The Kerala School & infinite series calculus

In the 14th century, a flourishing centre of mathematical astronomy developed in the river basins of Kerala (South India).¹

Founded by the astronomer-mathematician Mādhava of Saṅgamagrāma (c. 1340–1425 CE) and carried forward by Parameśvara, Nīlakaṇṭha Somayājī, and Jyeṣṭhadeva, the Kerala School of Astronomy and Mathematics achieved what some historians describe as the transition from finite algebra to infinite mathematical analysis and calculus.²

Mādhava of Saṅgamagrāma

Mādhava is credited with some of the earliest known infinite power series for trigonometric functions and π:

1. The Mādhava-Gregory Series for Inverse Tangent:

The series expresses an angle through successive odd powers of its tangent. Written in modern notation, with x = tan(θ) and −1 ≤ x ≤ 1:

arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + …

Setting θ = π/4 yields the famous infinite series for π (later rediscovered by James Gregory in 1671 and Gottfried Leibniz):

π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − …

2. The Mādhava-Newton Series for Sine and Cosine:

sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + …

cos(x) = 1 − x²/2! + x⁴/4! − x⁶/6! + …

Here, angles are measured in radians, and the factorial sign means multiplying the positive integers up to that number: 3! = 3 × 2 × 1.

Verses attributed to Mādhava and quoted by later Kerala authors state these results in concise Sanskrit; some numerical values in such verses are encoded in the kaṭapayādi alphanumeric notation.³

3. Accelerated Series and the Value of π:

Recognizing that the standard Leibniz series converges very slowly, Mādhava introduced correction terms (antyasaṃskāra) and transformed the series to achieve rapid convergence. Another, faster-converging series for π is also credited to him:

π = √12 × [1 − 1/(3 × 3) + 1/(5 × 3²) − 1/(7 × 3³) + …]

Mādhava gave a value of π accurate to 11 decimal places, a value that can be obtained from this formula:⁴

π ≈ 3.14159265359

Jyeṣṭhadeva’s Yuktibhāṣā (c. 1530 CE): An Early Text of Calculus

While Sanskrit scientific texts typically stated mathematical results as condensed verse aphorisms without step-by-step proofs, Jyeṣṭhadeva broke with convention by composing the Yuktibhāṣā (Rationale in the Language of the Land) in Malayalam prose.⁵

The Yuktibhāṣā provides detailed geometric and analytical derivations:

• It treats an arc of the circle as made up of a very large number of very small arc-bits.
• It performs term-by-term summation of infinitesimal elements. For a non-negative integer k, the sum of the kth powers of the integers from 1 to n, divided by n to the power k + 1, approaches 1/(k + 1) as n grows. In modern integral notation, this corresponds to the area under x to the power k between 0 and 1.⁶

Nīlakaṇṭha Somayājī’s Revised Planetary Model (1500 CE)

In his Tantrasaṅgraha (1500 CE) and his commentary on the Āryabhaṭīya, Nīlakaṇṭha Somayājī revised the planetary model:

• He revised the traditional geocentric epicyclic model.
• In the reading of some modern scholars, he proposed a model where the five planets (Mercury, Venus, Mars, Jupiter, Saturn) orbit the Sun, while the Sun itself, carrying the planets with it, orbits the Earth.⁷
• This model was similar to the system published by Tycho Brahe in Europe nearly a century later (1588).

Four terms to carry forward

What this chapter does not claim

1. That the Kerala School developed modern general infinitesimal calculus with formal epsilon-delta limit proofs; they developed geometric series analysis for specific trigonometric functions.
2. That European calculus was definitively stolen from Kerala; while Jesuit missionaries (such as Matteo Ricci) were active in Cochin, transmission to Europe remains unproven.
3. That Kerala astronomers rejected observational corrections; Parameśvara is reputed to have made astronomical observations, including of eclipses, over some 55 years to refine planetary parameters (Dṛggaṇita).
4. That mathematics died with Mādhava; the direct lineage continued after him through Parameśvara, Nīlakaṇṭha, Jyeṣṭhadeva and Acyuta Piṣāraṭi, and the wider tradition continued into the 18th century.

Sources and further reading

Version 0.4 · September 2026 · Editorial orientation · Reading list provided · Not expert-reviewed. Corrected after an editorial audit in September 2026.