S.G. Rajeev, physics professor at the University of Rochester, says not enough has been done to study and understand the Kerala School of Mathematics, an illustrious epoch of our ancient history. Dr. Rajeev is one of the many who are trying to unravel the findings and studies of the scholars of yore
Centuries ago, along the river valleys of Bharatapuzha, existed a great tradition of learning in mathematics and astronomy. What the brilliant minds of that medieval age accomplished in a span of few centuries remains a source of endless wonder. Never after has Kerala produced greater triumphs in these fields, as it was truly the golden age of discoveries. Know to the modern world as the Kerala School of Mathematics, this illustrious epoch of our ancient history remains insufficiently explored.
S.G. Rajeev is one of many interested in unscrambling this past from its rich and varied sources. Trained as a physicist, Rajeev always had a keen interest in the history of science, particularly mathematics. “Tantrasangraha and Yukthibhasha are two important texts that give us access to this remarkable time period. I first became interested in the topic when I came across the late K.V. Sarma’s editions of these texts. Yukthibhasa’s vocabulary is old Malayalam which I could read with some effort. The first five chapters of it are rather elementary mathematics. The surprises start from there. The sixth chapter describes a system based on geometry to calculate the value of the irrational number pi to great accuracy. It is done by introducing the concept of a convergent infinite series, not an easy idea to conceive of considering that it has been a recent invention even in modern mathematics,” says Rajeev.
The genesis of the Kerala school is traced to Madhava who lived during mid 14th century CE at Sangama-grama, what could be present-day Irinjalakuda. Madhava developed, among much else, the foundations of modern calculus, one of the most versatile branches of mathematics. The same ideas were later discovered independently by Newton and Leibniz, but centuries later. A similar feat of genius was the discovery of infinite series expansions for trigonometric functions, proofs of which are given in both Yukthibhasa and Tantrasangraha.
The astronomy branch of the Kerala School became prominent with Madhava’s student Vatasseri Parameswara (1360 - 1455 CE). “He was the first to put down in writing a mathematical model of astronomy based on observations. In his Drkganitha are calculations that describe with remarkable precision the passage of planets in the sky. The calculations are based on long five decades of observations. These were the most accurate astronomical data till the time of Tycho Brahe, but I doubt if they were commonly known to the outside world at that time,” explains Rajeev.
Following Parameswara was his son Damodara whose original works are less known. His disciple Neelakanta Somayaji (1444 - 1545 CE) wrote an exhaustive commentary on Aryabhateeyam, and authored the Tantrasangraha. Jyestadeva (1500 – 1610 CE), who was also a student of Damodara continued the train of thought with Yukthibhasa. The Yukthibhasa, which can rightfully be described as the first book on calculus, was an unusual attempt from earlier texts in providing elaborate proofs for theorems.
The last among this lineage of original thinkers was Melpathur Narayana Bhattathiri, known for his greatly popular devotional hymn Narayaneeyam. What is perhaps less known is that Narayana Bhattathiri was also a brilliant mathematical linguist along the lines of the grammatical tradition of Panini in developing precise theoretical frameworks for Sanskrit grammar. “The book Prakriya Sarvaswam presents an elaboration of and an alternative to Panini’s axiomatic theory of grammar,” explains Rajeev.
What comes through as surprising is why such an inventive and glorious past is rarely discussed by modern Kerala that otherwise likes to bask in the shadow of its past. An account of the accomplishments of the Kerala School of Mathematics is yet to find its way into our mainstream history text books.
One reason could be the lack of simple and accessible popular history writings from scholars. Over the years there has been a slow but growing interest to go back and look, in detail, at this enigmatic past. But a lot of research is to be done as Rajeev describes the challenges, “to write authentically one has to go to the source. The content in these original texts are written in very terse forms. It is not enough to be a good historian. One has to have good knowledge in mathematics and Sanskrit. The words are totally foreign to us now. So we have to create our own glossary. K.V. Sarma worked till the end of his life to complete his English translation of Yukthibhasa. It is hard work, which is probably why very few are attempting it.”
One can follow the trail of the Kerala School’s tradition up to the 17th century, after which the flow of original ideas thin down. How the thread of continuity broke, is a question that looms. The answer is probably draped in the social milieu of those times. This tradition was largely of a small group of Namboodiri Brahmins, sometimes members of the same family. They lead a life that was far too insular, of little transactions with the outside world.
Reflecting on this, Rajeev says “even in the midst of opulence, these scholars perhaps lived like hermits, doing little else than devoting their lives to intellectual pursuits, rarely traveling outside of home to preach their ideas. This isolation could have been forced by external factors like a growing colonial presence, outbreaks of epidemics, and also struggles from within. The curious thing about knowledge is that if you try to confine it, you will fail in one of two ways. One, the knowledge will somehow get leaked to the outside world because there is value to be made out of it. That kind of failure is in fact good for knowledge. The other is that the knowledge will die with you. We often fail to see this, but knowledge is perishable. It can be forgotten like everything else.”