### Cardano formula for cubics

The Old Babylonian Strassburg tablet seeks the solution of a quadratic elliptic equation. Cardano formula for cubics Plimpton 322 tablet gives a table of Pythagorean triples in Babylonian Cuneiform script. Euclid’s Elements 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.

It is now well known that the general cubic has no such solution using Euclidean tools. Hero of Alexandria gives the earliest fleeting reference to square roots of negative numbers. Greek mathematician Hero of Alexandria, treats algebraic equations in three volumes of mathematics. Hellenistic mathematician Diophantus, who 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. 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. Chinese mathematician Wang Xiaotong finds numerical solutions to certain cubic equations.

Dates vary from the 3rd to the 12th centuries. Brahmagupta invents the method of solving indeterminate equations of the second degree and is the first to use algebra to solve astronomical problems. Brahmagupta writes the Brahmasphuta-siddhanta, where zero is clearly explained, and where the modern place-value Indian numeral system is fully developed. Persian mathematician al-Mahani conceives the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. 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, . Thabit ibn Qurra: the only surviving fragment of his original work contains a chapter on the solution and properties of cubic equations. Chinese mathematician Jia Xian finds numerical solutions of polynomial equations of arbitrary degree. Omar Khayyám begins to write Treatise on Demonstration of Problems of Algebra and classifies cubic equations. 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. Leonardo Fibonacci of Pisa publishes his Liber Abaci, a work on algebra that introduces Arabic numerals to Europe. 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.