# Timeline of mathematics

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the 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.

## Syncopated stage

#### 1st millennium AD

• 1st century – Greece, Heron of Alexandria, (Hero) the earliest fleeting reference to square roots of negative numbers.
• c 100 – Greece, Theon of Smyrna
• 60 – 120 – Greece, Nicomachus
• 70 – 140 – Greece, Menelaus of Alexandria Spherical trigonometry
• 78 – 139 – China, Zhang Heng
• c. 2nd century – Greece, Ptolemy of Alexandria wrote the Almagest.
• 132 – 192 – China, Cai Yong
• 240 – 300 – Greece, Sporus of Nicaea
• 250 – Greece, Diophantus uses symbols for unknown numbers in terms of syncopated algebra, and writes Arithmetica, one of the earliest treatises on algebra.
• 263 – China, 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.
• 234 – 305 – Greece, Porphyry (philosopher)
• 300 – 360 – Greece, Serenus of Antinouplis
• 335 – 405– Greece, Theon of Alexandria
• c. 340 – Greece, Pappus of Alexandria states his hexagon theorem and his centroid theorem.
• 350 – 415 – Byzantine Empire, Hypatia
• c. 400 – India, 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.
• 300 to 500 – the Chinese remainder theorem is developed by Sun Tzu.
• 300 to 500 – China, a description of rod calculus is written by Sun Tzu.
• 412 – 485 – Greece, Proclus
• 420 – 480 – Greece, Domninus of Larissa
• b 440 – Greece, Marinus of Neapolis "I wish everything was mathematics."
• 450 – China, Zu Chongzhi computes 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 – 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 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 – 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 Hindu mathematicians give zero a numeral representation in the positional notation 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 – India, Bhaskara I gives a rational approximation of the sine function.
• 7th century – India, Brahmagupta invents the method of solving indeterminate equations of the second degree and is the first to use algebra to solve astronomical problems. He also develops methods for calculations of the motions and places of various planets, their rising and setting, conjunctions, and the calculation of eclipses of the sun and the moon.
• 628 – Brahmagupta writes 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.
• 721 – China, Zhang Sui (Yi Xing) computes the first tangent table.
• 8th century – India, 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 – India, Shridhara gives the rule for finding the volume of a sphere and also the formula for solving quadratic equations.
• 773 – Iraq, 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 – India, 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 – Iran, Al-Mahani conceived the idea of reducing geometrical problems such as doubling the cube to problems in algebra.
• c. 850 – Iraq, Al-Kindi pioneers cryptanalysis and frequency analysis in his book on cryptography.
• c. 850 – India, 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 – Syria, 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 – Egypt, Abu Kamil had begun to understand what we would write in symbols as $x^{n}\cdot x^{m}=x^{m+n}$ • 940 – Iran, 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 – 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 monomials $x$ , $x^{2}$ , $x^{3}$ , ... and $1/x$ , $1/x^{2}$ , $1/x^{3}$ , ... 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 – Mesopotamia, 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 $\cos \alpha =1/{\sqrt {1+\tan ^{2}\alpha }}$ .

## Symbolic stage

#### 1000–1500

##### 15th century
• 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.



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