Timeline of algebra and geometry

A timeline of algebra and geometry

Before 1000 BC

* ca. 2000 BC — Scotland, Carved Stone Balls exhibit a variety of symmetries including all of the symmetries of Platonic solids.
* 1800 BC — Moscow Mathematical Papyrus, findings volume of a frustum
* 1650 BC — 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
* 1300 BC — Berlin papyrus (19th dynasty) contains a quadratic equation and its solution. [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html#berlin]

1st millennium BC

* 800 BC — Baudhayana, author of the Baudhayana Sulba Sutra, a Vedic Sanskrit geometric text, contains quadratic equations, and calculates the square root of 2 correct to five decimal places
* ca. 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
* 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
* 530 BC — Pythagoras studies propositional geometry and vibrating lyre strings; his group also discover the irrationality of the square root of two,
* 370 BC — Eudoxus states the method of exhaustion for area determination
* 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
* 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.
* 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.

1st millennium

* 1st century — Heron of Alexandria, the earliest fleeting reference to square roots of negative numbers.
* 250 — Diophantus uses symbols for unknown numbers in terms of syncopated algebra, and writes "Arithmetica", one of the earliest treatises on algebra
* ca. 340 — Pappus of Alexandria states his hexagon theorem and his centroid theorem
* 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)
* 600s — Bhaskara I gives a rational approximation of the sine function
* 600s — 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
* 700s — 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
* 700s — Shridhara gives the rule for finding the volume of a sphere and also the formula for solving quadratic equations
* 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.
* 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).
* ca. 900 — Abu Kamil of Egypt had begun to understand what we would write in symbols as $x^n\; cdot\; x^m\; =\; x^\{m+n\}$
* 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 $x$, $x^2$, $x^3$, … 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 — Al-Batani — Extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tan­gent, secant and their inverse functions. Derived the formula: $sin\; alpha\; =\; an\; alpha\; /\; sqrt\{1+\; an^2\; alpha\}$ and $cos\; alpha\; =\; 1\; /\; sqrt\{1\; +\; an^2\; alpha\}$.

1000–1500

*ca. 1000 — Abū Sahl al-Qūhī (Kuhi) solves equations higher than the second degree.
*ca. 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.
* 1070 — Omar Khayyám begins to write "Treatise on Demonstration of Problems of Algebra" and classifies cubic equations.
* ca. 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).
* 1100s — Bhaskara Acharya writes the “Bijaganita” (“Algebra”), which is the first text that recognizes that a positive number has two square roots
* 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 which “represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.” [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]
* ca. 1250 — Nasir Al-Din Al-Tusi attempts to develop a form of non-Euclidean geometry.
* 1400s — Nilakantha Somayaji, a Kerala school mathematician, writes the “Aryabhatiya Bhasya”, which contains work on infinite-series expansions, problems of algebra, and spherical geometry

16th century

* 1520 — Scipione dal Ferro develops a method for solving “depressed” cubic equations (cubic equations without an x² term), but does not publish.
* 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.

17th century

* 1600s - Putumana Somayaji writes the "Paddhati", which presents a detailed discussion of various trigonometric series
* 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.
* 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.

18th century

* 1722 - Abraham de Moivre states de Moivre's formula connecting trigonometric functions and complex numbers,
* 1733 - Giovanni Gerolamo Saccheri studies what geometry would be like if Euclid's fifth postulate were false,
* 1796 - Carl Friedrich Gauss proves that the regular 17-gon can be constructed using only a compass and straightedge
* 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

* 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,
* 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,
* 1829 - Bolyai, Gauss, and Lobachevsky invent hyperbolic non-Euclidean geometry,
* 1832 - Évariste Galois presents a general condition for the solvability of algebraic equations, thereby essentially founding group theory and Galois theory,
* 1837 - Pierre Wantsel 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
* 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 now is called Boolean algebra,
* 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,
* 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,
* 1873 - Charles Hermite proves that e is transcendental,
* 1878 - Charles Hermite solves the general quintic equation by means of elliptic and modular functions
* 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,
* 1899 - David Hilbert presents a set of self-consistent geometric axioms in "Foundations of Geometry",

20th century

* 1901 - Élie Cartan develops the exterior derivative,
* 1905 - Einstein's theory of special relativity.
* 1912 - Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem,
* 1916 - Einstein's theory of general relativity.
* 1930 - Casimir Kuratowski shows that the three-cottage problem has no solution,
* 1931 - Georges de Rham develops theorems in cohomology and characteristic classes,
* 1933 - Karol Borsuk and Stanislaw Ulam present the Borsuk-Ulam antipodal-point theorem,
* 1955 - H. S. M. Coxeter et al. publish the complete list of uniform polyhedron,
* 1981 - Mikhail Gromov develops the theory of hyperbolic groups, revolutionizing both infinite group theory and global differential geometry,
* 1983 - the classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed,
* 1991 - Alain Connes and John W. Lott develop non-commutative geometry,
* 1998 - Thomas Hales (almost certainly) proves the Kepler conjecture,

21st century

* 2003 - Grigori Perelman proves the Poincaré conjecture,
* 2007 - a team of researches throughout North America and Europe used networks of computers to map E8 (mathematics). [ Elizabeth A. Thompson, MIT News Office, "Math research team maps E8" http://www.huliq.com/15695/mathematicians-map-e8 ]