Stolen Credit? Transmission Routes and Historical Injustice
How Kerala mathematics may have reached Europe and why credit went elsewhere
Examine the evidence for possible transmission of Kerala mathematics to Europe via Jesuit missionaries and trade routes, and why this history was suppressed.
Three Dates, One Series
In 1671, James Gregory, a Scottish mathematician, was credited with discovering the infinite series for the arctangent. In 1676, Gottfried Leibniz wrote down the same series for arctan(1), giving the famous expansion that yields π divided by four. Three centuries earlier, in roughly 1370, Madhava of Sangamagrama, a Brahmin mathematician working in a small town in Kerala, had already written down both. His verses are preserved in Sankaravariyar's commentary on the Lilavati, in Putumana Somayaji's Karanapaddhati, and in Jyesthadeva's Yuktibhasa, the world's first textbook of calculus, completed around 1530 CE.
The question is not whether Madhava's work came first. It did, by an unambiguous margin. The question is whether Newton and Leibniz invented calculus independently, or whether the foundational ideas reached them through a chain of contact, translation, and transmission that historians of mathematics have until recently chosen not to look at. This lesson is about that question, the evidence on each side, and what an honest history of mathematics would require us to admit.
The Three Tests of Transmission
Historians of science usually ask three questions when investigating possible transmission of an idea from one culture to another. First, was there opportunity? Could one party have reached the other physically, with enough sustained contact for ideas to move? Second, was there motive? Did the receiving party need exactly what the sending party possessed? Third, was there means? Did the receivers have the linguistic, technical, and institutional capacity to translate and absorb the source material?
If all three tests are passed, transmission becomes the simpler explanation than independent rediscovery. Independent discovery of the same complex result by different cultures is possible. It is also rare. When opportunity, motive, and means all line up, a historian who insists on independent discovery is making a stronger claim, not a weaker one.
For the Kerala school and Europe in the period from 1500 to 1700, all three tests are passed.
Opportunity: The Jesuit Foothold in Malabar
In 1498, Vasco da Gama landed at Calicut. Within forty years, the Portuguese had a permanent presence at Cochin, Cranganore, and Quilon, the three port cities of the Malabar coast. From 1540 onward, the Society of Jesus, the Jesuits, established colleges and missions throughout this region. The Jesuit mission to Cochin and the later Madurai mission attracted exactly the kind of personnel needed to absorb a foreign mathematical tradition: trained mathematicians, classical linguists, and men under instructions to learn the local languages thoroughly.
Matteo Ricci, the most famous Jesuit of this generation, was trained at the Collegio Romano in Rome under Christoph Clavius, the chief mathematician of the order. Ricci wrote home from his Asian mission asking specifically for information on the calendar and astronomical knowledge of his hosts, including the Indians. Antonio Rubino learned Malayalam. Johann Schreck (Terrentius), another Clavius student, traveled through India before settling in Beijing. The Jesuit network had men on the ground in Kerala, in Goa, in Cochin, and in the Tamil hinterland for over a century before Newton was born.
The opportunity was not occasional. It was structural and sustained.
Motive: Europe Needed What Kerala Had
In 1582, Pope Gregory XIII issued the bull Inter Gravissimas, replacing the Julian calendar with the Gregorian one. The reform was forced on Europe because the Julian year had drifted too far from the actual tropical year. Performing the reform required an accurate value for the length of the tropical year, and a usable model for predicting the date of Easter centuries into the future. Christoph Clavius, the lead mathematician of the reform, repeatedly admitted that European astronomy alone could not produce the precision needed. He instructed missionaries to gather what they could from the East.
India had this knowledge. The siddhantic tradition had been measuring the tropical and sidereal year to high precision since Aryabhata. The Kerala school of Madhava, Nilakantha, and Jyesthadeva had refined sine and cosine tables to a level of accuracy that European trigonometry would not match for another century. And, critically, Europe in the same period was about to be forced into a new science of navigation. The Portuguese and Dutch fleets needed accurate longitude, which required accurate trigonometry, which required exactly the infinite series for sine and cosine that Madhava had already published.
The receiving end was not merely curious. It was desperate.
Means: Translation, Transcription, and the Quiet Path Home
The Jesuit mission to Madurai, founded by Roberto de Nobili in 1606, eventually included priests fluent in Sanskrit and Tamil who openly engaged with Brahmin scholars. Manuscripts were copied. Astronomical tables were collected. Some of this material was sent back to Rome. The archives of the Jesuit order in Rome, in Lisbon, and in regional collections still contain unpublished material from this period that has never been fully catalogued by historians of mathematics.
The striking fact is that historians did not look. For centuries, the standard history of European mathematics treated the period from 1500 to 1700 as a story of internal European progress, beginning with the rediscovery of Greek geometry and culminating in Newton and Leibniz. The possibility that mathematical content arrived from outside Europe and was quietly absorbed into the work of European mathematicians simply did not appear in the standard textbooks. C.K. Raju, in Cultural Foundations of Mathematics (2007), argues that this absence is itself the evidence. A history that does not even ask whether the Jesuit archives in Rome contain Indian mathematical material is not a neutral history. It is one that has decided, in advance, that the question is not worth asking.
Why Credit Went Elsewhere
There are three reasons the standard history obscured this story for so long. The first is institutional. The history of mathematics as an academic discipline took shape in nineteenth-century Europe, at the height of colonial confidence, when the assumption that India had nothing to teach Europe was simply ambient. Historians did not lie. They mostly did not look. The result is the same.
The second is technical. The verses of Madhava are written in Sanskrit, in a compressed katapayadi-encoded notation, embedded in a culture that did not separate mathematics from astronomy from ritual. To even recognize that a verse contains the arctangent series, a European historian would need Sanskrit, Malayalam, and the technical vocabulary of Kerala astronomy. Few had any one of the three. Almost none had all three until the late twentieth century.
The third is political. Once Newton and Leibniz had become national heroes of England and Germany respectively, the question of whether their calculus had non-European sources became sensitive in a way that made it easier not to ask. The same dynamics that delayed the recognition of Aryabhata's heliocentric tendencies, the rules for zero in Brahmagupta, and the algorithmic origins of algebra in al-Khwarizmi (who himself credited his Indian sources) operated here.
What Honest History Requires
Nothing in this lesson asks for the deletion of Newton or Leibniz. They were extraordinary mathematicians. They built a notation, a synthesis, and a physical theory that transformed science. The honest claim is narrower and stronger. The infinite series at the heart of calculus existed in continuous use in Kerala for two and a half centuries before Newton and Leibniz wrote them down. The opportunity, motive, and means for transmission existed. The archives have not been fully searched. The question deserves to be asked openly.
Madhava-Gregory series. Madhava-Leibniz series. Madhava-Newton power series. These are now the proper names in technical literature, and they are slowly working their way into textbooks. The correction is happening. It is happening because the evidence is too weighty to ignore once anyone bothers to look. What it took was the willingness to look. That is the only thing the Indian mathematical tradition has been asking for, since the Yuktibhasa was first printed in 1948 and finally translated into English in 2008. Look at the verses. Read the dates. Then decide what you call the series.
Key figures
Christoph Clavius
1538 to 1612, Bamberg, Rome
Matteo Ricci
1552 to 1610, Macerata, Macao, Beijing
C.K. Raju
born 1954, India
Case studies
The Jesuit Foothold in Cochin: Opportunity for Transmission
From 1540 onward, the Society of Jesus established colleges and missions throughout the Malabar coast. The Jesuit College of St Paul in Cochin trained generations of priests, including a number who had received mathematical instruction at the Collegio Romano in Rome under Christoph Clavius before being sent to India. Matteo Ricci spent four years in Cochin between 1578 and 1582 before continuing to Macao. Antonio Rubino learned Malayalam. Roberto de Nobili founded the Madurai mission in 1606 and immersed himself in Sanskrit and Tamil literature, openly engaging with Brahmin scholars. By the early 1600s, Jesuits fluent in local languages, with mathematical training and explicit instructions to gather astronomical knowledge, were resident in exactly the cultural region where the Kerala school of Mādhava and Nīlakaṇṭha had been active for two centuries.
The Indian intellectual tradition records the Kerala school in clear paramparā: Mādhava taught Parameśvara, who taught Dāmodara, who taught Nīlakaṇṭha, who taught Jyeṣṭhadeva, with named teachers, named students, dated works, and identified locations. This is not folklore. It is documented knowledge of how mathematical teaching was organized in 14th to 16th century Kerala. The texts of this paramparā were openly available to anyone who could read Sanskrit and Malayalam. The Jesuit mission, by design, included men who could. The cultural and physical opportunity for transmission is therefore not speculation. It is established fact. What the standard history of European mathematics has long avoided is the next question: what did those men do with what they could read?
By 1700, the Jesuit network had been operating in Kerala for over 150 years. Manuscripts had been collected, astronomical tables compiled, and material had been sent back to Rome. The archives of the Society of Jesus in Rome and Lisbon retain unpublished documents from this period. C.K. Raju, George Joseph, and others have called for a systematic search of these archives for Indian mathematical material. As of 2026, the search has been only partially undertaken. The opportunity is documented. The means is documented. The motive (calendar reform, navigation) is documented. What is missing from the standard history is a concerted effort to look in the archives where the answer would most plausibly be found.
When opportunity, motive, and means for transmission of an idea are all in place over a long period, the burden of proof shifts. It becomes harder, not easier, to argue independent rediscovery. The question is no longer whether transmission could have happened, but why historians of mathematics have so consistently chosen not to look for it. Honest history requires looking in the places where uncomfortable answers are most likely to live.
The Jesuit mission to Cochin operated continuously from 1540 onward. Matteo Ricci alone spent four years (1578 to 1582) in Cochin during the height of the Kerala school's activity. Over 150 years of sustained, mathematically literate, linguistically capable European presence in the region preceded Newton's birth in 1643.
The Gregorian Calendar Reform of 1582: Europe's Need for Eastern Astronomy
By the 16th century, the Julian calendar had drifted approximately ten days from the actual position of the equinox. Easter, whose date is tied to the spring equinox and the lunar cycle, was being celebrated at the wrong astronomical moment. Pope Gregory XIII commissioned a reform. The lead mathematician was Christoph Clavius. The technical task was to determine the true length of the tropical year to high precision and to design a leap-year rule that would prevent the drift from recurring. European astronomy of the period could not produce the precision required from its own resources. Clavius openly acknowledged this and instructed missionaries in the Jesuit network to gather astronomical knowledge from their host cultures. The reform that resulted, the Gregorian calendar, used a tropical year value matching values that had been in continuous use in Indian panchanga astronomy since the time of the Sūryasiddhānta.
Indian astronomy had been measuring the tropical and sidereal year for over a thousand years before 1582. The Sūryasiddhānta, the Āryabhaṭīya, and the working panchanga tradition all carried high-precision year values that were checked against observation continuously. The values needed for a calendar reform of the kind Clavius was attempting were sitting on the desks of every panchanga-maker in Kerala and across India. The Kerala school, in particular, had refined astronomical constants further than any Indian tradition before it. The historical question is not whether the values existed in India when Clavius needed them. It is whether he or his students knew where to look. The Jesuit pipeline through Cochin gave him exactly the means to find out.
The Gregorian calendar was issued by papal bull in October 1582. It replaced the Julian calendar in Catholic Europe immediately and was adopted by Protestant and Orthodox countries over the next three and a half centuries. It is the calendar in worldwide civil use today. The astronomical constants embedded in it match values in continuous use in India for centuries before the reform. Whether Clavius obtained them through the Jesuit network or arrived at them by independent calculation is a question that the standard history of European astronomy has rarely asked, despite Clavius's own statements about needing Eastern sources. The convergence is, at the very least, suspicious enough to deserve serious investigation.
When a culture publicly admits that it needs knowledge from elsewhere, has the institutional means to acquire it, and shortly afterward produces work that exactly matches the knowledge it sought, the simplest historical hypothesis is acquisition. Independent rediscovery in such circumstances is the more elaborate claim, not the simpler one. Honest history follows the simpler hypothesis until evidence forces it to abandon it.
The Gregorian calendar of 1582 used a tropical year value of approximately 365.2425 days. The value implicit in Indian panchanga astronomy, in continuous use since the Sūryasiddhānta era, had been refined to comparable precision centuries earlier. Christoph Clavius repeatedly admitted that European astronomy alone could not produce the precision needed and instructed his missionaries to gather material from the East.
The Mādhava-Gregory Series: A Naming Correction in Progress
James Gregory, a Scottish mathematician, published the infinite series for the arctangent in 1671. Gottfried Leibniz wrote down the special case for π in 1676. For the next three hundred years, undergraduate textbooks in Europe and the United States referred to the result as the Gregory series, the Leibniz series, or sometimes the Gregory-Leibniz series. None of these names mentioned that Mādhava of Saṅgamagrāma had stated the same series, in the same form, in roughly 1370. From the 1990s onward, as scholars including K.V. Sarma, Roddam Narasimha, and others published authoritative editions and translations of the Yuktibhāṣā and related Kerala texts, the technical literature began to use the name 'Mādhava-Gregory series'. The renaming has spread unevenly. Specialist papers and modern history of mathematics texts now use it routinely. Most undergraduate textbooks still do not.
The Indian mathematical tradition does not fight for credit by polemic. It fights for credit by publishing the texts. The Yuktibhāṣā, which gives systematic proofs of the series, was first printed in Malayalam in 1948. K.V. Sarma's critical edition appeared in the 1970s. The first complete English translation, by Sarma, Ramasubramanian, Srinivas, and Sriram, was published in 2008 by Hindustan Book Agency and Springer. The renaming of the series is the mathematical world's gradual response to the simple fact that the texts now exist in forms anyone can read. The correction is not ideological. It is bibliographic. Once the verses are visible, the credit follows.
As of 2026, technical mathematical literature increasingly uses 'Mādhava-Gregory series' for the arctangent expansion and 'Mādhava-Leibniz series' for the special case at x = 1. The renaming is reflected in encyclopedic references, in research papers in the history of mathematics, and in some advanced undergraduate texts. Older textbooks have not yet been revised. Mainstream science journalism is mixed. The slow propagation of the corrected name is itself a study in how academic recognition works: it is not an event but a long process of erosion, in which each citation of the new name slightly undermines the old, until finally the new name becomes default. The process is unfinished.
Historical credit is not granted in a single act. It accumulates through small, repeated decisions by scholars, editors, and teachers about which name to use. If you are persuaded that a renaming is justified, the most effective thing you can do is use the corrected name in your own writing, your own teaching, your own conversations. The cumulative effect of such usage is greater than any single argument. Mādhava-Gregory series. Mādhava-Newton power series. The names will become standard when enough people decide to use them.
Mādhava of Saṅgamagrāma stated the arctangent series in approximately 1370 CE. James Gregory rediscovered it in 1671 CE, exactly 301 years later. Gottfried Leibniz wrote the special case in 1676, exactly 306 years later. The complete English translation of the Yuktibhāṣā, with proofs of the series, was published in 2008.
C.K. Raju's 2007 Challenge to the Standard History
In 2007, the Indian mathematician and historian C.K. Raju published Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th c. CE. The book makes a single sustained argument: that the foundational concepts of calculus were transmitted from the Kerala school to Europe via Jesuit missionaries operating in Cochin and the Madurai mission, and that this transmission, rather than independent rediscovery, is the simplest explanation of the historical record. Raju assembled the documentary case in detail. He named the missionaries, identified the relevant Jesuit institutions, listed the manuscripts that had been collected, and pointed to specific archives in Rome and Lisbon that had not been searched. The reception in Western academic history of mathematics was sharply divided. Some scholars engaged with the argument seriously. Others dismissed it as polemic. The book was awarded the Telesio Galilei Academy Gold Medal in 2010.
Raju's contribution belongs to a longer Indian intellectual move that includes George Gheverghese Joseph's Crest of the Peacock (1991), K.V. Sarma's editions of Kerala texts, and the work of historians at the Indian Institute of Science and the Tata Institute of Fundamental Research. Together, these scholars have done what the Indian tradition has always asked for: produced the texts, given the dates, and let the mathematics speak. The dispute over whether transmission occurred is now a question about archival evidence, not about whether the Indian work existed first. That much is settled.
Almost two decades after the publication of Cultural Foundations of Mathematics, the question Raju raised has neither been answered nor dismissed. No systematic search of the Jesuit archives has been undertaken to either confirm or refute his hypothesis. Several of his specific claims have been challenged by historians of European science. Several of his specific claims have not been answered at all. The standard textbook history of calculus continues to treat Newton and Leibniz as independent inventors, while specialist literature treats the question as open. This is an unstable equilibrium. Either the archives will eventually be searched and the question settled, or the standard textbook history will continue to drift further from what specialist scholars know. The corrected version is on its way. It is being delayed, not prevented.
A controversial historical claim that cannot be answered by argument alone must eventually be answered by evidence. The right response to Raju's book is neither acceptance nor rejection in advance. It is to fund the archival search he has called for, in the places he has identified, with the linguistic and technical competence the search requires. Until that search is undertaken, the dispute is not really a dispute. It is a polite agreement not to look.
C.K. Raju's Cultural Foundations of Mathematics was published in 2007 by Pearson Education and Pearson Longman. It was awarded the Telesio Galilei Academy Gold Medal in 2010. As of 2026, no systematic archival search of the Jesuit collections in Rome and Lisbon for Indian mathematical material has been completed.
Historical context
From the Kerala School to the European Calculus Crisis (1370 to 1700 CE)
The Kerala school flourished in a region of dense temple towns, brahmin settlements (such as Saṅgamagrāma, Tṛkkaṇṭiyūr, and Ālattūr), and active Sanskrit and Malayalam literary culture. The school's mathematical paramparā ran continuously from Mādhava in the late 14th century through Acyuta Piṣāroṭi and Putumana Somayājī in the 18th century. Throughout this period, manuscripts were copied, taught, and commented upon. The texts were not hidden or esoteric. They were accessible to anyone who could read the languages, which by 1600 included a substantial number of Jesuit missionaries working in the same region.
The history of mathematics is also the history of who is allowed to be remembered as a mathematician. For three centuries, the standard story of calculus had room for Newton and Leibniz but not for Mādhava, Nīlakaṇṭha, or Jyeṣṭhadeva. This was not because the Kerala work was unknown to specialists. It was because the question of whether it had reached Europe had not been seriously asked. To restore the missing names is not to subtract from the Europeans. It is to admit a longer and more accurate story.
Living traditions
The renaming of the arctangent series as the Mādhava-Gregory series, the arctangent special case as the Mādhava-Leibniz series, and the Taylor series for sine and cosine as the Mādhava-Newton series is in active progress in modern technical literature. Specialist journals, advanced textbooks, and reference works increasingly use the corrected names. The K.V. Sarma Research Foundation continues to publish critical editions of Kerala school texts. The first complete English translation of the Yuktibhāṣā, by Sarma, Ramasubramanian, Srinivas, and Sriram, was published in 2008 and has made the Kerala school's proofs available to the global mathematical community for the first time. C.K. Raju's Cultural Foundations of Mathematics (2007) and George Gheverghese Joseph's Crest of the Peacock (in its expanded 2011 edition) are now standard references in any serious modern history of mathematics. The correction of the historical record is unfinished, but it is no longer in doubt that it will be completed. The verses exist. The dates are fixed. The mathematics speaks for itself.
- Saṅgamagrāma (Irinjalakuda) and Tṛkkaṇṭiyūr: Irinjalakuda, identified with the ancient Saṅgamagrāma, is the traditional birthplace of Mādhava. Tṛkkaṇṭiyūr, near Tirur in Malappuram district, is where Nīlakaṇṭha Somayāji completed the Tantrasaṅgraha in 1500 CE. Both villages remain quiet temple towns with active Brahmin settlements. There are few formal monuments to the Kerala school. What survives is the living memory of the towns themselves and the manuscripts collected by the K.V. Sarma archives. A visit is less a museum experience than a pilgrimage to an unmarked but real place where calculus was first written down.
- St Paul's College Site, Cochin (Fort Kochi): Fort Kochi was the centre of Jesuit activity in Kerala from 1540 onward. The Santa Cruz Cathedral Basilica and St Francis Church (where Vasco da Gama was originally buried) date from this period. The Jesuit College of St Paul, where Matteo Ricci spent four years before continuing to Macao, no longer survives as a single building, but the site and the surviving Portuguese-era churches give a vivid sense of the European foothold from which the transmission story would have unfolded. Walking the old streets of Fort Kochi is the closest you can come to standing on the ground where Kerala mathematics could have crossed into European hands.
Reflection
- Have you ever been credited for an idea that came from someone else, or seen someone else credited for an idea that was yours? How did the misattribution feel, and what would have made it right?
- Why do you think the standard history of mathematics found it so difficult, for so long, to ask the simple question of whether Indian work had reached Europe?
- If a piece of knowledge survives by being used, does it matter who is credited for inventing it? Is the credit a matter of justice, of accurate history, of identity, or all three?