The Kerala School of Mathematics
The Indian Mathematicians Who Worked With Infinity
HISTORY
9/2/20267 min read
The Kerala School of Mathematics: The Indian Mathematicians Who Worked With Infinity
When we think about the history of mathematics, a familiar story often comes to mind: ancient Greek mathematicians laid the foundations, European mathematicians later developed modern mathematics, and Newton and Leibniz created calculus in the 17th century.
But there is another important story that deserves much more attention.
Between roughly the 14th and 17th centuries, a remarkable tradition of mathematicians and astronomers flourished in Kerala, in southern India. They developed sophisticated ideas about trigonometry, infinite series, algebra and astronomy.
At the centre of this tradition was a mathematician named Madhava of Sangamagrama.
His work, and that of the generations that followed him, included infinite series equivalent to the modern mathematical expansions for sine, cosine and arctangent. They also developed methods for calculating π with remarkable accuracy.
This tradition is now known as the Kerala School of Mathematics, or more fully, the Kerala School of Astronomy and Mathematics.
It was not a school in the modern sense
The first thing to understand is that the Kerala School was not a university or a single building. There was no "Kerala School of Mathematics" campus where all these mathematicians sat together. Instead, it was an intellectual tradition.
Knowledge passed from teacher to student, from one generation to another. Scholars wrote commentaries on earlier works, corrected calculations, developed new methods and preserved mathematical ideas in Sanskrit and Malayalam texts.
The tradition was closely connected with astronomy. Mathematicians needed accurate calculations to understand the movements of the Sun, Moon and planets, predict eclipses and construct astronomical tables.
This need for accuracy helped drive some extraordinary mathematical discoveries.
Madhava of Sangamagrama: The Beginning of a Mathematical Revolution
At the beginning of this story stands Madhava of Sangamagrama, generally regarded as the founder of the Kerala School.
Madhava probably lived around the late 14th and early 15th centuries, although his exact dates are uncertain. He was both a mathematician and an astronomer. His name is particularly important because of his work with infinite series. To understand why this was revolutionary, consider a simple question:
How can you calculate the value of π? Today, we know π begins: 3.14159265358979...
The digits continue forever. Modern computers can calculate millions, billions or even trillions of digits of π. But Madhava lived around 600 years ago, long before calculators or computers. Yet he developed mathematical methods that could produce highly accurate approximations of π.
One of the series associated with the Kerala tradition is: π/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − ...
This is an infinite series.
It means that by continuing the calculation with more and more terms, you can get closer and closer to π. Madhava also developed correction terms that made such calculations much more efficient. This is one of the reasons he is considered one of the great mathematicians of medieval India.
The Mathematics of Infinity
The most remarkable part of Madhava's work was not simply calculating π. It was his use of infinite series. A series is a sequence of mathematical terms added together.
For example: 1 + 1/2 + 1/4 + 1/8 + ...
This series has infinitely many terms. Yet it approaches a finite value.
The Kerala mathematicians developed similar ideas for trigonometric functions. They obtained series equivalent to the modern expansions:
sin x = x − x³/3! + x⁵/5! − x⁷/7! + ... and cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + ...
They also obtained the series: arctan x = x − x³/3 + x⁵/5 − x⁷/7 + ...
Today, these equations are basic tools in calculus, physics, engineering and mathematics.
The important historical point is that the Kerala tradition developed these results centuries before they became familiar in European mathematics.
Why Did Kerala Mathematicians Need Such Mathematics?
The answer is largely astronomy. For medieval astronomers, mathematics was not simply an abstract intellectual exercise. They needed mathematics to calculate where celestial objects would appear in the sky.
They worked on problems involving:
the movement of planets
the Sun and Moon
eclipses
angles
arcs and chords
calendars
astronomical tables
trigonometric calculations
The more accurate their calculations became, the better their astronomical predictions could be. This created a powerful relationship:
Astronomy created difficult mathematical problems, and mathematics provided better tools for astronomy.
The Kerala tradition pushed both subjects forward together.
Parameshvara: The Astronomer Who Looked to the Sky
After Madhava came a generation of scholars who preserved and expanded his work. One of the most important was Parameshvara, generally dated to around the 14th–15th centuries. He was an astronomer and mathematician and is traditionally regarded as having been connected directly with Madhava's school. Parameshvara was particularly notable for his interest in astronomical observation.
This is significant because Indian astronomy had a long tradition of mathematical models, but Parameshvara also emphasized checking astronomical calculations against actual observations. He carried out observations over many years and wrote commentaries on important astronomical works. In this way, he helped maintain the Kerala tradition's combination of: observation + mathematics + astronomy.
Damodara: Passing the Knowledge Forward
Another important scholar was Damodara, the son of Parameshvara. Damodara played an important role in transmitting the mathematical and astronomical tradition to the next generation. This is one of the fascinating features of the Kerala School. Its achievements were not the work of a single mathematician. Instead, knowledge accumulated over generations.
Madhava developed important mathematical ideas. Parameshvara preserved and expanded the tradition. Damodara passed it forward. And from this lineage came some of the Kerala School's most important later scholars.
Nilakantha Somayaji: Mathematics Meets Astronomy
One of the greatest successors to Madhava was Nilakantha Somayaji, who lived around 1444–1550. Nilakantha was an astronomer, mathematician and Sanskrit scholar.
His most famous work is the Tantrasangraha, written around the beginning of the 16th century. The work deals with mathematical astronomy and contains sophisticated calculations involving trigonometry and planetary motion. Nilakantha did something particularly interesting: he did not simply repeat the astronomical models inherited from earlier Indian astronomers. He developed and modified them.
His astronomical model for the movements of the planets was an important development within the Indian astronomical tradition. His work also preserved and extended mathematical ideas associated with Madhava and the Kerala School.
Jyeshthadeva and the Yuktibhāṣā
Perhaps the most valuable surviving source for understanding the Kerala School's mathematics is the Yuktibhāṣā, written by Jyeshthadeva in the 16th century. Jyeshthadeva is particularly important because he did not merely state mathematical rules. He explained how and why they worked.
The Yuktibhāṣā was written in Malayalam and contains detailed mathematical reasoning. It discusses results associated with Madhava and other Kerala mathematicians and gives explanations for mathematical procedures involving infinite series and trigonometry. This is extremely valuable to historians. Many ancient and medieval mathematical texts simply give a rule and an answer. The Yuktibhāṣā gives us something more: the reasoning behind the mathematics.
It allows modern historians to reconstruct the mathematical thinking of the Kerala tradition.
A Mathematical Family Tree
The Kerala School can be thought of as an intellectual chain. A simplified version looks like this: Madhava of Sangamagrama -> Parameshvara -> Damodara -> Nilakantha Somayaji -> Jyeshthadeva and other later scholars
There were several branches and many other important mathematicians, including Chitrabhanu, Sankara Variyar and Achyuta Pisharati. The important thing is that the knowledge did not disappear with one person. It was transmitted. Each generation inherited mathematics from the previous one and added something of its own.
Chitrabhanu and the Study of Algebra
The Kerala tradition was not limited to infinite series and astronomy. Scholars such as Chitrabhanu also worked on algebraic problems. Chitrabhanu is associated with methods for solving certain types of equations. His work was later discussed by other Kerala scholars, including Sankara Variyar. This shows that the Kerala School had a broad mathematical interest.
It included:
arithmetic
algebra
geometry
trigonometry
infinite series
astronomy
Sankara Variyar: The Commentator
Sankara Variyar was another important mathematician and astronomer of the Kerala tradition. One of his major contributions was through commentaries. Commentaries may sound less exciting than new discoveries, but they were extremely important.
A mathematical tradition survives only if its knowledge can be understood by the next generation. Sankara Variyar explained earlier mathematical and astronomical works, helping preserve the ideas of the Kerala School. This culture of writing commentaries was one of the reasons mathematical knowledge could continue across several centuries.
The Kerala School and Calculus
The Kerala School unquestionably developed mathematical ideas that are closely related to later calculus. Its mathematicians worked with:
infinite series
infinite processes
limiting approximations
trigonometric expansions
methods for calculating π
These are important ingredients in the mathematics that later became calculus.
The Kerala School developed sophisticated mathematics involving infinite series and trigonometric expansions centuries before these ideas became central to European calculus.
The Mystery of Madhava's Lost Works
There is another fascinating part of the story. Much of Madhava's original mathematical work has not survived in complete form. We know about many of his discoveries because later scholars preserved them, discussed them and explained them. This makes the Kerala tradition somewhat like a puzzle.
Historians look at later texts such as the Yuktibhāṣā and Tantrasangraha and work backwards to reconstruct what earlier mathematicians such as Madhava must have known.
In other words, some of Madhava's mathematics survives not because we possess all of Madhava's books, but because his intellectual descendants preserved his ideas.
The greatest lesson of the Kerala School may therefore not be a particular equation. It is the idea of cumulative knowledge. Madhava did not work in isolation. He belonged to a mathematical culture that had developed over centuries. His successors did not simply copy him.
They:
preserved his results,
explained them,
corrected them,
applied them to astronomy,
developed new mathematical methods,
and passed the knowledge to another generation.
This is how scientific traditions grow.
The Legacy of the Kerala School
The Kerala School of Mathematics occupies a special place in the history of mathematics. Its scholars developed sophisticated techniques involving infinite series and trigonometry centuries before these ideas became standard tools in Europe.
Madhava's work on infinite series and π is particularly striking. Parameshvara strengthened the connection between mathematical theory and astronomical observation. Nilakantha Somayaji developed mathematical astronomy further. Jyeshthadeva's Yuktibhāṣā preserved detailed explanations of the mathematics. And generations of later scholars continued the tradition.
The story therefore isn't simply about one Indian mathematician who "beat Europe to calculus." It is a much richer story. It is about a centuries-long mathematical tradition in medieval Kerala, where teachers and students explored the mathematics of circles, planets, angles and infinity.
And perhaps the most remarkable thing about it is that they were doing this at a time when, in much of the world, the mathematical tools we now take for granted were still centuries away from being developed.
In one sentence
The Kerala School of Mathematics was a centuries-old Indian mathematical and astronomical tradition whose scholars, beginning with Madhava of Sangamagrama, developed remarkably advanced methods involving infinite series, trigonometry, π and mathematical astronomy.
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