# Timeline of mathematics

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Timeline of mathematics

## 1st millennium BC

• c. 1000 BC — Vulgar fractions 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.[6]
• first half of 1st millennium BC — Vedic India — Yajnavalkya, in his 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.
• c. 8th century BC — the Yajur Veda, one of the four Hindu Vedas, contains the earliest concept of infinity, and states that “if you remove a part from infinity or add a part to infinity, still what remains is infinity.”
• 800 BC — 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.
• early 6th century BC — Thales of Miletus has various theorems attributed to him.
• c. 600 BC — 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 BC — The Lo Shu Square, the unique normal magic square of order three, was discovered in China.
• 530 BC — Pythagoras studies propositional geometry and vibrating lyre strings; his group also discovers the irrationality of the square root of two.
• c. 500 BC — 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.
• 5th century BC — Hippocrates of Chios utilizes lunes in an attempt to square the circle.
• 5th century BC — 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.
• c. 400 BC — Jaina mathematicians in India write the “Surya Prajinapti”, a mathematical text which classifies 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.
• 4th century BC — Indian texts use the Sanskrit word “Shunya” to refer to the concept of ‘void’ (zero).
• 370 BC — Eudoxus states the method of exhaustion for area determination.
• 350 BC — Aristotle discusses logical reasoning in Organon.
• 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 (the first positional base 10 numeral system) are conceived in India.
• 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.
• 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.
• 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.
• 140 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 millennium AD

• 1st century — Heron of Alexandria, the earliest fleeting reference to square roots of negative numbers.
• c. 3rd century — Ptolemy of Alexandria wrote the Almagest
• 250 — Diophantus uses symbols for unknown numbers in terms of syncopated algebra, and writes Arithmetica, one of the earliest treatises on algebra
• 300 — the earliest known use of zero as a decimal digit is introduced by Indian mathematicians
• c. 340 — Pappus of Alexandria states his hexagon theorem and his centroid theorem
• 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,
• 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)
• 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
• 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
• 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 also 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 Fazaii 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.
• 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 had begun to understand what we would write in symbols as $x^n \cdot x^m = x^{m+n}$
• 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 was first to define the monomials x, x2, x3, … and 1 / x, 1 / x2, 1 / x3, … and to give rules for 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 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. Derived the formulae: $\sin \alpha = \tan \alpha / \sqrt{1+\tan^2 \alpha}$ and $\cos \alpha = 1 / \sqrt{1 + \tan^2 \alpha}$.

## 1000–1500

• 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 — Al-Karaji writes a book containing the first known proofs by mathematical induction. He used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.[7] He was “the first who introduced the theory of algebraic calculus.”[8]
• 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 this famous formula: sin (α + β) = sin α cos β + sin β cos α. Also discussed the quadrature of the parabola and the volume of the paraboloid.
• 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.[9]
• 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.”[10]
• 1135 — Sharafeddin Tusi followed al-Khayyam's application of algebra to geometry, and wrote a treatise on cubic equations which “represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.”[10]
• 1202 — Leonardo Fibonacci demonstrates the utility of Hindu-Arabic numerals in his Book of the Abacus.
• 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 which have also been joint attributed to Fermat as well as Thabit ibn Qurra.[11]
• 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 p 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 of Sangamagrama|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 Ruffini and 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.[10]
• 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.
• 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.

## 16th century

• 1501 — Nilakantha Somayaji writes the “Tantra Samgraha”, which lays the foundation for a complete system of fluxions (derivatives), and expands on concepts from his previous text, the “Aryabhatiya Bhasya”.
• 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 — Niccolo 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”.
• 1550 — Jyeshtadeva, a Kerala school mathematician, writes the “Yuktibhasa”, the world's first calculus text, which gives detailed derivations of many calculus theorems and formulae.
• 1572 — Rafael Bombelli writes "Algebra" teatrise and uses imaginary numbers to solve cubic equations.
• 1596 — Ludolf van Ceulen computes π to twenty decimal places using inscribed and circumscribed polygons.

[12]

## Notes

1. This article is based on a timeline developed by Niel Brandt (1994) who has given permission for its use in Wikipedia. (See Talk:Timeline of mathematics.)
2. In 1966 IBM printed a famous timeline poster called Men of Modern Mathematics for the years 1000 AD to 1950 AD. It was based on personal stories about (mainly Western) mathematicians and their mathematical achievements. The poster was designed by the famous Charles Eames, with the content concerning mathematicians contributed by Professor Raymond Redheffer of UCLA.

## References

1. ^ Art Prehistory, Sean Henahan, January 10, 2002.
2. ^ How Menstruation Created Mathematics, Tacoma Community College, archive link
3. ^ OLDEST Mathematical Object is in Swaziland
4. ^ an old Mathematical Object
5. ^ a b Egyptian Mathematical Papyri - Mathematicians of the African Diaspora
6. ^ Carl B. Boyer, A History of Mathematics, 2nd Ed.
7. ^ Victor J. Katz (1998). History of Mathematics: An Introduction, p. 255–259. Addison-Wesley. ISBN 0321016181.
8. ^ F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris.
9. ^ O'Connor, John J.; Robertson, Edmund F., "Abu l'Hasan Ali ibn Ahmad Al-Nasawi", MacTutor History of Mathematics archive, University of St Andrews .
10. ^ a b c Arabic mathematics, MacTutor History of Mathematics archive, University of St Andrews, Scotland
11. ^ a b Various AP Lists and Statistics
12. ^ Paul Benacerraf and Hilary Putnam, Cambridge U.P., Philosophy of Mathematics: Selected Readings, ISBN 0-521-29648-X
13. ^ Elizabeth A. Thompson, MIT News Office, Math research team maps E8 Mathematicians Map E8, Harminka, 2007-03-20
14. ^ Laumon, G.; Ngô, B. C. (2004), Le lemme fondamental pour les groupes unitaires, arXiv:math/0404454

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