Madhava: The Kerala Mathematician Who Invented Calculus 200 Years Before Newton

In 1666 Isaac Newton invented calculus.

This is what most of the world was taught and what most of the world still believes.

There is a problem with this version of events.

A mathematician in Kerala had already done it. In the 14th century. Using infinite series expansions for trigonometric functions that are so mathematically sophisticated that Cambridge University Professor of Mathematics George Gheverghese Joseph has stated that the priority of Kerala developments in the calculus over that of Newton and Leibniz is now beyond doubt.

His name was Madhava of Sangamagrama. He was born around 1340 CE in a small town near what is now Thrissur in Kerala. He died around 1425 CE, more than two centuries before Newton was born. And the mathematical tradition he founded in Kerala, the Kerala School of Astronomy and Mathematics, produced infinite series expansions for sine, cosine, tangent and arctangent that are today formally known by the names of the European mathematicians who rediscovered them centuries after Madhava had already solved them.

Some of the infinite series discovered by Madhava were rediscovered by European mathematicians about two to three centuries later. Some of such infinite series discovered by Madhava are the Madhava-Gregory series, the Madhava-Newton series, the Madhava-Leibniz series, the Madhava-Gregory-Leibniz series, and the Madhava-Taylor series.

Madhava-Newton. Madhava-Leibniz. Madhava-Gregory. Madhava-Taylor.

The names that appear in every university mathematics syllabus worldwide contain the name of a 14th-century Kerala mathematician as the first author. And almost nobody outside the mathematical history community knows this.

But the story does not end with the mathematics. It ends with a question that historians of science are still actively debating.

Did Madhava’s work reach Europe? Did the Jesuit missionaries who were active on the Kerala coast in the 15th and 16th centuries, trading from the ancient port of Muziris near Madhava’s own birthplace, carry the Kerala calculus back to Europe in their manuscripts and letters? Did Newton and Leibniz build on mathematical foundations that an Indian mathematician had already laid?

The answer is not yet proven. But the circumstantial evidence is extraordinary.

This is the complete story of Madhava of Sangamagrama.

Madhava the Kerala Mathematician: Who Was He and What Did He Discover

Who Was Madhava of Sangamagrama and How Does His Work Build on Bhaskara II

Madhava of Sangamagrama was a mathematician and astronomer from the town of Sangamagrama, believed to be present-day Aloor, Irinjalakuda in Thrissur District, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity.

Madhava did not work in isolation from the Indian mathematical tradition that preceded him. Two centuries earlier, Bhaskara II of Bijapur in Karnataka had described specific foundational concepts of calculus in his Siddhanta Shiromani of 1150 CE. Bhaskara II demonstrated instantaneous velocity, described functions approaching limits in specific astronomical calculations and applied what we now recognise as differential calculus techniques to solve specific problems in planetary motion. He demonstrated calculus ideas in application, using the concepts brilliantly within the context of particular problems.

What Madhava did was take the decisive step further. He studied and explicitly extended the work of Bhaskara II and formalised the underlying mathematical machinery into a complete and generalisable system. Where Bhaskara II had used calculus concepts in specific applications, Madhava built the infinite series framework that allows those concepts to be applied universally to any trigonometric function. Bhaskara II described calculus concepts in application. Madhava formalised calculus as a complete mathematical system. The two traditions, Karnataka and Kerala, are sequential chapters of the same extraordinary story rather than competing claims and together they represent the most significant contribution to the development of calculus made anywhere in the world before Newton.

Little is known about Madhava’s personal life with certainty. His house name was associated with the Bakula tree, the Spanish Cherry, which grew abundantly in his village. He taught mathematics through a tradition of oral transmission, which is one reason why so much of his original work exists only through references in the writings of his students and their students rather than in his own surviving manuscripts.

Although almost all of Madhava’s original work is lost, he is referred to in the work of later Kerala mathematicians as the source for several infinite series expansions including the sine, cosine, tangent and arctangent functions and the value of pi, representing the first steps from the traditional finite processes of algebra to considerations of the infinite, with its implications for the future development of calculus and mathematical analysis.

The loss of most of Madhava’s original manuscripts is one of the great tragedies in the history of science. Palm leaf manuscripts in the Kerala climate are vulnerable to humidity, insects and the passage of time. What survived did so because Madhava’s students and their students copied, quoted and referenced his work in their own writings, preserving the mathematical content if not always the original form.

What those references reveal is a mind of extraordinary originality that took the decisive step, in the words of the MacTutor History of Mathematics at St Andrews University, onwards from the finite procedures of ancient mathematics to treat their limit-passage to infinity. This is the step that defines modern mathematical analysis. And Madhava took it in 14th-century Kerala.

What Is an Infinite Series and Why Madhava’s Discovery Changed Mathematics Forever

To understand what Madhava discovered it helps to understand what an infinite series is and why it matters.

A finite mathematical procedure gives you an exact answer using a fixed number of steps. The area of a rectangle is length times width. Two steps. Exact answer.

But some quantities cannot be calculated exactly using a finite number of steps. The value of pi. The sine of an angle. The arctangent of a number. These are irrational quantities that can be approached with increasing precision but never reached exactly by any finite calculation.

Madhava’s insight was that these quantities could be expressed as infinite series, sums of infinitely many terms that get progressively smaller, such that adding more and more terms produces an answer of progressively greater precision. The series never terminates. But it converges to the exact value as its limit.

This is the foundational concept of calculus. And it is what Madhava was doing in Kerala in the 14th century, building on the calculus concepts that Bhaskara II had demonstrated in Karnataka two centuries earlier and extending them into a formal universal system.

Madhava is credited with discovering infinite series expansions for trigonometric functions such as sine, cosine, and arctangent. These series, known as Madhava series or Kerala series, laid the foundation for later developments in calculus.

The specific series that Madhava discovered for pi is particularly celebrated. It states that pi divided by four equals one minus one third plus one fifth minus one seventh plus one ninth and so on indefinitely, each term being the reciprocal of a successive odd number with alternating signs. This series, which allows the value of pi to be calculated to any desired degree of precision by simply adding more terms, is today known as the Madhava-Leibniz series. Leibniz independently discovered it in 1676. Madhava had already derived it in the 14th century.

Madhava used the infinite series formula to evaluate the value of pi correctly to 11 decimal places as 3.14159265359.

Eleven decimal places of pi. In 14th-century Kerala. Using infinite series methods that European mathematics would not independently develop for another two to three centuries.

Madhava the Kerala Mathematician and the School That Changed the World

How Madhava Founded the Kerala School of Mathematics and Astronomy

Madhava, sometimes called the greatest mathematician-astronomer of medieval India, came from the town of Sangamagrama in Kerala near the southern tip of India and founded the Kerala School of Astronomy and Mathematics in the late 14th century.

The Kerala School that Madhava founded continued to produce world-class mathematics for over two hundred years after his death, an extraordinary institutional legacy for a tradition rooted in a small town in what is now the Thrissur district of Kerala.

The school’s method was the guru-shishya tradition, the direct transmission of knowledge from teacher to student across generations. Madhava’s students included Parameshvara who developed the circumradius formula for cyclic quadrilaterals. Parameshvara’s student was Nilakantha Somayaji who wrote the Tantrasangraha, one of the most important astronomical texts in Indian history, in which he attributes multiple mathematical discoveries to Madhava. Nilakantha’s students continued the tradition through the 16th century, maintaining the mathematical heritage that Madhava had established and extending it into new territory.

The Kerala School was well known in the 15th and 16th centuries in the period of the first contact with European navigators on the Malabar Coast. The school’s fame extended throughout the Kerala region and its mathematical results were discussed and built upon by every serious mathematician working in South India during this period.

The Mathematical Series That Should Carry Madhava of Sangamagrama’s Name Alone

The mathematical series that most clearly illustrates the gap between what Madhava achieved and what he has been credited with is the Gregory-Leibniz series for the arctangent function.

This series states that the arctangent of x equals x minus x cubed over three plus x to the fifth power over five minus x to the seventh power over seven and so on. It is one of the foundational results of calculus, used in the calculation of pi and in numerous applications in physics and engineering.

James Gregory, a Scottish mathematician, published this series in 1671. Gottfried Leibniz published it independently in 1676. Both received credit. The series is taught in every university calculus course in the world as the Gregory-Leibniz series.

This is wrongly attributed to James Gregory who discovered it three centuries after Madhava. Madhava laid the foundations for the development of calculus which were further developed by his successors at the Kerala school of astronomy and mathematics.

Three centuries after Madhava. The series that every mathematics student encounters as the Gregory-Leibniz series was known in Kerala three hundred years before either Gregory or Leibniz was born.

The mathematical historians who have worked on the Kerala School’s manuscripts over the past twenty-five years have been careful not to overstate the case. The attribution question is complex and the transmission question, whether European mathematicians knew of Madhava’s work, remains unresolved. But the priority question is not complex at all. Madhava was there first. By a very long way.

Did Madhava the Kerala Mathematician’s Calculus Travel to Europe Via the Jesuits

How Madhava of Sangamagrama’s Kerala Mathematics May Have Reached Europe Through Jesuit Missionaries

Here is the question that makes Madhava’s story more than a story about mathematical priority. It makes it a story about the transmission of knowledge across cultures, about the routes by which ideas travel and about what we do and do not know about the origins of the modern scientific tradition.

Some scholars have also suggested that Madhava’s work, through the writings of the Kerala School, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time. As a result, it may have had an influence on later European developments in analysis and calculus.

The ancient port of Muziris, near Madhava’s own birthplace of Sangamagrama in the Thrissur district of Kerala, was one of the most commercially active ports on the Indian Ocean coast in the 15th and 16th centuries. Jesuit missionaries arrived in Kerala in the 16th century. They were extraordinarily well educated men, often with strong mathematical backgrounds. They were in active communication with the leading mathematical centres of Europe. And they were working in the same region where the Kerala School’s mathematical manuscripts were circulating among scholars.

The priority of Kerala developments in the calculus over those of Newton and Leibniz is now beyond doubt. Madhava is credited with the original ideas in the Kerala mathematics. These ideas led to derivation for the infinite series for pi and to infinite series for a range of trigonometric functions.

The question is not whether Madhava’s mathematics preceded Newton and Leibniz. That priority is now beyond doubt. The question is whether Newton and Leibniz knew about it. Whether the Jesuit missionaries who studied Sanskrit manuscripts in Kerala, who had both the mathematical education to understand what they were reading and the communication networks to transmit it to Europe, were the channel through which Madhava’s infinite series reached the mathematicians of 17th-century England and Germany.

The honest answer is that this has not been definitively proved. The circumstantial evidence is substantial. The mathematical similarities between Kerala school results and early European calculus are striking. The communication routes existed. The Jesuits had both the motivation and the capability.

But the documentary evidence of direct transmission, the letter, the annotation, the manuscript that says specifically here is this Indian series, has not yet been found.

It may still be sitting in a Jesuit archive somewhere in Europe. Unread. Unremarked. Waiting.

What Madhava’s Pi Calculation Tells Us About the Sophistication of 14th-Century Kerala Mathematics

The calculation of pi to eleven decimal places by Madhava in the 14th century is the single most concrete demonstration of the sophistication of Kerala mathematics available to a non-specialist reader.

Pi to eleven decimal places is 3.14159265359.

Modern GPS systems require pi to approximately fifteen decimal places to function accurately. The eleven decimal places that Madhava calculated in the 14th century are sufficient for almost every practical application of pi in engineering and physics.

The method Madhava used to achieve this precision, the infinite series for pi divided by four, requires adding a large number of very small terms with alternating positive and negative signs and understanding precisely how the series converges as more terms are added. This requires a complete and formal understanding of the concept of a limit, which is the conceptual foundation of calculus.

Madhava had this understanding. In 14th-century Kerala. In a small town near Thrissur. Teaching his students through the oral tradition of the guru-shishya relationship. Using palm leaf manuscripts that the Kerala monsoon would eventually destroy.

The GPS satellite that helped you navigate to wherever you are reading this right now is using mathematics whose foundational concepts were worked out in Kerala six hundred years ago.

Experience Madhava the Kerala Mathematician’s Heritage With 5 Senses Tours

Irinjalakuda Thrissur: Visiting the Birthplace of Madhava of Sangamagrama and Modern Calculus

Madhava’s birthplace of Sangamagrama is believed to be present-day Irinjalakuda in the Thrissur district of Kerala, approximately 25 kilometres north of Thrissur city and approximately 55 kilometres north of Kochi.

The Thrissur district of Kerala is one of the most culturally extraordinary landscapes in South India. The ancient port of Muziris, through which Madhava’s mathematical ideas may have travelled to Europe, is accessible from the same district. The extraordinary Vadakkumnathan Temple at Thrissur, one of the oldest Shiva temples in Kerala, was a functioning centre of Sanskrit learning and astronomical observation in Madhava’s era. The Kerala backwaters, the ancient spice trade routes and the complete heritage of the Malabar Coast converge in this single district.

5 Senses Tours takes international travellers to the complete Kochi and Kerala heritage experience, connecting the ancient port of Muziris and its extraordinary Roman and Arab trading history to the living cultural traditions of the Kerala coast that Madhava inhabited. If this story has moved you, the landscape where Madhava worked out the infinite series for pi on palm leaf manuscripts in the 14th century is the same landscape you look out over from a houseboat on the Kerala backwaters today. Plan your Kerala heritage experience with 5 Senses Tours.

The Muziris Connection: Ancient Kerala’s Most Extraordinary Heritage Landscape

The ancient port of Muziris, through which Madhava’s mathematical ideas may have travelled to Europe via Jesuit missionaries, is one of the most historically significant sites in the entire Kochi and Kerala heritage circuit.

Madhava of Sangamagrama’s writings were later transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Muziris at the time.

The Pattanam archaeological site near Kodungallur in the Thrissur district, identified as the probable location of ancient Muziris, has produced Roman amphorae, Mediterranean glass beads and coins confirming the extraordinary commercial connections of this Kerala coast two thousand years ago. The same port that Roman merchants used to import Indian pepper in the 1st century CE was the port through which Jesuit missionaries may have exported Indian mathematics to Europe in the 16th century.

The complete Muziris heritage circuit, the archaeological site at Pattanam, the ancient synagogue at Kodungallur, the Cheraman Mosque which tradition holds is the oldest mosque in India and the extraordinary layered heritage of the Thrissur district, creates one of the most intellectually rich heritage experiences available anywhere in South India for an international traveller who arrives with the complete story of what happened here.

Our Kochi tours cover the complete Muziris heritage circuit alongside the Jewish, Portuguese, Dutch and British heritage of Fort Kochi, creating a complete encounter with the most cosmopolitan trading landscape in the history of Asian commerce and the landscape through which Madhava’s calculus may have changed the world.

The Complete Kerala Mathematical Heritage Journey

For travellers who want to experience the complete arc of the Kerala mathematical tradition, 5 Senses Tours can create a customised private heritage journey through the landscape where Madhava lived, taught and made the discoveries that changed the history of mathematics.

The journey covers Irinjalakuda and the Thrissur district where Madhava was born and where the Kerala School of Mathematics flourished, the Pattanam Muziris archaeological site through which his ideas may have travelled to Europe, the Vadakkumnathan Temple at Thrissur which was an active centre of Sanskrit learning in his era and the complete Fort Kochi heritage of the most cosmopolitan port city in the history of the Indian Ocean.

For travellers who want to experience the complete story of Indian mathematical genius from its Karnataka origins to its Kerala culmination, our Aurangabad tours cover the Deccan heritage of the Rashtrakuta dynasty that patronised Mahavira and produced the cultural tradition that Bhaskara II of Bijapur inherited. The Karnataka and Kerala mathematical traditions together form the most extraordinary continuous story of mathematical achievement in the pre-modern world.

Our Kochi tours hub covers the complete Kerala heritage experience with expert cultural guides who bring the full depth of the Madhava story to life at the physical landscapes where it happened.

5 Senses Tours is recognised by India’s Ministry of Tourism, winner of the Tripadvisor Travellers Choice Award and the Outlook Responsible Tourism Award. Every tour is private, expert-guided and completely customised for your group.

Talk to a 5 Senses Tours expert about your Kerala mathematical heritage experience today. Contact us and we will build the journey around your specific interests, travel dates and time available.

For travellers who want to experience the complete living culture of this extraordinary coast alongside its ancient mathematical heritage, our Fort Kochi heritage walk and our Kochi food walk bring the Jewish, Portuguese, Dutch and British layers of the most cosmopolitan port city in Asian history to life on foot, one historic street and one extraordinary dish at a time.

Contact 5 Senses Tours to begin planning your Kerala heritage journey today