The Kerala School of Mathematics: Madhava and the Roots of Calculus

Open any IIT-JEE preparation book. Flip to the chapter on infinite series. The first formula you meet is the expansion for arctan, often credited to James Gregory in 1671 or Gottfried Leibniz a few years later. Almost no Indian student is told that the same series appears, with proof, in a Sanskrit verse composed in a Kerala village around the year 1400. The mathematician's name was Madhava of Sangamagrama. He computed pi to eleven decimal places using methods that would not appear in Europe for nearly three centuries. His work was preserved in palm-leaf manuscripts written in Sanskrit and Malayalam, copied by hand, used by a small lineage of astronomer-priests, and then forgotten by the wider world until the nineteenth century. This is the story of the Kerala School: a mathematical tradition that built the analytical heart of calculus while Europe was still working out long division.

The Kerala School did not appear out of nowhere. It stood on a thousand years of mathematical building. Aryabhata, born in 476 CE, wrote the Aryabhatiya at age twenty-three. In just thirty-three verses of the Ganitapada, he set down algebra, geometry, trigonometry, and a value of pi accurate to four decimal places. Brahmagupta, in the seventh century, gave the world the first systematic rules for arithmetic with zero and negative numbers. Bhaskara I refined Aryabhata's sine table. Bhaskara II, writing in 1150 from the observatory at Ujjain, took the next step: he handled instantaneous rates of change, recognized that a function's derivative vanishes at its maximum, and stated what amounts to an early form of Rolle's theorem. By the time Madhava picked up the thread two hundred years later, the conceptual ground had been laid. Indian mathematics had moved from arithmetic to algebra, from algebra to trigonometry, from trigonometry to the brink of analysis. Madhava walked across that brink first.

What stands out in this verse is not just the number. The formula 62832 over 20000 is not a guess from measuring strings on circles. It comes from a computational method, possibly involving polygon-based approximation, that Aryabhata does not write down explicitly but his commentators reconstruct from his other verses. Equally important is the choice of word. He calls his pi value asanna, meaning close to, near, approaching. Other Indian and Greek mathematicians of the period used words meaning equal to or exactly. Aryabhata's word choice has led several historians to argue that he understood pi was incommensurable -- that is, irrational. The formal European proof came from Johann Lambert in 1761. Aryabhata's hint, if it was a hint, was twelve hundred years early. He also gave a sine table accurate to four decimal places, used a place-value decimal system that included zero, and proposed that the earth rotates on its axis -- a claim Aryabhata I made in the fifth century, and one Western astronomy would not accept until Copernicus in the sixteenth.

Six hundred years after Aryabhata, Bhaskara II took up the question of motion. In the Grahaganita section of his Siddhanta Shiromani, he asked a question that today every physics student answers without thinking: what is the speed of a planet at one specific instant? Average speed over a day is easy. You take the change in position and divide by the time. But the speed at this exact moment, this single instant, is harder. You cannot divide by zero. Bhaskara built a concept he called tatkalika gati, meaning instantaneous motion. He compared it with sthula gati, the gross or average motion, and sukshma gati, the fine motion. He showed that for a small change in angle, the change in sine is approximately the cosine of the angle multiplied by that small change -- which in modern notation reads as the derivative of sine equals cosine. He stated that the maximum value of a function occurs where its rate of change vanishes. This is differential calculus in everything but name. The name came from Newton and Leibniz five centuries later.

Around 1340, in a small village near present-day Cochin in Kerala, a Brahmin boy named Madhava was born into a Namboothiri family. We know almost nothing about his life. His own writings have been mostly lost. What we do know comes from later mathematicians of his lineage who quote him by name. Madhava founded what historians today call the Kerala School of Astronomy and Mathematics. The school continued for nearly two hundred years through his students and their students: Parameshvara, Damodara, Nilakantha Somayaji, Jyeshthadeva, Achyuta Pisharati, Shankara Variyar. They worked in Sanskrit and Malayalam, attached themselves to local temples and patron families, and produced a body of mathematical work that today's Princeton historian Kim Plofker has called one of the most important developments in pre-modern mathematics anywhere in the world. They derived infinite series for sine, cosine, and arctangent. They computed pi to eleven decimal places. They used something equivalent to integration of polynomials. They stated the mean value theorem. And they did all of it as a side activity to their main job: computing planetary positions for Hindu astronomical calendars used in temple rituals across Kerala.

Madhava's most famous result is the infinite series for pi divided by four: pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... and so on, alternating forever. Today this is taught in college mathematics as the Gregory-Leibniz series, named for the Scottish mathematician James Gregory who derived it in 1671 and the German philosopher Gottfried Leibniz who published a version in 1674. Madhava had it around 1400 -- nearly three hundred years before either European. In honest mathematical history today, the formula is increasingly called the Madhava-Gregory-Leibniz series. Madhava also derived the related series for arctangent: arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ... He used these series to compute pi to eleven correct decimal places: 3.14159265359. Compare this with Archimedes who reached three decimal places, and Ludolph van Ceulen of the Netherlands who reached thirty-five decimal places in 1596 using polygons of more than four billion sides. Madhava reached his eleven decimals not through the brute force of polygon counting but through an analytical method that converged faster as you took more terms. This was the conceptual leap. Pi was no longer something you measured. It was something you computed.

If the Kerala School achieved so much, why did the world not know about it for centuries? Three reasons. First, the texts were in Sanskrit and a medieval form of Malayalam. Yuktibhasa, the most important calculus text written by Jyeshthadeva around 1530, was in Malayalam prose -- not the standard scholarly Sanskrit that Indian and foreign scholars could read. Until the 1830s when Charles Whish, an East India Company magistrate stationed in Cochin, published the first English-language paper on the Kerala School, no Western mathematician had any way to access these results. Second, the school was geographically isolated. Kerala in the fourteenth and fifteenth centuries was not Banaras or Nalanda. It was a coastal region with strong local Brahmin networks but limited circulation of texts to North India, let alone the Islamic world or Europe. Third, the work was preserved as oral guru-shishya tradition. Madhava's own writings were largely lost. We know about him because Nilakantha Somayaji wrote a detailed commentary called Tantrasangraha in 1501 that quotes Madhava by name and credits him with specific formulas. Without Nilakantha's careful citation practice, Madhava might have been forgotten entirely.

There is a debate that has heated up in recent decades. Did Kerala mathematics reach Europe before Newton and Leibniz formalized calculus? The case for transmission rests on several pieces of circumstantial evidence. Jesuit missionaries arrived in Kerala in the sixteenth century. They learned local languages including Malayalam. Some Jesuits had mathematical training. The Collegio Romano in Rome was a major hub for both Jesuit mathematicians and the wider European mathematical correspondence network managed by Father Marin Mersenne, a friend of Descartes and Fermat. Cavalieri, Fermat, Pascal, and Gregory all worked on infinite series in the same decades that Jesuits returning from Kerala were filing reports in Rome. The timing fits. The motivation existed -- Europe needed accurate trigonometric tables for ocean navigation, exactly what Kerala mathematicians had perfected. The argument against transmission is equally weighted. No specific Kerala manuscript has been found in any European archive predating Whish's 1830s discovery. No European mathematician of the seventeenth century cites a Kerala source by name. The historian David Bressoud put it directly: there is no evidence that Indian work on series was known beyond India until the nineteenth century. Both positions are held by serious scholars. The transmission question remains open. The Eternal Raga position is honesty: Madhava's priority is established, transmission is plausible, transmission is not proved.

The Kerala School's legacy is alive in India today, though most Indians have never heard the name Madhava. Manjul Bhargava, born in Hamilton, Ontario to Indian parents, won the Fields Medal in 2014 -- mathematics' Nobel equivalent -- partly for work that revisited eighteenth-century European number theory using methods that have a Madhava-like flavour of computational craft. The Indian Mathematical Olympiad sends teams to compete internationally where infinite series problems regularly appear. ISRO's calculations for Chandrayaan-3's lunar landing in August 2023 used the same arctan series Madhava derived in 1400, now embedded inside standard floating-point arithmetic libraries that every Indian software engineer at TCS or Infosys uses without thinking. The IIT-JEE Advanced exam regularly tests problems on infinite series convergence -- the same mathematical territory the Kerala School pioneered. When you watch the next ISRO launch from Sriharikota, when your friend cracks JEE Advanced, when you see an Indian win a Fields Medal, the line back to Kerala matters. Not because we need to claim ownership over modern mathematics. We do not. But because the modern Indian mathematical tradition is not a borrowed thing. It has indigenous roots that go back at least fifteen hundred years.

How should an honest Indian student think about the Kerala School in 2026? Three principles. First, celebrate the genuine priority. Madhava beat Gregory and Leibniz to the infinite series for pi by nearly three centuries. Bhaskara II beat Newton to the basic ideas of differentiation by five centuries. These are not nationalist boasts. They are facts now accepted in mainstream history of mathematics. Second, drop the conspiracy framing. Whether Jesuits stole the work and brought it to Europe is unproved. The honest position is that transmission is possible, debated, and not necessary to make Indian mathematics important. The achievements stand on their own. Third, do not collapse into the opposite trap of dismissing what was done. The Kerala School did not invent calculus in the full Newton-Leibniz sense -- the unification of differentiation and integration as inverse operations is genuinely a European synthesis. But the Kerala School did derive the analytical core of calculus, the infinite series for transcendental functions, centuries before Europe. Both things can be true. A mature reading of mathematical history allows for both. The world is full of independent discovery, partial transmission, parallel development, and ideas that travel on routes we cannot fully reconstruct.

Among all the texts of the Kerala School, one stands out: the Yuktibhasa, written around 1530 by Jyeshthadeva. It is an unusual document for several reasons. Most Indian mathematical texts of the medieval period were composed in compressed Sanskrit verse, built for memorization but cryptic to read. Yuktibhasa broke that pattern. It is written in Malayalam prose, in flowing readable sentences, and it does something almost no earlier Indian mathematical text had done: it provides full proofs. Where Madhava's verses state results, Yuktibhasa shows the derivations step by step. The text derives the infinite series for sine, cosine, and arctangent in full. It presents what amounts to the integration of polynomial functions. It explains the convergence properties of these series. Some modern historians of mathematics now call Yuktibhasa the first systematic textbook of calculus written anywhere in the world -- a full century and a half before Newton's Principia Mathematica. The text was unknown to the West until Whish translated portions of it in the 1830s. The original palm-leaf manuscripts are still preserved in libraries across Kerala, and a complete English translation by K.V. Sarma was published only in 2008. For centuries the world's first calculus textbook sat in plain sight, waiting to be read.

Walk into any temple in Kerala today -- the Vadakkunnathan in Thrissur, the Krishna temple at Guruvayur, the small Shiva shrines that dot the backwaters. The astronomer-priests who computed planetary positions for those temples were the inheritors of Madhava's tradition. The mathematics was never separate from the worship. Computing pi was an act of careful attention, of ganita -- the same Sanskrit word that gives us today's word for mathematics. The Kerala School worked because their patrons valued precision in calendar-making, because their students copied texts on palm leaves with unusual care, because the gurukul system kept the chain unbroken for two centuries. None of this was framed as a race against Europe. Madhava did not know Newton would be born. The work was its own reward. There is something to learn here for anyone studying mathematics today, anyone preparing for JEE or competitive exams or college math: the deepest discoveries often come from people who were not chasing fame, not chasing priority, not chasing a Western audience. They came from people who looked carefully at the sky, asked precise questions, and computed. Madhava is the patron saint of that quiet, patient, accurate way of working.