Brahmagupta to the Kerala School
Medieval Indian mathematics — from zero to infinite series
9.1 Brahmagupta
Brahmagupta (598–668 CE) formulated the first explicit rules for arithmetic with zero and negative numbers. He also discovered the formula for the area of a cyclic quadrilateral and made significant contributions to the solution of Pell's equation.
9.2 Bhaskara II
Bhaskara II (1114–1185) developed concepts of derivatives and differentials — recognizing that the derivative of the sine function is the cosine function. These ideas represent genuine precursors to differential calculus, developed independently and centuries before Newton and Leibniz.
9.3 The Kerala School
The Kerala school (c. 1350–1600) achieved results strikingly anticipatory of European calculus. Madhava of Sangamagrama (c. 1340–1425) discovered infinite series expansions for π, sine, cosine, and arctangent — including π/4 = 1 − 1/3 + 1/5 − 1/7 + ... roughly 250 years before Leibniz.