Kerala school of astronomy and mathematics

Based on Wikipedia: Kerala school of astronomy and mathematics

The Mathematicians Who Found Calculus Two Centuries Before Newton

In the southwestern corner of India, along the lush Malabar Coast where the Arabian Sea meets dense coconut groves and winding backwaters, a group of astronomers and mathematicians quietly discovered the foundations of calculus. This was around the year 1500. Isaac Newton wouldn't be born for another 142 years.

The Kerala school, as historians now call them, developed infinite series expansions for trigonometric functions—the mathematical machinery that would later become central to physics, engineering, and virtually every branch of modern science. They worked out methods for calculating pi to nine decimal places. They used concepts that look remarkably like differentiation and integration.

And then, for reasons that remain somewhat mysterious, their discoveries stayed in Kerala. The work was written in Sanskrit verse and Malayalam prose, studied by generations of students, and largely unknown to the wider world until an English civil servant stumbled upon it in the 1830s.

Who Were They?

The school began with Madhava of Sangamagrama, who lived in what is now the Tirur region of Kerala during the late 14th and early 15th centuries. We know frustratingly little about Madhava himself—none of his original mathematical works survive. What we know of his discoveries comes from his students and their students, who quoted him extensively and credited him as the source of their most important results.

This was how knowledge moved in medieval India. A guru would teach a small group of students, who would commit the teachings to memory, often in verse form designed for memorization. These students would become teachers themselves, passing knowledge down through generations like a chain of careful hands.

The chain from Madhava ran something like this: Madhava taught Parameshvara, who taught Damodara, who taught Nilakantha Somayaji, who taught Jyesthadeva. Each link added new results while preserving the old ones. The chain remained unbroken for roughly three centuries, from the 1300s into the 1600s.

Nilakantha Somayaji, working around 1500, compiled much of the school's knowledge into a work called the Tantrasangraha. This was written in Sanskrit verse—a common format for technical writing in India, since verse is easier to memorize than prose. The theorems were stated clearly, but without proofs.

The proofs came later. Around 1530, Jyesthadeva wrote the Yuktibhasa, which means something like "discourse on rationales." This is one of the most remarkable mathematical texts in history. Written in Malayalam rather than Sanskrit—making it accessible to a wider audience—it provides detailed proofs for the infinite series that the school had discovered. It explains the reasoning step by step, in a way that any mathematician today could follow.

What They Actually Discovered

To understand what the Kerala mathematicians achieved, you need to know a bit about what an "infinite series" is.

Consider the number pi, the ratio of a circle's circumference to its diameter. Pi is roughly 3.14159, but those digits go on forever without repeating. How do you pin down such a number?

One answer: you write it as a sum of infinitely many fractions, each one smaller than the last. The Kerala school found that pi divided by four equals one minus one-third plus one-fifth minus one-seventh, and so on forever. In modern notation:

π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

This is beautiful. It connects the mysterious number pi to the humble odd numbers. But it's also practical: if you add up enough terms, you get as close to pi as you want.

The catch is that this particular series converges slowly. You need to add thousands of terms to get just a few decimal places. So the Kerala mathematicians didn't stop there. They found ways to speed up the convergence, using what we would now call "error correction terms." With their improved series, they calculated pi as 104348 divided by 33215, which equals 3.141592653—correct to nine decimal places.

But infinite series for pi were just part of the story. The school also found infinite series for the sine and cosine functions, which describe the relationship between angles and ratios in triangles. In Europe, these would later be called Taylor series or Maclaurin series, after the Scottish and English mathematicians who developed them in the early 1700s. The Kerala versions predate the European ones by two centuries.

The Concepts Behind the Series

Here's what makes the Kerala work so striking: to derive these series, they had to use ideas that would later become the core of calculus.

Calculus is fundamentally about two related problems. The first is finding rates of change—how fast is something growing or shrinking at any given moment? This is called differentiation. The second is finding accumulated totals—if something changes at a given rate, how much total change builds up over time? This is called integration. Newton and Leibniz, working independently in Europe in the late 1600s, realized that these two problems are inverses of each other, and they developed a systematic notation and method for solving both.

The Kerala mathematicians never quite made that final synthesis. They didn't create a general theory of calculus applicable to any problem. But they did use both differentiation and integration in specific contexts, particularly for trigonometric functions.

They computed what we would call the derivative of the sine function. They found areas under curves by methods equivalent to integration. They used something very close to mathematical induction—a technique for proving that a statement holds for all whole numbers by showing it holds for the first number and showing that if it holds for any number, it holds for the next.

They even had an intuitive notion of a limit, the idea that a sequence of numbers can approach some value as closely as you like without ever quite reaching it. This concept, which European mathematicians spent centuries trying to make rigorous, was used by the Kerala school to justify their infinite series.

The Astronomical Connection

Why were medieval Indian mathematicians so interested in trigonometric functions in the first place?

The answer is astronomy. In medieval India, as in medieval Europe and the Islamic world, astronomy was one of the most important applications of mathematics. Predicting the positions of the sun, moon, and planets required sophisticated calculations. Religious festivals were tied to lunar phases. Agriculture depended on understanding the seasons. Navigation at sea relied on reading the stars.

Trigonometry—the mathematics of triangles and angles—is essential for astronomical calculations. If you want to know where the moon will be next week, you need to understand the geometry of circles and spheres. The Kerala school's mathematical innovations arose directly from their attempts to solve astronomical problems more accurately.

This wasn't unusual. Throughout history, many mathematical advances have come from astronomy. The Babylonians developed sophisticated mathematics partly to predict eclipses. Greek astronomers like Hipparchus and Ptolemy created trigonometry for the same reason. In the Islamic world, mathematicians refined these techniques further. The Kerala school was part of this long tradition.

What set them apart was how far they pushed the mathematics. Previous astronomers had used finite approximations. The Kerala school asked: what if we could write exact answers as infinite sums?

A Parallel Discovery in the Islamic World

The Kerala school wasn't working in complete isolation. The Islamic world had made its own advances toward calculus-like ideas centuries earlier.

Around the year 1000, the Iraqi mathematician Ibn al-Haytham (known in Europe as Alhazen) found formulas for summing the fourth powers of consecutive integers. This is a step toward integration—finding the area under a curve that represents the fourth power function. Islamic scholars could apparently find such formulas for any polynomial they cared about.

But they didn't seem to care about polynomials of very high degree, and they didn't extend their methods to trigonometric functions. The Kerala mathematicians did both. They took formulas similar to what Ibn al-Haytham had developed and applied them to calculate power series for sines and cosines.

Some historians have suggested that ideas might have flowed from the Islamic world to Kerala through trade routes. The Malabar Coast was a major hub of maritime commerce, with Arab traders visiting regularly. But no one has found definitive evidence of such transmission.

Lost and Found

In 1825, a British civil servant named John Warren published a memoir about traditional methods of timekeeping in southern India. Almost as an aside, he mentioned that Kerala astronomers had discovered infinite series.

A decade later, another Englishman named Charles Matthew Whish wrote a more detailed account. Whish had learned Malayalam and studied manuscripts held by traditional scholars. He was astonished by what he found. The Kerala mathematicians, he wrote, had "laid the foundation for a complete system of fluxions"—the term Newton had used for calculus. Their works "abounded with fluxional forms and series to be found in no work of foreign countries."

This should have been explosive news. Here was evidence that fundamental ideas of calculus had been developed in India two centuries before Newton and Leibniz. But Whish's paper, published in 1835, was almost completely ignored.

Why? Partly, perhaps, because European scholars in the Victorian era were not inclined to credit major mathematical achievements to non-European peoples. Partly because the claim seemed too extraordinary to believe without independent verification. And partly because Whish's paper was published in an obscure journal and simply didn't circulate widely.

It wasn't until the 1940s that Indian mathematicians, particularly C. T. Rajagopal and his colleagues, began seriously investigating the Kerala school's work. They translated the Sanskrit and Malayalam texts, analyzed the proofs, and confirmed that yes, medieval Indians had indeed discovered the foundational ideas of calculus.

Did It Reach Europe?

This is the question that hovers over the whole history: did the Kerala discoveries somehow make their way to Europe?

The chronology is suggestive. The Kerala school flourished from roughly 1400 to 1600. Portuguese traders arrived in Kerala in 1498, and Jesuit missionaries followed soon after. The Jesuits were specifically interested in Indian astronomy and mathematics—they saw it as potentially useful for predicting eclipses, which was important for establishing the correct dates of religious festivals.

There were trade routes. There were missionaries who learned local languages and studied local texts. There was definitely opportunity for transmission.

But no one has found a smoking gun. No European manuscript has been discovered that clearly shows knowledge derived from Kerala sources. Newton's notebooks show his own development of calculus step by step, with no indication that he had access to Indian work. The same is true for Leibniz.

David Bressoud, a mathematician who has studied this question carefully, puts it bluntly: "There is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century."

But absence of evidence isn't always evidence of absence. Historians are still searching through manuscript collections in Spain and the Maghreb, looking for possible links. The archives of the Jesuits might hold clues. This remains an active area of research.

Why It Matters

Even if the Kerala discoveries never left India, they matter profoundly for how we understand the history of mathematics.

The standard story, still told in many textbooks, is that calculus was invented by Newton and Leibniz in Europe during the scientific revolution. The Kerala school complicates this narrative. It shows that the core ideas of calculus—infinite series, differentiation, integration—were discovered independently at least twice, possibly three times if we count the Islamic contributions.

This suggests that calculus wasn't a unique product of European genius but rather an idea whose time had come. When people work hard on certain kinds of problems—in this case, astronomical problems requiring precise calculations with trigonometric functions—they may converge on similar mathematical solutions.

The Kerala story also reminds us how much has been lost or forgotten. The Yuktibhasa is a sophisticated mathematical text, with careful proofs that any modern mathematician can follow. Yet it was unknown to Western scholarship for centuries. How many other traditions of mathematical thought have been lost entirely?

The Transmission of Knowledge

There's something poignant about the Kerala school's mode of transmission. Knowledge passed from teacher to student through memorized verses, through handwritten manuscripts, through a chain of human relationships stretching across generations.

This method was remarkably robust in some ways. The chain held together for three hundred years, preserving and building upon Madhava's original insights. But it was also fragile. When the chain broke—when, for whatever reason, students stopped learning and teaching the tradition—the knowledge didn't die exactly, but it became dormant. The manuscripts sat in temple libraries and private collections, unread and unstudied, until curious outsiders came looking.

K. V. Sarma, who published a comprehensive history of the Kerala school in 1972, noted the importance of this direct transmission. In an era "when there was not a proliferation of printed books and public schools," knowledge in technical disciplines like astronomy survived only through the personal relationship between teacher and student. Each link in the chain was essential.

We live now in a world of printed books and digital archives, where information is stored and transmitted in ways that would have seemed miraculous to Madhava. Yet somehow, despite all our technologies of preservation, the Kerala school remained obscure for centuries after Whish first described it.

Perhaps the lesson is that discovering something is only half the battle. The other half is convincing others that it matters—making sure the discovery enters the collective consciousness of humanity rather than remaining a local treasure.

What Might Have Been

One of the intriguing counterfactuals of history is what might have happened if the Kerala discoveries had spread more widely, more quickly.

Some scholars have noted that around 1500, Nilakantha Somayaji seems to have proposed a partially heliocentric model of the solar system—one in which Mercury and Venus orbit the sun while the sun orbits the earth. This is similar to the model that Tycho Brahe would propose in Europe nearly a century later.

Imagine if European astronomers had learned of the Kerala school's work in the 1500s. Would the scientific revolution have happened faster? Would calculus have been systematized sooner? Would the credit for these discoveries be shared differently?

We can never know. History doesn't run experiments with control groups. What we can say is that human beings, facing similar problems in different parts of the world, sometimes arrive at similar solutions. The Kerala school is proof that mathematical genius flourishes wherever there are minds willing to push beyond the known.

The moon's phases, the sun's path across the sky, the stars wheeling overhead through the night—these were the same in Kerala as in Cambridge, the same in Baghdad as in Athens. People who watched them carefully, who wanted to predict them precisely, who refused to be satisfied with rough approximations—these people, wherever they lived, developed the mathematics of infinity.