Bhāskara II

Bhāskara II  (c. 1114–1185), also known as Bhāskarāchārya and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. His main work Siddhānta-Śiromaṇi, (Sanskrit for "Crown of Treatises") is divided into four parts called Līlāvatī, Bījagaṇita, Grahagaṇita and Golādhyāya which are also sometimes considered four independent works. These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karaṇā Kautūhala.

Some of Bhāskara's contributions to mathematics include the following:




 * A proof of the Pythagorean theorem by calculating the same area in two different ways and then cancelling out terms to get a2 + b2 = c2.


 * In Līlāvatī, solutions of quadratic, cubic and quartic indeterminate equations are explained.

Līlāvatī (meaning a beautiful woman) is based on Arithmetic. It is believed that Bhāskara named this book after his daughter Līlāvatī. Many of the problems in this book are addressed to his daughter. For example “Oh Līlāvatī, intelligent girl, if you understand addition & subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10 & 100, as well as [the remainder of] those when subtracted from 10000.” The book contains thirteen chapters, mainly definitions, arithmetical terms, interest computation, arithmetical & geometric progressions. Many of the methods in the book on computing numbers such as multiplications, squares & progressions were based on common objects like kings & elephants, which a common man could understand. The Siddhānta Śiromaṇi (written in 1150) demonstrates Bhāskara 's knowledge of trigonometry, including the sine table and        relationships between different trigonometric functions. He also developed spherical trigonometry, along with other interesting trigonometrical results. In particular Bhāskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhāskara, results found in his works include computation of sines of angles of 18 and 36 degrees, and the now well known formulae for  sin(a+b) and sin(a-b).
 * Solutions of indeterminate quadratic equations (of the type ax2 + b = y2).
 * The first general method for finding the solutions of the problem x2 − ny2 = 1 (so-called "Pell's equation") was given by Bhāskara II.
 * Preliminary concept of mathematical analysis.
 * Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus.
 * Calculated the derivatives of trigonometric functions and formulae.
 * In Siddhānta-Śiromaṇi, Bhāskara developed spherical trigonometry along with a number of other trigonometric results.