Bhaskara’s II contributions to mathematics – India

Bhaskara’s II  Mathematics

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 ………………