Some of the Bhaskara’s contributions to the mathematics include the following:
1. A general proof of the Pythagorean Theorem by method of calculating the same area in two or more different ways and then canceling out the terms to get the
a 2 + b 2 = c 2 .
2. In Lilavati, solutions of the quadric , cubic and the quartic indeterminate equation are well explained.
3. Solutions of the indeterminate quadratic equations are of the type ax 2 + b = y 2 .
4. Integer solutions of the linear and many quadratic indeterminate equations (Kuttaka). The rules that he gives are (in effect) the same as those rules given by the famous Renaissance European mathematicians of the 17th century
5. A cyclic Chakravala method used for solving the indeterminate equations of the form such as ax 2 + bx + c = y. The solution to this kind of equation was traditionally attributed to the William Brouncker in 1657, though his used method was more kind of difficult than the chakravala method.
6. The first ever general method used for finding all the solutions of the problem x 2 − ny 2 = 1 (so-called as the “Pell’s equation “)was given by the Bhaskara II.
7. Solutions of the Diophantine Equations of the second order, such as the 61x 2 + 1 = y 2 . This type of equation was posed as a difficult problem in 1657 by the French mathematician called as Pierre de Fermat , but its all the solution was unknown in the Europe until the time of the Euler in the 18th century.
8. Solved quadratic equations with more than one type of equation unknown, and have found several negative and many irrational i solutions.
9. Preliminary concept of the mathematical analysis.
10. Preliminary concept of the infinitesimal Calculus, along with all the notable contributions towards the integral calculus .
11. Conceived differential calculus, after the discovering of the derivative and the differential coefficient.
12. Stated Roll’s theorem, a special type case of one of the most valuable and important theorems in the analysis, the mean value theorem. Traces of this general mean value theorem are also easily found in his works.
13. Calculated the derivatives of the trigonometric functions and several formula.
14. In Siddhanta Shiromani, the Bhaskara had developed spherical trigonometry along with various type of numbers of all other trigonometric results.
