Madhava, the Kerala School, and the long road to pi that loops back to modern string theory

 

Copyright: Sanjay Basu

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There are mathematicians who feel like a line in a textbook. Clean. Finished. Filed away.

Then there is Madhava of Sangamagrama.

Madhava is the opposite of filed away. He is the kind of figure who keeps reappearing, like a recurring character in a very serious novel that refuses to end. He shows up whenever we talk about infinite series, fast numerical approximations, and that slippery boundary between computation and proof. He shows up again, unexpectedly, in a modern physics story where two string theorists set out to tame scattering amplitudes and accidentally bump into a new family of formulas for pi.

This is an article about Madhava and his school. It is also an article about a certain style of thinking. A style that treats infinity as something you can work with, not merely admire. A style that cares about speed, error, and practicality. A style that does not panic when a calculation refuses to terminate.

It is a very Indian story. It is also a universal one.

First, who was Madhava

Madhava of Sangamagrama is usually dated to roughly the late 14th and early 15th centuries, often given as about 1340 to about 1425. He lived in Kerala and is widely regarded as the founding figure of what historians call the Kerala school of astronomy and mathematics.

If you are expecting a neat biography with portraits, letters, and a shelf of surviving manuscripts signed in his own hand, you will be disappointed. Much of what we know about Madhava comes through later works in the Kerala tradition that attribute results to him and preserve methods that clearly belong to the same lineage. Historians take this seriously but carefully. There is attribution. There is continuity. There is also the ordinary messiness of medieval intellectual history.

Still, one fact stands firm.

By the time Europe is still centuries away from the calculus textbooks that traumatized us in college, the Kerala school is already working with infinite processes that look uncannily like modern power series expansions for trigonometric functions and inverse trigonometric functions.

That sentence should make you sit up straighter.

Why Kerala cared so much about trig and series

Kerala mathematics is not best understood as a detached exercise in abstract purity. It is bound up with astronomy. In that world, you do not do trigonometry because triangles are pretty. You do it because you need to compute angles, arcs, and celestial positions with high precision, again and again, in a form that can be tabulated and used.

This pushes you toward three habits.

One, you want formulas that are computationally friendly.

Two, you want to know how fast your approximations improve.

Three, you want a way to control error.

Those are not modern habits. Those are human habits. They emerge whenever people must compute difficult quantities reliably.

The big idea that Madhava represents

Madhava is most famous for series that, in modern notation, line up with expansions like

arctan(x) = x − x^3/3 + x^5/5 − x^7/7 + …
sin(θ) = θ − θ^3/3! + θ^5/5! − …
cos(θ) = 1 − θ^2/2! + θ^4/4! − …

Modern scholarship discusses these as part of the Kerala school’s “power series” tradition and points to Madhava as the central early figure behind it.

Now pause. Notice what is happening.

These are infinite expressions. They do not end. So on paper, they are useless unless you also invent the art of stopping.

So the real breakthrough is not merely writing an infinite series. The breakthrough is turning infinity into a tool, where you can stop after N terms and still know something trustworthy about what you have.

That is the difference between poetry and engineering.

Madhava’s legacy is that the Kerala tradition learned to make infinity behave.

Pi enters the stage and refuses to leave

Pi is a constant that has a talent for dragging entire civilizations into its orbit. It looks innocent. It is just a ratio of circumference to diameter. Then it ruins your afternoon.

The Kerala tradition approached pi through inverse tangent series.

Set x = 1 in the arctan series and you get
pi/4 = 1 − 1/3 + 1/5 − 1/7 + 1/9 − …

This alternating series is sometimes called the Madhava series in the Indian context, and in later European context it is associated with Gregory and Leibniz. Kerala sources credit Madhava earlier, and modern overviews discuss his use of such a series centuries before it appears in Europe’s published record.

So far, so good.

But here is the catch that every numerical person immediately sees.

This series converges slowly. Painfully slowly. It is the kind of slow that makes you regret being born with curiosity.

If you add 100 terms, you are still not spectacularly close. Scientific American’s discussion of the modern rediscovery story makes that point bluntly, and it is correct in spirit.

So if the Kerala school actually used these ideas for computation, they needed more than the headline series. They needed acceleration.

And they had it.

Madhava’s correction terms and the birth of practical convergence control

A remarkable piece of the Kerala tradition is what is often described today as Madhava’s correction term, a systematic way to improve the truncated alternating series for pi/4.

In modern summaries, you will often see three candidate forms recorded in the tradition for a correction term F(n) added to the partial sum, such as

F1(n) = 1/(4n)
F2(n) = n/(4n^2 + 1)
F3(n) = (n^2 + 1)/(4n^3 + 5n)

The idea is simple and profound.

You compute the partial sum up to n terms. Then you add a small adjustment with the correct alternating sign to compensate for the remainder.

If you are a modern analyst, you can read this as an early form of remainder estimation and convergence acceleration. If you are a modern numerical person, you read it as something even more basic.

This is a craftsman’s move.

It is what you do when you refuse to waste time.

Modern historical discussions tie these correction terms to Kerala texts and later expositions, and the Cambridge excerpt by Ranjan Roy highlights the broader Kerala power series achievements.

Even if you do not care about who discovered what first, the methodological point matters.

The Kerala school did not merely find a series. They found a way to use it efficiently.

That is the part people forget when they reduce this story to a national pride poster.

The Kerala School as a lineage, not a lone genius

Madhava is often treated as the spark, but the fire is the lineage.

The Kerala school includes major later figures such as Nilakantha Somayaji and Jyesthadeva, among others. Their works preserve, extend, and justify methods that historians connect to the Madhava origin point.

This matters for a subtle reason.

When we say “Madhava discovered series,” we are compressing decades of thinking into a single name. What really happened is closer to what we see in any living research community.

Someone makes a leap. Others formalize it. Someone else generalizes it. Another person writes the textbook version. Then centuries later, people argue about priority while ignoring the fact that the real miracle was the existence of a sustained intellectual ecosystem.

Kerala had that ecosystem.

And that is why this school keeps surprising modern readers.

A quick word about transmission myths

At this point, every discussion attracts a familiar question.

Did Europe steal calculus from Kerala.

It is a tempting story because it has drama, villains, and an easy moral. It is also a hard historical claim that requires hard evidence.

There are writers who argue strongly for transmission paths, including arguments that lean on Jesuit contact and colonial era channels.

Mainstream historical caution is that direct influence on Newton or Leibniz is not established as a settled fact in the strong form. It is one thing to show that Kerala developed powerful series methods early. That is well supported. It is another thing to prove a documented chain of transmission into the specific European calculus creators. That is a higher bar.

You can hold both truths at once.

Kerala was brilliant.

And history is strict about evidence.

I will explore the story of Calculus and the roles of Newton, Leibniz, and other European mathematicians in a later series.

Now leap forward, into physics, and watch pi fall out of a string calculation

I am researching Strings again, after it fell out of favor about a decade back. And now again resurfacing. I came across an interesting Phys.org piece linking the result to Physical Review Letters. This widely reported modern work is by Arnab Priya Saha and Aninda Sinha, both of whom are researching at the Indian Institute of Science in Bengaluru.

Now the fun part.

Their work begins in high energy theory. They are looking at representations of the Euler Beta function and string scattering amplitudes. The math of the Beta function is old and familiar. The way it appears inside scattering calculations is part of the daily grind of theoretical physics.

In the course of reorganizing these expressions, they obtain a new parametric representation that leads to fast converging series for constants, including pi. Their arXiv preprint describes this as “parametric representations” that “show fast convergence.”

This result was published in Physical Review Letters, and it generated a small media storm because the pi angle is irresistible.

The key structural idea in the Saha and Sinha pi formula

One of the striking features reported in Scientific American is that their pi representation depends on a free parameter, often called lambda in expositions. For many choices of that parameter, the series converges far faster than the old Madhava alternating series.

There is also a beautiful limiting link.

As lambda becomes very large, their general formula collapses back down to the Madhava style alternating series with odd denominators. Scientific American explicitly states this relationship and walks through the simplification idea at a popular level.

So the modern discovery is not a replacement of Madhava.

It is a generalization that contains Madhava as a special case.

That is exactly how good mathematics behaves.

A concrete expression, without the hype

A clean public discussion of their series appears in a MathOverflow thread, which writes an explicit form where pi is expressed as 4 plus an infinite sum involving factorial terms and a rising factorial, also called the Pochhammer symbol.

You do not need to memorize that formula to grasp the deeper connection.

The connection is philosophical and technical.

Madhava gave the world a simple alternating series that is easy to compute but slow.

The modern physics derived family introduces a tunable parameter that can accelerate convergence dramatically.

It is like taking a bullock cart route that works and then discovering a highway that contains the old road as a limiting case.

The old road is still real. It is still correct. It is simply not always the fastest way to get home.

The shared DNA, Madhava and modern pi series

If we strip away centuries, language, and academic fashion, what do we see.

We see three common themes.

Theme one, express a hard constant as a sum of simpler things

This is the oldest trick in the analytic book. You take something geometric like pi and you turn it into an arithmetic process. A recipe. Add, subtract, repeat.

Madhava’s pi series does that.

Saha and Sinha do it too, but their “simpler things” are not just odd reciprocals. They are terms that arise naturally from residues and analytic structure in Beta function expansions, shaped by the demands of crossing symmetry and field theory intuition.

Different kitchens. Same instinct.

Theme two, convergence is not an afterthought

This is where Madhava stops being a quaint pre modern anecdote and starts sounding like a computational scientist.

The naive alternating series for pi is slow.

So the Kerala tradition uses correction terms and smarter approximations.

The modern formula is newsworthy largely because it can converge fast for certain parameter choices, as popular coverage emphasizes.

In both cases, the story is not “here is a series.”

The story is “here is a series you can actually use.”

Theme three, parameterization is a form of power

Madhava’s correction term tradition is, in a sense, an early example of shaping the remainder. You keep the core series, but you add a structured adjustment that depends on how far you went.

That is a primitive but powerful parameterization of error.

Saha and Sinha go further. Their representation has an explicit free parameter, and by tuning it you can drastically change convergence behavior while keeping the sum equal to pi.

This is the modern way of thinking.

Do not accept the first correct representation as the final one.

Search the space of representations.

Pick the one that behaves best for your problem.

That is what numerics people do. That is what physicists do when the integrals fight back. That is also what the Kerala school did, because they were trying to compute, not merely admire.

Madhava’s pi is the opening move, not the whole game

There is a tendency in popular storytelling to treat Madhava’s pi series as the Kerala school’s single achievement.

That is like treating the computer as the calculator app.

The deeper Kerala contributions revolve around building a computational theory of trigonometric functions.

The sin and cos expansions matter because they enable tables and interpolation for astronomy. The inverse tangent expansions matter because they connect arcs to ratios. The overall power series viewpoint is what allows systematic approximation of quantities that are not algebraic.

Ranjan Roy’s historical overview [Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century (published by Cambridge University Press)] emphasizes precisely this point, that the Kerala school developed infinite power series methods well before Newton’s era.

So if pi is the celebrity in this story, trig is the working professional.

And the Kerala school was full of working professionals.

A short guided tour of what makes Kerala mathematics feel modern

Let me describe a few features that modern readers recognize immediately, even when the notation is unfamiliar.

Algorithmic thinking

Kerala mathematics is full of procedures. Steps you can follow. Steps you can teach. Steps that lead to numbers you can put in a table.

That is a culture of computation.

Approximation with accountability

The correction term tradition is not just clever. It is accountable. It tells you that the series is not merely converging in some abstract sense. It is converging in a way you can exploit.

This matters because astronomy is unforgiving. Your eclipse prediction does not accept excuses.

A comfort with infinity as a process

This is the hardest part for many students. Infinity is not a number. It is a behavior.

The Kerala school treated infinite expansions as behaviors you can harness.

This is the conceptual move that later becomes central in calculus and analysis.

It does not matter whether you call it calculus. The move is real.

How the modern Saha and Sinha story amplifies Madhava rather than replacing him

Media framing loves the trope of “new formula beats old formula.” It is lazy. It is also not the interesting part.

The interesting part is that modern theoretical physics, while chasing its own dragons, keeps rediscovering structures that earlier mathematicians uncovered under entirely different motivations.

Saha and Sinha were not trying to honor Madhava. They were trying to rewrite amplitude expansions in a way that behaves like field theory while preserving stringy features.

Yet the Madhava series appears as a limit inside their family of representations, as Scientific American highlights.

That is a form of influence that is deeper than citation.

It is structural influence.

It means Madhava’s way of expressing pi is not a historical accident. It is a natural attractor in the space of identities. A simple fixed point that other more general identities can flow toward in a limit.

If you like physics metaphors, Madhava’s series is a low energy effective description. It emerges when the fancy parameters go to infinity and the extra machinery decouples.

The analogy is not perfect. But it is emotionally accurate.

And it is a wonderful tribute.

A Bengali thread in a pan Indian mathematical tapestry

India has a deep tradition of mathematical thought that is not confined to one region, one language, or one century. Kerala’s astronomical mathematics, Bengal’s later analytical and scientific culture, the pan Indian engagement with series through figures like Ramanujan, and the modern global scientific ecosystem where Indian theorists publish in PRL and post on arXiv, all of it forms a continuous texture.

What does Madhava contribute to that texture.

He contributes a model of boldness.

He shows that you can take an irrational constant and represent it as an infinite arithmetic process.

He shows that you can refine that process with correction terms.

He shows that convergence is not merely a proof obligation. It is a practical design parameter.

Those ideas travel, even when the manuscripts do not.

They travel as methods. As instincts. As mathematical taste.

And when Saha and Sinha derive a tunable series representation for pi out of Beta function machinery, they are acting in a spirit that Madhava would immediately recognize.

Not because of nationalism.

Because of craft.

What to take away, if you only remember five things

One, Madhava is a late medieval Kerala mathematician and astronomer, widely credited as the founding figure of the Kerala school.

Two, the Kerala school developed powerful infinite series methods for trig and inverse trig, centuries before similar expansions become standard in European texts.

Three, Madhava’s pi series pi/4 = 1 − 1/3 + 1/5 − … is simple but slow, and the Kerala tradition includes correction term ideas to accelerate practical computation.

Four, modern string theory work by Arnab Priya Saha and Aninda Sinha produced a parametric family of pi series with fast convergence for certain parameter choices, and Madhava’s series appears as a limiting case.

Five, the real continuity is not a rumor about who stole what. The continuity is methodological. It is the shared insistence that infinity should be usable.

Madhava as a living ancestor

Madhava is not a footnote. He is not a trivia answer. He is not a nationalist token.

He is a living ancestor in the modern culture of computation.

Every time you see someone accelerate a series, tune a parameter for convergence, or turn a constant into a controllable infinite process, you are watching a family resemblance. You are watching a style of thought that Madhava helped make respectable.

And the next time a physicist claims they “accidentally” discovered a pi formula, smile politely.

Nothing about pi is accidental.

Pi is a magnet.

Madhava simply learned how to machine it into gears.