actually prove the theorem, but rather worked out examples using his method. The first example he presented was: Example (Brahmagupta): Find integers such that: In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician." The solution he provided was: Bhaskara I Bhaskara I (c. 600–680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya. He produced: Solutions of indeterminate equations. A rational approximation of the sine function. A formula for calculating the sine of an acute angle without the use of a table, correct to two decimal places. Ninth to twelfth centuries Virasena Virasena (8th century) was a Jain mathematician in the court of Rashtrakuta King Amoghavarsha of Manyakheta, Karnataka. He wrote the Dhavala, a commentary on Jain mathematics, which: Deals with the concept of ardhaccheda, the number of times a number could be halved, and lists various rules involving this operation. This coincides with the binary logarithm when applied to powers of two, but differs on other numbers, more closely resembling the 2-adic order. The same concept for base 3 (trakacheda) and base 4 (caturthacheda). Virasena also gave: The derivation of the volume of a frustum by a sort of infinite procedure. It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE. Mahavira Mahavira Acharya (c. 800–870) from Karnataka, the last of the notable Jain mathematicians, lived in the 9th century and was patronised by the Rashtrakuta king Amoghavarsha. He wrote a book titled Ganit Saar Sangraha on numerical mathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of: Zero Squares Cubes square roots, cube roots, and the series extending beyond these Plane geometry Solid geometry Problems relating to the casting of shadows Formulae derived to calculate the area of an ellipse and quadrilateral inside a circle. Mahavira also: Asserted that the square root of a negative number did not exist Gave the sum of a series whose terms are squares of an arithmetical progression, and gave empirical rules for area and perimeter of an ellipse. Solved cubic equations. Solved quartic equations. Solved some quintic equations and higher-order polynomials. Gave the general solutions of the higher order polynomial equations: Solved indeterminate quadratic equations. Solved indeterminate cubic equations. Solved indeterminate