Centuries before Isaac Newton and Gottfried Wilhelm Leibniz developed calculus in Europe, an Indian mathematician from Kerala made pioneering discoveries involving infinite series, including mathematical formulas for calculating pi (π), sine, cosine and arctangent.
Madhava of Sangamagrama (c. 1340–1425) is credited with discovering these series in the 14th century and founding the Kerala School of and Mathematics.
His work marked an important advance in mathematics, moving beyond finite calculations to explore the properties of infinity — concepts that would later become central to mathematical analysis and calculus.
According to mathematician Ranjan Roy’s , published by Cambridge University Press, Madhava discovered infinite power series roughly two and a half centuries before Newton independently developed similar methods.
Most of Madhava’s original mathematical writings have been lost, and his discoveries are preserved in the works of later Kerala scholars.
Madhava was a mathematician and astronomer from Sangamagrama, a historical settlement near present-day in Kerala.
He is regarded as the founder of the Kerala School, a mathematical tradition whose prominent scholars included Parameshvara, Nilakantha Somayaji and Jyeshthadeva.
and astronomy were closely connected in India’s intellectual traditions. Astronomy formed part of Jyotisha, one of the six Vedangas, or auxiliary disciplines of Vedic learning.
Mathematical techniques were essential for astronomical calculations, including determining planetary movements and the positions of celestial bodies.
One of Madhava’s most significant contributions was an infinite series for calculating pi, the ratio of a circle’s circumference to its diameter.
By repeatedly adding and subtracting fractions with odd-numbered denominators, the series approaches pi.
Madhava also developed correction terms to improve the accuracy of the calculation and address the slow convergence of the basic series.
According to a historical account published by the Indian Knowledge Systems division at , Madhava calculated pi to 11 decimal places using infinite series and correction techniques — an extraordinary achievement for the 14th century.
His discoveries extended beyond pi. Madhava is credited with deriving infinite series for sine, cosine and arctangent, allowing increasingly accurate calculations of trigonometric values.
These mathematical techniques had important applications in astronomy, where precise calculations were needed to determine angles, arcs and celestial positions.
He also developed methods to estimate the errors in successive approximations, demonstrating an understanding of how series approach their limiting values.
These contributions are documented in the later mathematical works of the Kerala School, as discussed in Kim Plofker’s in India.
Madhava’s mathematical ideas were developed further by successive generations of scholars.
Nilakantha Somayaji’s Tantrasangraha, written around 1500, recorded important mathematical and astronomical findings.
Another significant work was Jyeshthadeva’s Yuktibhāṣā, composed in Malayalam around 1530.
It provided detailed mathematical explanations and derivations of infinite series, including those associated with Madhava.
Some historians have suggested that mathematical knowledge from Kerala could have reached Europe through traders and Jesuit missionaries operating along India’s coast. However, there is no conclusive documentary evidence that Madhava’s discoveries directly influenced Newton or Leibniz.
As Press research notes, the Kerala and European mathematical traditions developed similar results using different methods, with no established connection between them.
Madhava’s contribution nevertheless represents a significant chapter in the history of mathematics, demonstrating that sophisticated work on infinite series and the foundations of mathematical analysis was taking place in India centuries before similar developments in Europe.
