Jya-Shreni: Sine and Cosine Series Before Taylor
How Madhava of Sangamagrama wrote down the power series for sine and cosine in Kerala around 1400, three centuries before Brook Taylor
Around 1400 CE, Madhava of Sangamagrama wrote down the infinite power series for sine and cosine as a sequence of corrective refinements to the angle itself. The series are what modern mathematics calls the Taylor expansions. Madhava had them three centuries before Taylor, derived them rigorously, and used them to build an 8-decimal-accurate sine table.
The Sine In Your Phone
Open the calculator on your phone. Type the sine of thirty-seven degrees. The screen shows 0.6018150232. That number did not come from a lookup table. The phone's math library computed it, in a few microseconds, by evaluating a polynomial. That polynomial is the truncation of an infinite series for sine. And the earliest known author of that series was not Brook Taylor in 1715, nor Isaac Newton in 1669. It was a Nambudiri Brahmin named Mādhava of Saṅgamagrāma, working with an iron stylus on a palm-leaf folio in a Kerala village around 1400 CE, roughly three hundred years earlier. The series Mādhava scored onto the leaf, in modern notation, looks like this.
sin x = x − x³/3! + x⁵/5! − x⁷/7! + ...
And the companion for cosine.
cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + ...
Every calculus student who has ever opened a textbook has seen those two lines. They are almost always attributed to Brook Taylor, who published his general expansion theorem in 1715, or to Isaac Newton, who had the sine series by 1669. The attribution is historically wrong by about two and a half centuries. The person who first wrote those series down, in verse, in Sanskrit, in southern India, was a Nambudiri Brahmin named Mādhava of Saṅgamagrāma, working around 1400 CE.
Who Mādhava Was
Mādhava (c. 1340 to 1425 CE) lived in Saṅgamagrāma, a village near modern Irinjalakuda in central Kerala, about sixty kilometres from the port of Cochin. No original manuscript of his own survives. What we have of him is quotations preserved by his students and his students' students, a lineage that ran for the next two centuries through Parameśvara, Nīlakaṇṭha Somayājī, Jyeṣṭhadeva, Śaṅkara Vāriyar, and Acyuta Piṣāratī. Each of them names Mādhava as the source of the infinite series results. The Tantrasaṅgraha of Nīlakaṇṭha, composed in 1501, explicitly attributes the sine and cosine series to Mādhava. The Yuktibhāṣā of Jyeṣṭhadeva, composed around 1530 in Malayalam, preserves the full derivation and credits Mādhava at every step.
This citation chain is the only reason we can say the series are Mādhava's rather than Nīlakaṇṭha's or Jyeṣṭhadeva's. The Kerala tradition was scrupulous about acknowledgment. When the downstream texts say 'as stated by Mādhava', they are preserving a fact the tradition considered important to record.
The Series in Mādhava's Own Words
The jyā-śreṇī is set out in the Yuktibhāṣā and the Kriyākramakarī as a sequence of verses attributed to Mādhava. The verses describe the sine series not as a fixed formula but as a procedure for refinement. You start with the arc itself. This is your zeroth approximation to the sine. Then you compute a correction by multiplying the arc by a known factor and subtracting. Then you compute the next correction and add it back. You keep going. Each term is called a saṃskāra, a refinement or corrective term, and each is smaller than the last.
In modern notation, Mādhava's prescription reads like this. Let s be the length of the arc, measured as a multiple of the radius (so s = Rθ where θ is the angle in radians). Then:
jyā(s) = s − s³/(3! × R²) + s⁵/(5! × R⁴) − s⁷/(7! × R⁶) + ...
koṭi-jyā(s) = R − s²/(2! × R) + s⁴/(4! × R³) − s⁶/(6! × R⁵) + ...
The factors of R are there because Mādhava, like all Indian astronomers, worked with the chord rather than the abstract sine. When you divide out the radius, what you get is identical to what Taylor would write in 1715. The pattern of signs alternating plus and minus, the odd powers for sine, the even powers for cosine, the factorials in the denominators: all of it is already there in the Kerala verses.
Mādhava also gave the numerical values. He used the katapayadi system, a Sanskrit convention that encodes numbers as consonants arranged into meaningful words, to tabulate sine values at twenty-four intervals across a quadrant, each value accurate to about eight decimal places. The mnemonic verses that carry these values begin with the words vidvān, tunnabala, kavī-śanicaya and continue through the table. A Nambudiri Brahmin memorizing the verses in 1450 had, inside a few lines of Sanskrit, the contents of a modern eight-digit trigonometric table.
How the Derivation Worked
The Yuktibhāṣā does not simply state the series. It derives them. Jyeṣṭhadeva works through the argument in four steps. First, he sets up the geometry of a circular arc and its chord and shows how each small piece of the arc produces a small piece of the chord. Second, he replaces a finite sum of these small pieces with a more refined sum by taking the number of pieces to be very large and each piece to be correspondingly small. Third, he uses a procedure of successive correction: an initial approximation is replaced by a better one, which is replaced by a still better one, and the limiting process is carried to the point where the pattern of the corrections becomes clear and can be written down in closed form. Fourth, he justifies the resulting series by showing that each term is necessary and that their sum is the exact chord.
This is calculus. It is not a hint of calculus, or a precursor to calculus, or an ancient version of something that would later become calculus. The Yuktibhāṣā uses the concept of a limit, the concept of a variable summation with arbitrarily many terms, and the concept of a rigorous derivation of a general result. By any working definition of what calculus is, the Kerala school had it.
Why It Matters
The first reason the jyā-śreṇī matters is priority. The sine series was not independently discovered by Newton and then confirmed by Taylor. It was discovered, stated, proved, and tabulated in Kerala around 1400, two and a half centuries before Newton was born. The Kerala tradition did not make a noise about this. It recorded the result, cited the discoverer, built on the foundation, and moved on. The story became invisible to Europe because the Kerala texts were in Sanskrit and Malayalam, were held in family libraries in Nambudiri households, and were not part of the curriculum at the institutions that eventually wrote the global history of science.
The second reason is pedagogical. Mādhava did not write the series down as a formula to be memorized. He wrote it down as a procedure of successive refinement, with each term a correction to the previous answer. This framing is arguably closer to what the series actually is. A truncation of an infinite series is never the final answer. It is a current best guess that the next term improves. Mādhava's saṃskāra vocabulary, the language of refinement rather than the language of formula, reflects the working mathematician's actual relationship with the object. The European textbook convention, which presents the series as a fixed equation, hides the iterative character that Mādhava made central.
The third reason is that the jyā-śreṇī is not a museum piece. When your phone computes a sine, it truncates Mādhava's series to as many terms as the precision of the hardware requires and evaluates the polynomial. Billions of times per second across every device on the planet, the procedure Mādhava set out in Saṅgamagrāma six hundred years ago is being run as the fastest path to a trigonometric value. The Kerala school's most important result is also, in purely practical terms, the most frequently used piece of mathematics in the world today.
Key figures
Mādhava of Saṅgamagrāma
c. 1340 to 1425 CE, Saṅgamagrāma (near modern Irinjalakuda), central Kerala
Jyeṣṭhadeva
c. 1500 to 1575 CE, central Kerala
Śaṅkara Vāriyar
c. 1500 to 1560 CE, Tṛkkuṭaveli, central Kerala
Case studies
The Vidvān Verse: An Eight-Decimal Sine Table in Four Lines of Sanskrit
Imagine you are a Nambudiri Brahmin astronomer in fifteenth-century Kerala. You need the sine of a particular angle for a planetary calculation. You do not have a printed book. You do not have a calculator. You do not even have reliable access to a manuscript, because manuscripts rot in the monsoon and every copy introduces errors. What you do have is a verse your teacher taught you to recite, beginning vidvān tunnabalaḥ kavīśa-nicayaḥ, which you can say from memory in about twenty seconds. The verse is ordinary devotional Sanskrit on its surface, something like 'the learned one of well-struck strength, the assembly of lords of poets, steadfast in all purposes...'. It is easy to memorize because it scans and because the words are meaningful. But the consonants of the verse, read in blocks from right to left and decoded by the kaṭapayādi convention where ka=1, kha=2, and so on, yield twenty-four six-digit numbers. Those numbers are the values of the sine of 3°45′, 7°30′, 11°15′, and so on up to 90°, each accurate to the second of arc. You recite the verse, you decode the block you need, and you have your sine value. The whole operation takes under a minute and has no scribal-error risk, because the Sanskrit verse survives intact in a way that a column of digits in a manuscript never does.
The Kerala school's solution to the manuscript-preservation problem was not better ink or better paper. It was a different storage medium altogether. A Sanskrit verse that a trained reciter holds in active memory is a piece of data that does not rot, does not get copied wrong, and does not need a library. The vidvān verse is a compression algorithm in a literal sense: it packs twenty-four eight-digit values into thirty-two syllables, uses the Sanskrit metrical structure as an error-correcting code, and hides the numerical content behind a surface meaning that is itself memorable and devotional. Modern computer science took a very long time to reinvent the conceptual idea of encoding data into something harder to corrupt than the raw data itself. The Kerala tradition had it as a standard technique in 1400.
The vidvān verse and its companions kept Mādhava's sine table accurate and available for roughly three centuries in family libraries and teacher-pupil lineages across Kerala. When European scholars finally accessed the Kerala texts in the nineteenth and twentieth centuries, they found Mādhava's tabulated values intact, still decodable from the Sanskrit consonants, still accurate to the second of arc. The compression method survived better than many printed tables of the eighteenth century.
Precision and preservation are not the same problem. A table can be precise but fragile, or it can be rough but durable. Mādhava and his school insisted on both. They computed the sine values to eight decimal places using the power series, and then they wrapped the result in a form that would survive the monsoon, the copyist, and the century. The discipline of asking how your result will be preserved, as a separate question from whether it is correct, is one of the most practical lessons the Kerala school has to offer a modern researcher.
Mādhava's vidvān verse encodes twenty-four sine values at intervals of 3°45′, each carrying six to eight digits of precision. The largest entry, for 90°, is 3437′44″48, which corresponds to R = 3437.746 minutes of arc for the full radius, accurate to the arc-second. The entire table occupies four lines of Sanskrit.
The Yuktibhāṣā: The World's Earliest Surviving Calculus Textbook
Around 1530 CE in central Kerala, a mathematician named Jyeṣṭhadeva sat down to write out the derivations underlying the results of his tradition. He chose to write in Malayalam prose rather than Sanskrit verse, because his goal was pedagogical rather than mnemonic. He wanted a student to be able to follow an argument step by step, check each move, and reach the same conclusion the original teachers had reached. The book he produced is called the Yuktibhāṣā, 'the treatise on rationales'. It is divided into two parts, one on mathematics and one on astronomy. The mathematics part opens with fundamentals and builds up to the derivation of the jyā-śreṇī, Mādhava's sine series. The derivation is not a pointing-out or a hand-wave. It is a full argument. Jyeṣṭhadeva begins with a circle of known radius, divides the arc into a very large number of equal small pieces, takes the limit as the pieces shrink and their count grows, and uses a repeated process of refinement to derive the coefficients of the series one at a time. He then checks the result by reversing the procedure and shows that the derived series is consistent with the known numerical sine values from Mādhava's table. Every step is stated in ordinary Malayalam sentences that a literate reader can follow without specialized training.
The Yuktibhāṣā is not a precursor to calculus or a near-miss or an intriguing parallel. By any working definition of calculus, it contains calculus. It uses the concept of a limit. It uses the concept of a sum with arbitrarily many terms. It derives a power-series expansion of a transcendental function from geometric first principles. It proves its results rather than stating them. And it does all of this a century before Cavalieri, a century and a half before Newton, and two centuries before Taylor. Kim Plofker's Mathematics in India (Princeton, 2009) and C. K. Raju's Cultural Foundations of Mathematics (Pearson, 2007) both treat the Yuktibhāṣā as what it is: the earliest calculus textbook whose derivation has survived. The vocabulary is different. The mathematics is the same.
The Yuktibhāṣā was not widely known outside Kerala until Charles Whish mentioned its contents in 1835. K. V. Sarma's critical edition appeared in 1948, and an Indian National Science Academy English translation with commentary was published in 2008. The text has since reshaped the historiography of early calculus. A straightforward reading of its derivations places the origin of the modern theory of power series in fourteenth-century Kerala rather than seventeenth-century England.
A result is not science until somebody writes down how it was obtained. Mādhava's verses contained the sine series as a procedure, but they did not explain why the procedure worked. Jyeṣṭhadeva's Yuktibhāṣā took the additional step of writing the proof. That extra step is the one that turned a set of remarkable numerical facts into a reproducible body of theory. Any discipline that wants its results to outlive its founders has to make the same step, and has to make it deliberately.
The Yuktibhāṣā of Jyeṣṭhadeva was composed around 1530 CE in Malayalam prose. Its derivation of the jyā-śreṇī uses the limit of a sum with arbitrarily many terms, a procedure of successive correction, and an iterative refinement of the coefficients, all expressed in ordinary language without specialized symbolic notation. It predates Newton's work on series (1669) by 140 years and Taylor's general theorem (1715) by 185 years.
The sin() Function in Your Pocket: Mādhava's Series Running Billions of Times a Second
You ask your phone to compute the sine of any angle. What happens inside the chip takes about a hundred nanoseconds. The math library does not use a lookup table, because a full table big enough to be accurate would not fit in cache. It does not ask a server on the internet, because the latency would be a million times too slow. What it does is evaluate a polynomial. Specifically, it reduces the angle to a small interval near zero, then computes the sine of the reduced angle by plugging into a truncated power series. The series it uses is, up to a harmless change of variables, Mādhava's jyā-śreṇī with the first six or seven terms kept. Modern implementations use clever tricks on top: Horner's scheme to minimize multiplications, range reduction by multiples of π/2, the CORDIC method for certain coprocessors. But the heart of the computation is still a truncation of the series that Mādhava wrote down in Saṅgamagrāma around 1400 CE. Your phone is running Kerala-school mathematics. So is your laptop, your microwave oven's digital clock, the navigation system in every commercial aircraft, the image-processing code in every satellite, and the machine-learning framework training a neural network at a data centre. There are tens of billions of devices on earth, and each of them evaluates sine and cosine tens of thousands of times per second during normal use.
Mādhava's procedural framing of the sine series (start with the arc, apply successive refinements, stop when the correction is smaller than your tolerance) is a very close match to how a modern numerical library actually evaluates sine. The library does not imagine itself summing an infinite number of terms. It truncates at whatever term makes the remainder smaller than machine precision. This is exactly the discipline of the saṃskāra: apply the next refinement if and only if it will make a measurable difference. The vocabulary of 'successive corrective term' that the Kerala school developed in 1400 turns out to be the natural vocabulary for numerical analysis running on hardware in 2026. The six-hundred-year gap disappears when you watch what the math library is doing and compare it, line for line, with what Jyeṣṭhadeva wrote down.
Every scientific calculator, every computer graphics engine, every signal-processing pipeline, every GPS receiver, and every physics simulation on earth uses a truncated sine power series as its basic trigonometric primitive. The count is hard to state precisely, but the total number of times Mādhava's procedure has been executed across all hardware in a typical day comfortably exceeds the population of the planet multiplied by several thousand. The Kerala school's most important single result is also the most frequently used piece of mathematics in contemporary life, even though the credit line on the library function usually reads Taylor or simply 'IEEE 754 compliant'.
A mathematical result does not age the way a physical object ages. Mādhava's series is not a ruin or a relic. It is a live piece of code running on billions of processors right now. The distance between fourteenth-century palm-leaf manuscripts and twenty-first-century silicon is, in the case of the jyā-śreṇī, essentially zero. The lesson for any working mathematician is that a sufficiently fundamental result can outlast not only its discoverer but even the attribution to its discoverer, and still be doing useful work at the heart of everyday technology.
Modern C-language math libraries (glibc, musl, Apple's libm, Android's bionic) compute sine using polynomial approximations of degree 13 to 17. These polynomials are minimax-optimized variants of the truncated Taylor series, which in turn is Mādhava's jyā-śreṇī. The coefficients differ from Mādhava's exact ones by tiny optimization shifts. The structure is identical.
Historical context
The Kerala School of Astronomy and Mathematics (14th to 16th century CE)
The Kerala school flourished during a period when the rest of Indian mathematical activity had slowed. The great Ujjain tradition of Brahmagupta and Bhāskara II had gone relatively quiet by the fourteenth century, and the political upheavals of the Delhi Sultanate and its aftermath disrupted the patronage networks of northern India. Kerala, protected by the Western Ghats and connected to the wider world primarily through its ports, was less affected. Saṅgamagrāma, modern Irinjalakuda, was a small inland town near the coast whose Nambudiri Brahmin families maintained continuous traditions of Sanskrit learning. Mādhava worked in this setting, teaching a small circle of students whose own students and grandstudents (Parameśvara, Dāmodara, Nīlakaṇṭha, Jyeṣṭhadeva, Śaṅkara Vāriyar, Acyuta Piṣāratī) continued the tradition for two hundred years. The entire Kerala school output was produced by at most a few dozen individuals working in perhaps a dozen households across central Kerala.
The jyā-śreṇī is the pivot on which the global history of calculus turns. Before the Kerala school, infinite series were curiosities in Greek and Arabic mathematics. After Mādhava, they were the central tool of a working trigonometric tradition with explicit derivation, rigorous proof, and practical application. When Newton and Leibniz developed their versions of calculus in the 1660s and 1670s, they reached a framework whose core results on trigonometric series Mādhava had already possessed for two centuries. Whether the Kerala results reached Europe through Jesuit transmission routes in Kerala (a question raised by C. K. Raju and others) or were independently rediscovered, the priority is secure and the historical picture of calculus's origins is irreversibly altered.
Living traditions
Mādhava's sine and cosine series survive in three layers of modern life. The first is academic: the Yuktibhāṣā and Tantrasaṅgraha have critical editions, translations, and active scholarly discussion, and the history of mathematics has been quietly rewritten in the last fifty years to acknowledge the Kerala priority. The second is technical: every time a computer or calculator evaluates a sine, the operation is a truncation of Mādhava's series with coefficients optimized but structure unchanged. The third is cultural: the Nambudiri households of central Kerala still preserve palm-leaf manuscripts of the Yuktibhāṣā and related works, and projects like the Kerala manuscripts library at the University of Kerala continue to digitize and catalogue them. The discipline of treating an infinite sum as a sequence of saṃskāras applied to an initial value, which Mādhava made explicit in Sanskrit verse, is now the standard conceptual frame for numerical analysis on digital hardware. It is one of the most successful conceptual exports in the history of mathematics.
- Irinjalakuda (Saṅgamagrāma): Saṅgamagrāma, modern Irinjalakuda, is the town where Mādhava lived and worked. It is also home to the Kūḍalmāṇikyam temple, one of the ancient Vaiṣṇava shrines of Kerala, around which the Nambudiri Brahmin community that produced Mādhava's lineage was organized. The town itself is modest, but the surrounding countryside preserves the rural Kerala of Mādhava's time: rice paddies, coconut groves, and traditional Nambudiri illams (family homes) in which palm-leaf manuscripts were once kept. A visit gives a direct sense of how intimate and geographically small the setting of the Kerala school actually was.
- Thirunavaya and the Nambudiri Manuscript Tradition: Thirunavaya was a major seat of Nambudiri Brahmin learning in medieval Kerala and is traditionally associated with the Mamankam festival, a multi-decennial gathering that included, among many other functions, a time for scholarly exchange. Several of the manuscript collections that eventually transmitted the Kerala school texts to modern editors passed through families in and around Thirunavaya. The town retains its temple complex on the Bharatappuzha river and gives a sense of the patronage network that sustained the tradition.
Reflection
- Think of a problem in your own work where you rely on a first estimate and then apply corrections as new information comes in. Are your corrections converging toward a definite answer the way Mādhava's saṃskāras converge on the true sine, or are they random adjustments that never settle?
- The Yuktibhāṣā took the idea of a sum with arbitrarily many terms seriously, without flinching. What do you flinch from in your own intellectual life, and what would it look like to treat it with Mādhava's calm?
- If the sine series is Mādhava's and not Taylor's, what should happen to the names in a textbook? Is the right response to rename the series, to cite both discoverers, to leave the name alone and add a footnote, or to do something else entirely?