Bhāskara IIBhāskara (also known as Bhāskara II and Bhāskarāchārya ("Bhāskara the teacher"), (1114–1185), was an Indian mathematician and astronomer. He was born near Vijjadavida (Bijāpur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of ancient India. He lived in the Sahyadri region (Patnadevi, Jalgaon, Maharashtra).
Some of Bhaskara's contributions to mathematics include the following: A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a² + b² = c². In Lilavati, solutions of quadratic, cubic and quartic indeterminate equations are explained. Solutions of indeterminate quadratic equations (of the type ax² + b = y²). Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method. The first general method for finding the solutions of the problem x² − ny² = 1 (so-called "Pell's equation") was given by Bhaskara II. Solutions of Diophantine equations of the second order, such as 61x² + 1 = y². This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century. Solved quadratic equations with more than one unknown, and found negative and irrational solutions. Preliminary concept of mathematical analysis. Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus. Conceived differential calculus, after discovering the derivative and differential coefficient. Stated Rolle's theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works. In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. |