## 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
*a*^{2}+*b*^{2}=*c*^{2}.

- In
*Lilavati*, solutions of quadric , cubic and quartic indeterminate equation are explained.

- Solutions of indeterminate quadratic equations (of the type
*ax*^{2}+*b*=*y*^{2}).

- 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
*ax*^{2}+*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
*x*^{2}−*ny*^{2}= 1 (so-called “Pell’s equation “)was given by Bhaskara II.

- Solutions of Diophantine Equations of the second order, such as 61
*x*^{2}+ 1 =*y*^{2}. 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.

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