Indian mathematician and astronomer (c. 1340 – c. 1425 CE), founder of the Kerala School of Astronomy and Mathematics. Discovered infinite series expansions for π, sine, and cosine roughly 250 years before their independent rediscovery in Europe by Gregory, Newton, and Leibniz. None of Mādhava's own writings survive directly — his work is known through the writings of his successors in the Kerala School.
Mādhava lived in Kerala, on the southwestern coast of India, in the late 14th and early 15th centuries. He founded a remarkable tradition of mathematics and astronomy — now called the Kerala School — that produced results usually credited to much later European mathematicians.
Mādhava's own writings have not survived directly. We know about his results only because later mathematicians in the Kerala School — particularly Nīlakaṇṭha Somayājī (c. 1500 CE) and Jyeṣṭhadeva (in the Yuktibhāṣā, c. 1530 CE) — explicitly credited these discoveries to Mādhava while recording them in their own texts. Western mathematical historiography, developing largely independently and without full awareness of the Kerala School's work until the 19th–20th centuries, ended up naming the results after the later, independent European discoverers.
Mādhava also found infinite series for the sine and cosine functions — series we now know as Taylor series (rediscovered by Brook Taylor in 1715) for these specific functions, again roughly three centuries ahead of the general Western theory. He also devised improved, rapidly-converging approximations for π, achieving accuracy far beyond his predecessors.
| Active | c. 1340 – c. 1425 CE |
| Location | Saṅgamagrāma (near modern Irinjalakuda), Kerala, India |
| School founded | Kerala School of Astronomy and Mathematics |
| Key successors | Parameśvara, Nīlakaṇṭha Somayājī, Jyeṣṭhadeva |
This series converges very slowly (needing hundreds of terms for even modest accuracy), so Mādhava also developed correction terms to accelerate convergence dramatically — a sophisticated numerical technique that modern historians consider genuinely impressive by any standard.
The π series above is the special case x = 1. This general arctangent series, in Europe, is credited to James Gregory (1671).
These are precisely the modern Taylor series for sine and cosine, derived by the Kerala School using geometric and iterative arguments rather than the differential calculus framework Taylor would later use.
Using his accelerated series, Mādhava computed π to 11 decimal places — 3.14159265359 — a remarkable achievement for the era, surpassing all previous known approximations from any civilisation up to that point.
No original manuscript by Mādhava survives. Our knowledge comes entirely from later Kerala School texts that explicitly attribute specific results to him: Nīlakaṇṭha's Tantrasaṅgraha (c. 1500 CE) and, most importantly, Jyeṣṭhadeva's Yuktibhāṣā (c. 1530 CE, written in Malayalam rather than Sanskrit — unusual for a mathematical text of this era), which provides detailed derivations ("yukti," meaning rationale or demonstration) of the series, going well beyond simply stating results and instead showing extended reasoning resembling a form of proof.
The derivations in the Yuktibhāṣā involve summing infinitesimal quantities in ways that bear a structural resemblance to integration, leading some historians (most prominently C.K. Raju, 2007) to argue the Kerala School developed something functionally equivalent to calculus centuries before Newton and Leibniz. This claim remains contested: other historians of mathematics (Plofker 2009; Katz 1995; Bressoud 2002) acknowledge the genuine sophistication and originality of the Kerala School's methods while arguing that a general, unifying theory comparable to European calculus — with a clear concept of a derivative as a limit, applicable to arbitrary functions, and a Fundamental Theorem connecting differentiation and integration — is not present in the surviving texts. The debate concerns how much conceptual generality and abstraction is required to constitute "calculus" versus sophisticated special-case techniques, and remains an active area of historical scholarship rather than a fully settled question.
Some historians (Raju, 2007) have speculated that Jesuit missionaries active in Kerala in the 16th–17th centuries might have transmitted knowledge of these series to Europe, potentially influencing later European discoveries. This transmission hypothesis remains speculative and is not accepted by mainstream historians of mathematics (Plofker and others regard the European rediscoveries as independent), given the lack of direct documentary evidence of such transmission and the plausibility of independent discovery using different methods on both sides.
Only since detailed scholarly work in the 19th and especially 20th century (beginning with C.M. Whish's 1835 paper drawing British attention to the Kerala texts, and substantially developed by later historians including Rajagopal, Sarma, and Plofker) has Mādhava's priority for these results become widely recognised in the international history of mathematics community.