A timeline of key algebraic developments are as follows:

Year | Event |
---|---|

Circa 1800 BC | The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation. |

Circa 1800 BC | The Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script. |

1300 BC | Berlin papyrus (19th dynasty) contains a quadratic equation and its solution. |

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 |

Circa 300 BC | In Book II of his Elements, Euclid gives a geometric construction with Euclidean tools for the solution of the quadratic equation for positive real roots. The construction is due to the Pythagorean School of geometry. |

Circa 300 BC | A geometric construction for the solution of the cubic is sought (doubling the cube problem). It is now well known that the general cubic has no such solution using Euclidean tools. |

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 |

Circa 100 BC | Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu (The Nine Chapters on the Mathematical Art), which contains solutions of linear equations solved using the rule of double false position, geometric solutions of quadratic equations, and the solutions of matrices equivalent to the modern method, to solve systems of simultaneous linear equations. |

1st century | Heron of Alexandria, the earliest fleeting reference to square roots of negative numbers. |

Circa 150 | Greek mathematician Hero of Alexandria, treats algebraic equations in three volumes of mathematics. |

Circa 200 | Hellenistic mathematician Diophantus lived in Alexandria and is often considered to be the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers. |

499 | Indian mathematician Aryabhata, in his treatise Aryabhatiya, obtains whole-number solutions to linear equations by a method equivalent to the modern one, describes the general integral solution of the indeterminate linear equation, gives integral solutions of simultaneous indeterminate linear equations, and describes a differential equation. |

Circa 625 | Chinese mathematician Wang Xiaotong finds numerical solutions to certain cubic equations. |

Circa 7th century Dates vary from the 3rd to the 12th centuries. |
The Bakhshali Manuscript written in ancient India uses a form of algebraic notation using letters of the alphabet and other signs, and contains cubic and quartic equations, algebraic solutions of linear equations with up to five unknowns, the general algebraic formula for the quadratic equation, and solutions of indeterminate quadratic equations and simultaneous equations. |

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 Brahmasphuta-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 |

Circa 800 | The Abbasid patrons of learning, al-Mansur, Haroun al-Raschid, and al-Mamun, had Greek, Babylonian, and Indian mathematical and scientific works translated into Arabic and began a cultural, scientific and mathematical awakening after a century devoid of mathematical achievements. |

820 | The word algebra is derived from operations described in the treatise written by the Persian mathematician, Muḥammad ibn Mūsā al-Ḵhwārizmī, titled Al-Kitab al-Jabr wa-l-Muqabala (meaning "The Compendious Book on Calculation by Completion and Balancing") on the systematic solution of linear and quadratic equations. Al-Khwarizmi is often considered the "father of algebra", for founding algebra as an independent discipline and for introducing the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which was what he originally used the term al-jabr to refer to. His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems." |

Circa 850 | Persian mathematician al-Mahani conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. |

Circa 990 | Persian mathematician Al-Karaji (also known as al-Karkhi), in his treatise Al-Fakhri, further develops algebra by extending Al-Khwarizmi's methodology to incorporate integral powers and integral roots of unknown quantities. He replaces geometrical operations of algebra with modern arithmetical operations, and defines the monomials x, x2, x3, .. and 1/x, 1/x2, 1/x3, .. and gives rules for the products of any two of these. He also discovered the first numerical solution to equations of the form ax2n + bxn = c. Al-Karaji is also regarded as the first person to free algebra from geometrical operations and replace them with the type of arithmetic operations which are at the core of algebra today. His work on algebra and polynomials, gave the rules for arithmetic operations to manipulate polynomials. The historian of mathematics F. Woepcke, in Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi (Paris, 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic calculus". Stemming from this, Al-Karaji investigated binomial coefficients and Pascal's triangle. |

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). |

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, … and, … 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.” |

ca. 1000 | Abū Sahl al-Qūhī (Kuhi) solves equations higher than the second degree. |

Circa 1050 | Chinese mathematician Jia Xian finds numerical solutions of polynomial equations of arbitrary degree. |

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

1072 | Persian mathematician Omar Khayyam gives a complete classification of cubic equations with positive roots and gives general geometric solutions to these equations found by means of intersecting conic sections. |

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: “ 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.” |

Circa 1200 | Sharaf al-Dīn al-Tūsī (1135–1213) wrote the Al-Mu'adalat (Treatise on Equations), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the "Ruffini-Horner method" to numerically approximate the root of a cubic equation. He also developed the concepts of the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the cubic equation and used an early version of Cardano's formula to find algebraic solutions to certain types of cubic equations. Some scholars, such as Roshdi Rashed, argue that Sharaf al-Din discovered the derivative of cubic polynomials and realized its significance, while other scholars connect his solution to the ideas of Euclid and Archimedes. |

1202 | Algebra is introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci. |

Circa 1300 | Chinese mathematician Zhu Shijie deals with polynomial algebra, solves quadratic equations, simultaneous equations and equations with up to four unknowns, and numerically solves some quartic, quintic and higher-order polynomial equations. |

Circa 1400 | Jamshīd al-Kāshī developed an early form of Newton's method to numerically solve the equation to find roots of N. |

Circa 1400 | Indian mathematician Madhava of Sangamagrama finds the solution of transcendental equations by iteration, iterative methods for the solution of non-linear equations, and solutions of differential equations. |

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 |

1412–1482 | Arab mathematician Abū al-Hasan ibn Alī al-Qalasādī took "the first steps toward the introduction of algebraic symbolism." He used "short Arabic words, or just their initial letters, as mathematical symbols." |

1535 | Niccolò Fontana Tartaglia and others mathematicians in Italy independently solved the general cubic equation. |

1545 | Girolamo Cardano publishes Ars magna -The great art which gives Fontana's solution to the general quartic equation. |

1572 | Rafael Bombelli recognizes the complex roots of the cubic and improves current notation. |

1591 | Franciscus Vieta develops improved symbolic notation for various powers of an unknown and uses vowels for unknowns and consonants for constants in In artem analyticam isagoge. |

1619 | René Descartes discovers analytic geometry (Pierre de Fermat claimed that he also discovered it independently), |

1631 | Thomas Harriot in a posthumous publication is the first to use symbols < and > to indicate "less than" and "greater than", respectively. |

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. |

1682 | Gottfried Wilhelm Leibniz develops his notion of symbolic manipulation with formal rules which he calls characteristica generalis. |

1683 | Japanese mathematician Kowa Seki, in his Method of solving the dissimulated problems, discovers the determinant, discriminant, and Bernoulli numbers. |

1685 | Kowa Seki solves the general cubic equation, as well as some quartic and quintic equations. |

1693 | Leibniz solves systems of simultaneous linear equations using matrices and determinants. |

1722 | Abraham de Moivre states de Moivre's formula connecting trigonometric functions and complex numbers, |

1750 | Gabriel Cramer, in his treatise Introduction to the analysis of algebraic curves, states Cramer's rule and studies algebraic curves, matrices and determinants. |

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, |

1806 | Jean-Robert Argand publishes proof of the Fundamental theorem of algebra and the Argand diagram, |

1824 | Niels Henrik Abel proved that the general quintic equation is insoluble by radicals. |

1832 | Galois theory is developed by Évariste Galois in his work on abstract algebra. |

1847 | George Boole formalizes symbolic logic in The Mathematical Analysis of Logic, defining what now is called Boolean algebra, |

1873 | Charles Hermite proves that e is transcendental, |

1878 | Charles Hermite solves the general quintic equation by means of elliptic and modular functions |

1981 | Mikhail Gromov develops the theory of hyperbolic groups, revolutionizing both infinite group theory and global differential geometry, |

2007 | a team of researches throughout North America and Europe used networks of computers to map E8 (mathematics). |

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