Mādhava and the Kerala School:
Calculus in India Before Newton
In European textbooks, calculus was invented by Isaac Newton and Gottfried Leibniz, independently, in the late 17th century. The infinite series expansions for sine, cosine, and arctangent are attributed to James Gregory (1671) and Newton (1676). These attributions are accurate as far as they go — but they are incomplete.
Mādhava of Saṅgamagrāma — a mathematician from Kerala who lived approximately 1340–1425 CE — derived these exact same infinite series 200–300 years before the European mathematicians. His results are preserved in later texts by his successors Nīlakaṇṭha Somayāji and Jyeṣṭhadeva, and the mathematical history is thoroughly documented by modern scholars.
The question of whether the Kerala School's results influenced European mathematics through transmission routes (Portuguese trade contacts with Kerala in the 16th century are documented) is actively debated by historians. What is not in dispute is the priority of the mathematical discoveries themselves.
Mādhava's Series — Before Gregory and Newton
The Mādhava–Leibniz Series (π/4)
π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − ...
Derived by Mādhava c. 1400 CE. Attributed in European history to Leibniz (1673) and James Gregory (1671) — 270 years later. Preserved in Yuktibhāṣā by Jyeṣṭhadeva (c. 1530 CE).
Mādhava Sine Series
sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + ...
Derived by Mādhava, preserved in verse form in the Yuktibhāṣā. Attributed in Europe to Newton (1676). The Kerala version precedes it by approximately 250 years.
Mādhava Cosine Series
cos(x) = 1 − x²/2! + x⁴/4! − x⁶/6! + ...
Derived by Mādhava. Attributed in Europe to Newton. Kerala priority documented by modern scholars including George Gheverghese Joseph and K.V. Sarma.
Source: K.V. Sarma (ed.), Yuktibhāṣā, Hindustan Book Agency (2008); George Gheverghese Joseph, The Crest of the Peacock, Princeton University Press (2011)
Mādhava's Value of π
Mādhava derived π to 11 decimal places — 3.14159265359 — using his infinite series. This was the most precise value of π computed anywhere in the world until the 16th century, when European mathematicians using Mādhava-equivalent series (independently rediscovered) extended it further.
His correction terms to the series — designed to accelerate convergence — show a sophisticated understanding of what we now call error analysis, and anticipate 20th century computational techniques for accelerating slowly converging series.
Source: Kim Plofker, Mathematics in India, Princeton University Press (2009), pp. 220–226
"The Kerala mathematicians had, in effect, invented calculus before Newton and Leibniz. The evidence is contained in texts that were largely unknown to Western scholars until the 20th century."
— George Gheverghese Joseph, The Crest of the Peacock, Princeton University Press (2011)
Sources
- • K.V. Sarma (ed.), Yuktibhāṣā, Hindustan Book Agency (2008)
- • George Gheverghese Joseph, The Crest of the Peacock, Princeton University Press (2011)
- • Kim Plofker, Mathematics in India, Princeton University Press (2009)
- • C.K. Raju, Cultural Foundations of Mathematics, Pearson Longman (2007)
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