A domain of mathematics, believed to have emerged around 600 years ago, in settlements, perhaps temples along Kerala’s main river, Bharatapuzha, continues to surprise maths historians around the world. This branch of mathematics, similar to modern-day calculus, took birth, advanced over a span of some 300 years, and faded away by the 17th century before Isaac Newton and German mathematician Gottfried Leibniz independently invented calculus — a bedrock of mathematics with myriad applications in physics to economics to biology.

While the roots of calculus in Kerala have been traced for decades, modern-day mathematicians say there is still burgeoning interest in the achievements of 14th and 15th century Kerala mathematicians. Researchers are gaining insights into the depth of their ideas, seeking to link the rise and fall of calculus in medieval Kerala to social and cultural factors, and looking for evidence of a Kerala-Europe connection.

The current evidence points to two parallel movements separated in time and place, said Kim Plofker, a maths historian at Union College, Schenectady, New York, and an invited delegate at the International Congress of Mathematicians (ICM) 2010 in Hyderabad.

An analysis of medieval writings suggest that a 14th century mathematician in Kerala named Madhava and his disciples developed a key component of calculus called infinite series: the representation of numbers as a sum of an endless number of terms.

“Their results in infinite series, calculation of infinitesimal quantities, and their proofs of earlier results in geometry were in many cases undreamed-of elsewhere in the world until the development of calculus and subsequent explosion of mathematics in early modern Europe,” Plofker said in a special paper on ancient Indian mathematics published in a journal Hyderabad Intelligencer circulated at the ICM.

Some scholars have suggested transmission of ideas from Kerala to Europe. Three years ago, a University of Manchester researcher George Gheverghese Joseph analysed the contributions of Madhava and others in Kerala from 1350 to 1540 and Jesuit documents scattered in archives in southern Europe to suggest that Jesuit scholars may have helped carry ideas from Kerala to Europe.

But most researchers believe there is no hard evidence yet for such transmission. “We have such a complete historical record of what Leibniz did. There is no indication he had encountered ideas from India,” said David Mumford, an emeritus professor of maths at Brown University, US, who has been studying medieval Kerala contributions for five years.

“Many of us are still trying to analyse the differences and similarities [between the Kerala and European schools of calculus],” said Mumford. “The world should know about this deep mathematical work from India — it shifts our perspective about the history of the subject.”

Scholars who have translated medieval mathematical texts from Kerala have found ideas embedded in verse. One set of verses attributed to Madhava, for instance, describes the relation between the circumference and diameter of a circle, said K. Ramasubramanian, a researcher at the Indian Institute of Technology, Mumbai, who has helped interpret a 16th century text called Yuktibhasa, dubbed the world’s first textbook of calculus.

In the Hyderabad Intelligencer, Ramasubramanian describes a section of the Yuktibhasa that outlines the derivation of a method to represent the number pi divided by four as the sum of an endless series of fractions: 1 - 1/3 + 1/5 - 1/7 ....

“The number pi has no fixed value — we truncate it with a decimal point,” said Anantoth Koyitti Vijayarajan, a researcher at the Kerala School of Mathematics, Kozhikode, a new institution set up jointly by the Kerala government and India’s Department of Atomic Energy. “Madhava and his successors discovered how to represent it through an infinite series that converges on pi.”

The origins and demise of calculus in medieval Kerala have remained a mystery, but physicist P.P. Divakaran has spent several years trying to understand the contributions as well as the social and cultural factors that might have shaped this movement.

“All I’ve done is try to juxtapose some events,” Divakaran said. His analysis — just published in a special issue of the Indian Academy of Sciences’ Current Science — suggests that some ideas of geometry and trigonometry that could be traced back to the [5th century] Aryabhata and arithmetical ideas might have influenced the Kerala School.

The migration into Kerala of vedic brahmins from northern India — beginning in the 11th century and continuing for up to 300 years — led to a flourishing of literature, arts and studies, including astronomy and architecture. Divakaran calls it “a radical transformation of the cultural matrix”. It’s in this milieu that calculus emerged.

But in the 16th century, soon after the arrival of the Portuguese, the tradition declined. “As European mathematics was hitting its stride, mathematics and its teaching in Kerala had already wasted away, never to be revived until modern times,” Divakaran wrote in his paper.

Divakaran points out that the demise of the tradition of learning wasn’t confined to astronomy and mathematics. One factor that may have contributed to this decline may be that access to learning and even basic education was restricted to a narrow, socially superior section of people. “As the relative number of brahmins began to fall due to their own short-sighted and socially regressive marital customs, the catchment area began to shrink to the point of vanishing,” said Divakaran.

“A history of mathematics is also a history of human intellectual development. We can try and understand how the developments in mathematics are related to culture,” Plofker said.