The Madhava series are infinite power-series expansions of trigonometric functions developed by the Kerala School of mathematics between roughly 1400 and 1550, attributed primarily to Madhava of Sangamagrama (c. 1340–c. 1425). They cover sine, cosine and arctangent, and were used to compute π to 11 decimal places by adding successive terms. The same series were independently rediscovered in Europe by Isaac Newton (1665–1666), James Gregory (1671) and Gottfried Leibniz (1673), roughly 250 years later. The Kerala results survive in three principal sources: Nilakantha Somayaji’s Tantrasamgraha (1501), Jyeshthadeva’s Yuktibhasha (c. 1530), and the anonymous Karanapaddhati. Western scholarship encountered them through Charles M. Whish’s 1834 paper to the Royal Asiatic Society.
The three principal Madhava series
The series, in modern notation:
- Madhava-Leibniz series for π: π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − …
- Madhava arctangent series: arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + … for x ≤ 1.
- Madhava sine series: sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + …
- Madhava cosine series: cos(x) = 1 − x²/2! + x⁴/4! − x⁶/6! + …
The Kerala texts express these in Sanskrit verses without modern symbolic notation. The arctangent series appears in the Tantrasamgraha (verse II.206 of the standard edition) and is attributed there explicitly to Madhava. The sine and cosine series appear in the same work and are similarly attributed.
How Madhava derived them: the geometry of the circle
The Yuktibhasha (chapter 7) presents a derivation of the arctangent series that historians have reconstructed in modern terminology as follows. Consider a circle of radius r and a chord intercepting arc length s. Divide the arc into n equal small pieces and use the small-arc approximation; then take n to infinity. The Yuktibhasha argues, in essence, that the limit of the sum of small triangle areas as n becomes large yields the integral expression that gives the arctangent series. The argument is a limit-style passage from finite to infinite without using the formal notion of a derivative or integral; the Kerala School worked in a pre-symbolic geometric register that nonetheless captured the analytic content.
Modern historians of mathematics including Ranjan Roy, K. V. Sarma, M. D. Srinivas, K. Ramasubramanian and M. S. Sriram have provided detailed reconstructions in the two-volume English Yuktibhasha edition (Hindustan Book Agency, 2008). The derivations meet contemporary standards of rigour when expressed in modern terms.
The π-correction terms
The Madhava-Leibniz series for π/4 (setting x=1 in the arctangent series) converges extremely slowly: roughly 5,000 terms are required to obtain four decimal places of accuracy. Madhava developed three increasingly accurate end-correction formulae that allow the series to be truncated at finite n with a residual term that absorbs most of the error. The simplest correction (added after the nth term) is approximately 1/(2(n+1)), and Madhava’s third correction reduces the truncation error to a much smaller residual. Using these corrections, the Kerala school computed π to 11 decimal places: 3.14159265359, an accuracy unmatched by any European or Chinese computation until the late 16th century when Ludolph van Ceulen reached 35 decimal places using a different (Archimedean polygon) method.
Madhava’s trigonometric tables
The series enabled computation of sine and cosine values at any angle. Madhava produced tables of sine values for 24 equally-spaced angles between 0° and 90° (3.75° steps), accurate to nine decimal places. These tables, preserved in the Karanapaddhati and quoted in later commentaries, are substantially more accurate than the sine tables in Aryabhata’s Aryabhatiya (c. 499 CE), which gave 24-step values to four decimal places. The Kerala School’s tables were used for planetary calculations throughout the 15th to 17th centuries.
The European rediscovery
The arctangent series was independently derived by:
- James Gregory in 1671 using a Taylor expansion approach. Gregory’s series for arctangent appears in a 1671 letter to John Collins.
- Gottfried Leibniz in 1673, who derived the π/4 = 1 − 1/3 + 1/5 − … series using his early calculus methods.
- Isaac Newton’s sine and cosine series in 1665–1666, derived from the binomial series and integration.
Whether the European derivations were truly independent or whether some Kerala results reached Europe through Jesuit channels is contested. The Jesuit College of St Paul at Goa and the Cochin College were active in the 16th and 17th centuries, and Jesuits had access to Indian astronomical work for calendar reform; but no surviving documentary evidence currently shows Kerala mathematical results travelling north. The hypothesis is plausible and remains a live research question; the priority itself is uncontested.
For what it’s worth, on translating the verses
For what it’s worth, the Sanskrit verse form is not a barrier to mathematical content; it is a compression strategy. A single Sanskrit sloka can carry a complete algebraic identity in a way that prose paragraphs cannot. The series for arctangent appears in two verses (vyase varidhinihate, in Sanskrit). For a reader who wants to understand the Kerala results in their own register rather than only through modern translation, the K. V. Sarma critical edition of the Tantrasamgraha provides the verses with detailed commentary; the 2008 Yuktibhasha edition does the same for the longer Malayalam treatise.
Why the Kerala results stayed in Kerala
The Kerala School transmitted its work through Brahmin teacher-student lineages within central Kerala. The treatises were composed in Sanskrit (the technical works) and Malayalam (the pedagogical Yuktibhasha). Outside the Kerala temple-Brahmin scholarly network, the results had no obvious institutional channel to travel to north India or beyond. By the 18th century the school’s institutional vitality had declined, and the manuscripts went into the private holdings of Brahmin families. The 19th century recovery began with Whish’s paper but did not produce significant scholarly engagement until C. M. Whish’s findings were taken up systematically by K. V. Sarma’s school in the 1950s.
Common questions
Is the Madhava-Leibniz series practical for computing π?
Not in its naive form. Setting x=1 in the arctangent series gives π/4 = 1 − 1/3 + 1/5 − 1/7 + …, which converges so slowly that obtaining ten decimal places would require tens of billions of terms. Madhava’s correction formulae substantially improve this. Modern π computation uses much faster-converging algorithms (Machin-like formulae, the Gauss-Legendre algorithm, the Chudnovsky algorithm). The historical importance of the Madhava series is the principle, not the practical efficiency.
Did Madhava also derive the binomial series?
The Yuktibhasha contains expansions related to the binomial series for specific cases needed in the trigonometric derivations, but not the general binomial expansion that Newton derived in 1664. The Kerala work is focused on series for trigonometric and circular functions, not on a general theory of power series.
Who computed π to higher precision after Madhava?
Within the Indian tradition, Madhava’s 11-decimal value remained the standard until colonial period contact with European methods. In Europe, Ludolph van Ceulen reached 35 decimal places in 1610 using a polygon method (Archimedean approach extended to 2⁶² sides). The Machin formula of 1706 enabled faster computation; modern algorithms have computed π to trillions of decimal places.
A limitation worth noting
The “Madhava series” terminology is a modern convention; the surviving Kerala texts attribute the results to Madhava but do not consistently use a single name for the formulae. Some of the results may have been partially developed by Madhava and completed by Parameshvara or Nilakantha; the Yuktibhasha presents the derivations as Kerala School consensus rather than separating individual contributions cleanly. The attribution to Madhava is conventional and supported by the Tantrasamgraha’s citation but should not be read as ruling out collaborative refinement.
For further reading, the Madhava series entry on Wikipedia covers the formulae and historical detail, and the Kerala School entry covers the institutional context. The 2008 two-volume Yuktibhasha translation by K. V. Sarma and colleagues remains the most detailed scholarly source in English.
