# Bhaskara’s II contributions to mathematics – India

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 quadric , cubic and quartic indeterminate equation 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 i 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 Roll’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.
• Calculated the derivatives of trigonometric functions and formula.
• In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results.

to be continue ………………