – Jurij Vega improves Machin’s formula and computes π to 140 decimal places, 136 of which were correct.
1794 – Jurij Vega publishes Thesaurus Logarithmorum Completus.
1796 – Carl Friedrich Gauss proves that the regular 17-gon can be constructed using only a compass and straightedge.
1796 – Adrien-Marie Legendre conjectures the prime number theorem.
1797 – Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms.
1799 – Carl Friedrich Gauss proves the fundamental theorem of algebra (every polynomial equation has a solution among the complex numbers).
1799 – Paolo Ruffini partially proves the Abel–Ruffini theorem that quintic or higher equations cannot be solved by a general formula. **

**19th century**

**1801 – Disquisitiones Arithmeticae, Carl Friedrich Gauss’s number theory treatise, is published in Latin.
1805 – Adrien-Marie Legendre introduces the method of least squares for fitting a curve to a given set of observations.
1806 – Louis Poinsot discovers the two remaining Kepler-Poinsot polyhedra.
1806 – Jean-Robert Argand publishes proof of the Fundamental theorem of algebra and the Argand diagram.
1807 – Joseph Fourier announces his discoveries about the trigonometric decomposition of functions.
1811 – Carl Friedrich Gauss discusses the meaning of integrals with complex limits and briefly examines the dependence of such integrals on the chosen path of integration.
1815 – Siméon Denis Poisson carries out integrations along paths in the complex plane.
1817 – 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.
1822 – Augustin-Louis Cauchy presents the Cauchy integral theorem for integration around the boundary of a rectangle in the complex plane.
1822 – Irisawa Shintarō Hiroatsu analyzes Soddy’s hexlet in a Sangaku.
1823 – Sophie Germain’s Theorem is published in the second edition of Adrien-Marie_Legendre’s Essai sur la théorie des nombres
1824 – Niels Henrik Abel partially proves the Abel–Ruffini theorem that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots.
1825 – Augustin-Louis Cauchy presents the Cauchy integral theorem for general integration paths—he assumes the function being integrated has a continuous derivative, and he introduces the theory of residues in complex analysis.
1825 – Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre prove Fermat’s Last Theorem for n = 5.
1825 – André-Marie Ampère discovers Stokes’ theorem.
1828 – George Green proves Green’s theorem.
1829 – János Bolyai, Gauss, and Lobachevsky invent hyperbolic non-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 of algebraic equations, thereby essentially founding group theory and Galois theory.
1832 – Lejeune Dirichlet proves Fermat’s Last Theorem for n = 14.
1835 – Lejeune Dirichlet proves Dirichlet’s theorem about prime numbers in arithmetical progressions.
1837 – 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 – Peter Gustav Lejeune Dirichlet develops Analytic number theory.
1841 – Karl Weierstrass discovers but does not publish the Laurent expansion theorem.
1843 – Pierre-Alphonse Laurent discovers and presents the Laurent expansion theorem.
1843 – William Hamilton discovers the calculus of quaternions and deduces that they are non-commutative.
1847 – George Boole formalizes symbolic logic in The Mathematical Analysis of Logic, defining what is now called Boolean algebra.
1849 – George Gabriel Stokes shows that solitary waves can arise from a combination of periodic waves.
1850 – Victor Alexandre Puiseux distinguishes between poles and branch points and introduces the concept of essential singular points.
1850 – George Gabriel Stokes rediscovers and proves Stokes’ theorem.
1854 – Bernhard Riemann introduces Riemannian geometry.
1854 – Arthur Cayley shows that quaternions can be used to represent rotations in four-dimensional space.
1858 – August Ferdinand Möbius invents the Möbius strip.
1858 – Charles Hermite solves the general quintic equation by means of elliptic and modular functions.
1859 – Bernhard Riemann formulates the Riemann hypothesis, which has strong implications about the distribution of prime numbers.
1870 – Felix Klein constructs an analytic geometry for Lobachevski’s geometry thereby establishing its self-consistency and the logical independence of Euclid’s fifth postulate.
1872 – Richard Dedekind invents what is now called the Dedekind Cut for defining irrational numbers, and now used for defining surreal numbers.
1873 – Charles Hermite proves that e is transcendental.
1873 – Georg Frobenius presents his method for finding series solutions to linear differential equations with regular singular points.
1874 – Georg Cantor proves that the set of all real numbers is uncountably infinite but the set of all real algebraic numbers is countably infinite. His proof does not use his diagonal 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 the Klein bottle.
1895 – 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 – Georg Cantor publishes a book about set theory containing the arithmetic of infinite cardinal numbers and the continuum hypothesis.
1895 – Henri Poincare publishes paper “Analysis Situs” which started modern topology.
1896 – Jacques Hadamard and Charles Jean de la Vallée-Poussin independently prove the prime number theorem.
1896 – Hermann Minkowski presents Geometry of numbers.
1899 – Georg Cantor discovers a contradiction in his set theory.
1899 – David Hilbert presents a set of self-consistent geometric axioms in Foundations of Geometry.
1900 – David Hilbert states his list of 23 problems, which show where some further mathematical work is needed.
20th century
1901 – Élie Cartan develops the exterior derivative.
1901 – Henri Lebesgue publishes on Lebesgue integration.
1903 – Carle David Tolmé Runge presents a fast Fourier transform algorithm
1903 – Edmund Georg Hermann Landau gives considerably simpler proof of the prime number theorem.
1908 – Ernst Zermelo axiomizes set theory, thus avoiding Cantor’s contradictions.
ng>1908 – Josip Plemelj solves the Riemann problem about the existence of a differential equation with a given monodromic group and uses Sokhotsky – Plemelj formulae.
1912 – Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem.
1912 – Josip Plemelj publishes simplified proof for the Fermat’s Last Theorem for exponent n = 5.
1915 – Emmy Noether proves her symmetry theorem, which shows that every symmetry in physics has a corresponding conservation law.
1916 – Srinivasa Ramanujan introduces Ramanujan conjecture. This conjecture is later generalized by Hans Petersson.
1919 – Viggo Brun defines Brun’s constant B2 for twin primes.
1921 – Emmy Noether introduces the first general definition of a commutative ring.
1928 – John von Neumann begins devising the principles of game theory and proves the minimax theorem.
1929 – Emmy Noether introduces the first general representation theory of groups and algebras.
1930 – Casimir Kuratowski shows that the three-cottage problem has no solution.
1930 – Alonzo Church introduces Lambda calculus.
1931 – Kurt Gödel proves his incompleteness theorem, which shows that every axiomatic system for mathematics is either incomplete or inconsistent.
1931 – Georges de Rham develops theorems in cohomology and characteristic classes.
1933 – Karol Borsuk and Stanislaw Ulam present the Borsuk–Ulam antipodal-point theorem.
1933 – Andrey Nikolaevich Kolmogorov publishes his book Basic notions of the calculus of probability (Grundbegriffe der Wahrscheinlichkeitsrechnung), which contains an axiomatization of probability based on measure theory.
1940 – Kurt Gödel shows that neither the continuum hypothesis nor the axiom of choice can be disproven from the standard axioms of set theory.
1942 – G.C. Danielson and Cornelius Lanczos develop a fast Fourier transform algorithm.
1943 – Kenneth Levenberg proposes a method for nonlinear least squares fitting.
1945 – Stephen Cole Kleene introduces realizability.
1945 – Saunders Mac Lane and Samuel Eilenberg start category theory.
1945 – Norman Steenrod and Samuel Eilenberg give the Eilenberg–Steenrod axioms for (co-)homology.
1946 – Jean Leray introduces the Spectral sequence.
1948 – John von Neumann mathematically studies self-reproducing machines.
1948 – Alan Turing introduces LU decomposition.
1949 – John Wrench and L.R. Smith compute π to 2,037 decimal places using ENIAC.
1949 – Claude Shannon develops notion of Information Theory.
1950 – Stanisław Ulam and John von Neumann present cellular automata dynamical systems.
1953 – Nicholas Metropolis introduces the idea of thermodynamic simulated annealing algorithms.
1955 – H. S. M. Coxeter et al. publish the complete list of uniform polyhedron.
1955 – 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 – Noam Chomsky describes an hierarchy of formal languages.
1957 – Kiyosi Itô develops Itô calculus.
1957 – Stephen Smale provides the existence proof for crease-free sphere eversion.
1958 – Alexander Grothendieck’s proof of the Grothendieck–Riemann–Roch theorem is published.
1959 – Kenkichi Iwasawa creates Iwasawa theory.
1960 – C. A. R. Hoare invents the quicksort algorithm.
1960 – Irving S. Reed and Gustave Solomon present the Reed–Solomon error-correcting code.
1961 – Daniel Shanks and John Wrench compute π to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer.
1961 – John G. F. Francis and Vera Kublanovskaya independently develop the QR algorithm to calculate the eigenvalues and eigenvectors of a matrix.
1961 – Stephen Smale proves the Poincaré conjecture for all dimensions greater than or equal to 5.
1962 – Donald Marquardt proposes the Levenberg–Marquardt nonlinear least squares fitting algorithm.
1962 – Gloria Conyers Hewitt becomes the third African American woman to receive a PhD in mathematics.
1963 – Paul Cohen uses his technique of forcing to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory.
1963 – Martin Kruskal and Norman Zabusky analytically study the Fermi–Pasta–Ulam–Tsingou heat conduction problem in the continuum limit and find that the KdV equation governs this system.
1963 – meteorologist and mathematician Edward Norton Lorenz published solutions for a simplified mathematical model of atmospheric turbulence – generally known as chaotic behaviour and strange attractors or Lorenz Attractor – also the Butterfly Effect.
1965 – Iranian mathematician Lotfi Asker Zadeh founded fuzzy set theory as an extension of the classical notion of set and he founded the field of Fuzzy Mathematics.
1965 – Martin Kruskal and Norman Zabusky numerically study colliding solitary waves in plasmas and find that they do not disperse after collisions.
1965 – James Cooley and John Tukey present an influential fast Fourier transform algorithm.
1966 – E. J. Putzer presents two methods for computing the exponential of a matrix in terms of a polynomial in that matrix.
1966 – Abraham Robinson presents non-standard analysis.
1967 – Robert Langlands formulates the influential Langlands program of conjectures relating number theory and representation theory.
1968 – Michael Atiyah and Isadore Singer prove the Atiyah–Singer index theorem about the index of elliptic operators.
1973 – Lotfi Zadeh founded the field of fuzzy logic.
1975 – Benoît Mandelbrot publishes Les objets fractals, forme, hasard et dimension.
1976 – Kenneth Appel and Wolfgang Haken use a computer to prove the Four color theorem.
1978 – Olga Taussky-Todd is awarded the Austrian Cross of Honour for Science and Art, 1st Class, the highest scientific award of the government of Austria.
1981 – 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 – Gerd Faltings proves the Mordell conjecture and thereby shows that there are only finitely many whole number solutions for each exponent of Fermat’s Last Theorem.
1983 – the classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed.
1985 – Louis de Branges de Bourcia proves the Bieberbach conjecture.
1986 – Ken Ribet proves Ribet’s theorem.
1987 – Yasumasa Kanada, David Bailey, Jonathan Borwein, and Peter Borwein use iterative modular equation approximations to elliptic integrals and a NEC SX-2 supercomputer to compute π to 134 million decimal places.
1991 – Alain Connes and John W. Lott develop non-commutative geometry.
1992 – 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 – Andrew Wiles proves part of the Taniyama–Shimura conjecture and thereby proves Fermat’s Last Theorem.
1994 – Peter Shor formulates Shor’s algorithm, a quantum algorithm for integer factorization.
1995 – Simon Plouffe discovers Bailey–Borwein–Plouffe formula capable of finding the nth binary digit of π.
1998 – Thomas Callister Hales (almost certainly) proves the Kepler conjecture.
1999 – the full Taniyama–Shimura conjecture is proven.
2000 – the Clay Mathematics Institute proposes the seven Millennium Prize Problems of unsolved important classic mathematical questions.
21st century
2002 – 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 (the AKS primality test).
2002 – Yasumasa Kanada, Y. Ushiro, Hisayasu Kuroda, Makoto Kudoh and a team of nine more compute π to 1241.1 billion digits using a Hitachi 64-node supercomputer.
2002 – Preda Mihăilescu proves Catalan’s conjecture.
2003 – Grigori Perelman proves the Poincaré conjecture.
2004 – Ben Green and Terence Tao prove the Green-Tao theorem.
2007 – a team of researchers throughout North America and Europe used networks of computers to map E8.
2009 – Fundamental lemma (Langlands program) had been proved by Ngô Bảo Châu.
2010 – Larry Guth and Nets Hawk Katz solve the Erdős distinct distances problem.
2013 – Yitang Zhang proves the first finite bound on gaps between prime numbers.
2014 – Project Flyspeck announces that it completed proof of Kepler’s conjecture.
2014 – Using Alexander Yee’s y-cruncher “houkouonchi” successfully calculated π to 13.3 trillion digits.
2015 – Terence Tao solves The Erdös Discrepancy Problem
2015 – László Babai found that a quasipolynomial complexity algorithm would solve the Graph Isomorphism Problem
2016 – Using Alexander Yee’s y-cruncher Peter Trueb successfully calculated π to 22.4 trillion digits**

#### Glossary

**Analogue** – analogue signals are continuous while digital signals are discrete – soanalog technologies record waveforms while digital technologies convert analog signals into numbers

**Frustum** – in geometry, a frustum is the portion of a solid that lies between one or two parallel planes cutting it. A right frustum is a parallel truncation of a right pyramid

**Integer** – a number that can be written without a fractional component e.g. 4,0, -2010 while 0.75 and 3/2 are not

**Irrational number** – a real number that cannot be written as a simple fraction, like pie. numbers that are not the ratio of two whole numbers

**Natural numbers** – are those used for counting (cardinal numbers) and ordering (ordinal numbers)

**Prime number** – a natural number>1 that has no positive divisors other than 1 and itself. A natural number that is >1 that is not a prime number is called a composite number

**Rational number** – any number that can be expressed as the fraction (quotient) of two integers, a numerator p and a non-zero denominator q. Since q maybe equal to 1 then every integer is a rational number

**Real number ** – a value that represents a quantity along a line

**Whole number** – a number without fractions; an integer

## Commentary

Though archaeological indications of numeracy date back into prehistory, mathematics developed in urban environments as a means of facilitating commerce, surveying, engineering, architecture, and building construction, and where academic interests could be shared by a community of people with like interests. This included a special interest in the motions of other-worldly astronomical objects of the night sky, a possible world of the gods. Though, at times, it was clearly treated as a form of intellectual recreation. These are also the kind of communities for which there is a written historical record.

The relationship between complexity of social organization and political and economic power is reflected in the regions and countries where mathematical innovation took place. Complex societies employed an intelligentsia of bureaucrats and academics and the geography of mathematical achievement has followed the geography of political and economic dominance.

The presence of a strong mathematical community is a measure of intellectual aspiration within that community since mathematics is generally recognized as making high intellectual demands.

Applied mathematics that depends on computational capacity has made major progress as once demanding calculations moved from abacus to slide rule to personal calculators. Analogue computers address continuously changing physical phenomena such as electrical, mechanical, or hydraulic quantities while digital computers represent varying quantities symbolically, as their numerical values change.

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First published on the internet – 1 March 2019

. . . substantive revision 31 August 2020