Madhava & the Kerala School
Madhava of Sangamagrama discovered the systematic use of infinite power series in mathematics — the foundation of calculus — approximately 250–300 years before Isaac Newton and Gottfried Wilhelm Leibniz in Europe. He calculated Pi to 13 decimal places. His methods are now called the Madhava-Leibniz series, Madhava-Newton series, and Madhava-Gregory series.
Infinite Series — 300 Years Before Europe
Series for Pi (π/4)
Madhava-Gregory-Leibniz Series
Madhava: c. 1375 CE
Europe: Leibniz, 1673 CE (298 years later); Gregory, 1671 CE
π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − ...
The first infinite series for Pi ever discovered. Madhava derived this from the power series expansion of arctan(1) — the arctangent of 1 equals π/4. He also developed sophisticated correction terms to accelerate convergence.
Power Series for Sine
Madhava-Newton Sine Series
Madhava: c. 1375 CE
Europe: Newton, 1669 CE (294 years later)
sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + ...
The infinite series expansion of the sine function — used in every scientific computation today. Madhava derived this and used it to compute accurate trigonometric tables for planetary position calculations in the Panchang.
Power Series for Cosine
Madhava-Newton Cosine Series
Madhava: c. 1375 CE
Europe: Newton, 1669 CE (294 years later)
cos(x) = 1 − x²/2! + x⁴/4! − x⁶/6! + ...
The companion series to sine, derived by Madhava in the same period. Together, these two series form the basis of all modern trigonometric computation, Fourier analysis, and digital signal processing.
Arctangent Series
Madhava-Gregory-Leibniz Arctangent
Madhava: c. 1375 CE
Europe: Gregory, 1671 CE; Leibniz, 1676 CE
arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + ...
The general arctangent series, of which the Pi series (arctan 1 = π/4) is a special case. Madhava used this with clever correction terms to compute Pi to 13 decimal places — setting a world record that stood for over a century.
Pi to 13 Decimal Places — c. 1400 CE
Madhava's Value (c. 1400 CE)
π = 3.14159265358979...
13 decimal places. The most accurate value of Pi anywhere in the world at that time. This remained the world record until the Chinese mathematician Zu Chongzhi's value was rediscovered in Europe.
Modern Value
π = 3.14159265358979323846...
Madhava's 13-digit value is identical to the modern value for the first 13 decimal places. Zero error in the digits he computed.
How It Was Preserved
Most of Madhava's direct works are lost. His series and results are preserved in the writings of his students and successors: Nilakantha Somayaji's Tantrasangraha (c. 1500 CE), Jyeshthadeva's Yuktibhasha (c. 1530 CE) — which also provides rigorous proofs for the series — and Sankara Variyar's Kriyakramakari. The Yuktibhasha is considered the world's first calculus textbook, predating Newton's Principia by over 150 years.
The Kerala School Lineage
Madhava of Sangamagrama
c. 1340–1425
Founder. Infinite series for π, sin, cos, arctan. Pi to 13 decimal places.
Parameshvara
c. 1370–1460
55 years of direct observations. Drgganita system. Circumradius of cyclic quadrilateral formula.
Nilakantha Somayaji
1444–1544
Tantrasangraha preserving Madhava. Partially heliocentric planetary model (predates Tycho Brahe by ~80 years).
Jyeshthadeva
c. 1500–1575
Yuktibhasha — first calculus textbook with rigorous proofs. Derivations of Madhava's series.
Citrabhanu
c. 1550
Systematised solutions of cubic and quartic equations.
Sankara Variyar
c. 1500–1560
Kriyakramakari — detailed commentary preserving multiple Madhava results.