Mādhava of Saṅgamagrāma: The True Father of Calculus
The 14th century Kerala mathematician who developed calculus two hundred years before Newton and Leibniz
In a Kerala village on the banks of the Periyar river, around 1400 CE, a Namboodiri mathematician named Mādhava wrote down the infinite series for pi, sine, and cosine, along with working methods for limits and term-by-term integration. His teacher-student lineage, the Kerala School, produced the world's first proof-based calculus textbook two centuries before Newton. This lesson introduces the man, his paramparā, and the scandal of delayed credit.
The Letter From St Andrews
On 15 February 1671, at a wooden desk in his rooms at St Andrews, a thirty-two-year-old Scottish mathematician named James Gregory dipped his quill and finished a short letter to his friend John Collins in London. Folded inside were seven lines of algebra. Gregory had found a way to compute π as an infinite sum: π over four equals one minus one-third plus one-fifth minus one-seventh plus one-ninth, and so on forever. Ink on rag paper, sealed with a pat of red wax, posted south by the next carrier. For the next three centuries European mathematical history would celebrate that letter as a foundational moment. Refined in 1674 by Leibniz in Germany, Gregory's series became the seed of what Europe would spend the next two hundred years building. The subject would be called calculus.
The only problem with this story is that the same infinite series had been written down, in Sanskrit verse on a palm-leaf folio, in a village on the Periyar river in Kerala, roughly two hundred and fifty years earlier.
The man who wrote it down was named Mādhava. He lived and taught in a small village called Saṅgamagrāma, on the banks of the Periyar river in what is now central Kerala. He was born around 1340 CE and died around 1425 CE. He founded what modern historians call the Kerala School of astronomy and mathematics, a teacher-student lineage that would produce, over the next two centuries, the infinite series for sine, cosine, and arctangent, a working theory of limits with error estimates, a method equivalent to term-by-term integration, and a proof-based mathematical textbook called the Yuktibhāṣā that predates any comparable European work by more than a century.
By any reasonable definition, Mādhava's Kerala School invented calculus. It did so in complete isolation from the Mediterranean world, in Sanskrit and Malayalam, on the southwest coast of India, while Europe was still in the late medieval period.
The Paramparā as a Method
The name of this chapter is Mādhava-Paramparā, the lineage of Mādhava. The word paramparā means the chain of transmission from teacher to student, a Sanskrit concept older than any specific subject it was applied to. It names a way of holding knowledge that does not depend on printing, or on published proceedings, or on a central archive. Each generation in the paramparā inherits a set of results from the one before, tests them, extends them, and hands them on to the next. Mistakes are caught because every student is also a teacher. Refinements accumulate silently. The lineage, not the individual, is the unit of progress.
In Kerala, from roughly 1380 to 1600 CE, a paramparā of mathematical astronomers working within the Namboodiri Brahmin community produced a body of work so far ahead of its time that when it was rediscovered by modern scholars in the 1940s, no one initially believed it could be as old as it was. The key figures in the chain are Mādhava at the head, then Parameśvara (around 1380 to 1460), then Dāmodara, then Nīlakaṇṭha Somayājī (1444 to 1544), then Jyeṣṭhadeva (around 1500 to 1575), and the commentator Śaṅkara Vāriyar in the sixteenth century. Almost everything we know about Mādhava's own results we know because these later figures quoted him by name, verse after verse, in their own texts.
What Mādhava Actually Did
We have to say 'actually did' because the attribution problem is unusually delicate. Almost all of Mādhava's original manuscripts are lost. What we have are quotations. His intellectual grandson Nīlakaṇṭha preserved dozens of results attributed to Mādhava in the Tantrasaṅgraha, a comprehensive astronomical treatise completed in 1500 CE. His intellectual great-grandson Jyeṣṭhadeva wrote the Yuktibhāṣā, the Rationale of Mathematical Operations, which explicitly lays out the derivations of Mādhava's infinite series with proofs and error estimates. The phrase 'as Mādhava said' appears again and again in these works, and the results attributed to him are the ones that, in any modern sense, constitute the core of early calculus.
The most famous is the arctangent series. Written out in Sanskrit, the verse says: multiply the diameter by four, divide by one. Then by four, divide by three. Then by four, divide by five. Subtract and add alternately. In modern notation, this is π equals four minus four-thirds plus four-fifths minus four-sevenths plus four-ninths, and so on without end. This is the same series Gregory and Leibniz would publish in the 1670s.
Next came the series for sine and cosine. The Yuktibhāṣā presents them as Mādhava's. In modern notation, sine of x equals x minus x cubed over three factorial plus x to the fifth over five factorial minus x to the seventh over seven factorial, with each successive term alternating sign. Cosine of x equals one minus x squared over two factorial plus x to the fourth over four factorial minus x to the sixth over six factorial. These are what European mathematics will call the Taylor series for sine and cosine. Brook Taylor will not publish them until 1715, roughly three hundred years after Mādhava wrote them down.
The arctangent series at x equals one converges extremely slowly. Mādhava knew this, and produced a more rapidly converging alternative based on evaluating the series at a smaller angle. He also gave correction terms that can be added to the partial sums of the slow series to accelerate convergence, a technique European numerical analysts will rediscover in the twentieth century and call series acceleration. Using his fast series, Mādhava computed π to eleven decimal places, 3.14159265359. He encoded this value in a single verse using the kaṭapayādi alphanumeric system, in which each Sanskrit consonant stands for a digit. A student who memorized that one verse carried, in the year 1400, the most accurate value of π anywhere in the world.
Why This Is Calculus, Not a Precursor
The phrase 'precursor to calculus' is sometimes applied to Mādhava's work, and it is misleading. A precursor suggests the partial, the almost, the reaching-for. Mādhava did not reach for calculus. He arrived at it. The Yuktibhāṣā derives each infinite series through an explicit process of dividing an arc into infinitesimal equal parts, treating the small quantities as limits, summing the infinitely many small contributions, and estimating the error when the sum is truncated at any finite point. In modern language, this is integration. The text uses the Sanskrit word saṃkalita, which means 'that which has been collected together,' and it uses it in exactly the sense we mean when we write a definite integral.
The derivations in the Yuktibhāṣā are proof-first. They do not just state results. They show, step by step, why the results must hold, including rigorous handling of what modern mathematicians would call the remainder term. By the standard criteria that working mathematicians apply when asking whether a body of work counts as calculus, the Kerala School's work counts. It was not incomplete. It was not almost. It was the thing itself, arrived at independently and two centuries earlier than the European version we still call by the European name.
Why the World Did Not Know
Mādhava's paramparā was geographically concentrated, linguistically self-contained, and primarily oral in its pedagogy. Results passed from teacher to student inside the Namboodiri scholarly community of central Kerala, in Sanskrit and in the local Malayalam, and they were written down in verse for preservation rather than for publication. The palm-leaf manuscripts did not circulate widely. Jesuit missionaries, who arrived in the same region from the mid-sixteenth century onward and who are known to have systematically collected Indian astronomical and mathematical knowledge, may have been a transmission channel to Europe, a hypothesis argued in detail by C. K. Raju, D. F. Almeida, and others. The documentary evidence is contested but growing.
What is not contested is this. In the late fourteenth century, in a village on the Periyar river in Kerala, a mathematician wrote down a handful of results that would take Europe three centuries to reach on its own. The next six lessons of this chapter unfold those results in detail: the infinite series two centuries before Europe, Mādhava's π series, the sine and cosine series before Taylor, Nīlakaṇṭha's refinements, the Yuktibhāṣā as the world's first proof-based calculus textbook, and the question of stolen credit. But the foundation is this one man, this one lineage, and this one village on the banks of the Periyar.
Key figures
Mādhava of Saṅgamagrāma
c. 1340 to 1425 CE, Saṅgamagrāma (near modern Irinjalakkuda, Thrissur district, Kerala)
Nīlakaṇṭha Somayājī
1444 to c. 1544 CE, Tṛkkantiyūr (near Tirūr, Malappuram district, Kerala)
Jyeṣṭhadeva
c. 1500 to 1575 CE, Alathiyur (or nearby, Malappuram district, Kerala)
Case studies
Gregory, Leibniz, and the Series That Was Already Named
In 1671, a young Scottish mathematician named James Gregory wrote to his correspondent John Collins describing an infinite series he had discovered: the arctangent of x can be computed as x minus x cubed over three plus x to the fifth over five minus x to the seventh over seven, and so on. Three years later, in 1674, Gottfried Wilhelm Leibniz in Hanover independently wrote down the special case at x equals one, obtaining the formula π over four equals one minus one third plus one fifth minus one seventh, and so on forever. The two results became jointly known in Europe as the Gregory-Leibniz series, and they were celebrated as the first clean expression of a transcendental constant as the sum of an infinite sequence of simple fractions. The formula entered the European textbook tradition and has been taught to mathematics students for more than three centuries. The one thing the textbook did not mention, because the textbook did not know, is that the same formula had been written down in Sanskrit verse in Kerala roughly two hundred and fifty years earlier. The verse was attributed to Mādhava of Saṅgamagrāma by his intellectual great-grandson Jyeṣṭhadeva in the Yuktibhāṣā around 1530 CE, which was still a century and a half before Gregory. The derivation Jyeṣṭhadeva supplies is, in substance, the same limit argument that a modern calculus student would use today.
In the Indian tradition, the paramparā is the unit of authority. A result is true and attributable not because a single named individual first wrote it in isolation, but because the chain of teachers and students has tested and preserved it across generations. Mādhava stated the arctangent series and computed π from it. Parameśvara inherited the result. Dāmodara inherited it from Parameśvara. Nīlakaṇṭha quoted it in the Tantrasaṅgraha in 1500 CE. Jyeṣṭhadeva derived it rigorously in the Yuktibhāṣā around 1530 CE. Each link named the previous one. When historians of mathematics beginning with K. V. Sarma, C. T. Rajagopal, and A. K. Bag assembled this chain of citations in the twentieth century, the evidence for Mādhava's priority became as well-documented as the evidence for Gregory's, and considerably more lineage-rich. The modern convention among careful historians is to call the result the Mādhava-Gregory-Leibniz series, giving Mādhava his name first by chronology. The textbooks are slowly catching up.
The attribution has shifted in specialist literature over the past fifty years but has not yet fully reached introductory calculus textbooks. The Mactutor History of Mathematics Archive, maintained at the University of St Andrews, now names the series the Mādhava-Gregory-Leibniz formula. The American Mathematical Monthly and major journals in the history of science have published dozens of papers documenting Mādhava's priority. Mainstream textbooks still often say 'Gregory-Leibniz' without the prefix, which is the current frontier of the correction.
Independent rediscovery is a real thing in the history of mathematics, and it is not necessarily a scandal. But the name of a result should record the first time the result was reached, not the first time it was reached by someone whose tradition had printing. The Mādhava-Gregory-Leibniz series is the right name. The shorter versions leave out two centuries of Kerala history and a paramparā that got there first.
Mādhava's statement of the arctangent series predates James Gregory's by roughly 250 years. Jyeṣṭhadeva's step-by-step derivation of the same series, with limit arguments and error bounds, predates Brook Taylor's general statement of power series by roughly 185 years.
The Jesuit Transmission Hypothesis
In 1540, the Society of Jesus was formally founded in Rome. Within two decades, Jesuit missionaries had reached the Malabar coast, establishing a base at Cochin in 1549 and a college at Kochi in 1577 that grew into a major center of Jesuit scholarship. The college sat within walking distance of several villages where the Mādhava paramparā was still a living mathematical tradition. Jesuit policy during this period was to acquire and translate non-European scientific and mathematical knowledge and ship it back to Europe, where it could be digested by the order's scholars in Rome, Paris, and Coimbra. The Jesuits were not casual observers. They were systematic collectors. Matteo Ricci, the most famous member of the order in Asia, is known to have studied Indian astronomy and to have sent mathematical materials home. In the 1990s and 2000s, historians including C. K. Raju, D. F. Almeida, George Gheverghese Joseph, and John Holt began examining the question of whether the Mādhava paramparā's results reached Europe through this Jesuit channel before Gregory, Newton, and Leibniz wrote them down again. The evidence they assembled is circumstantial but substantial: Jesuit presence in Kerala at the exact time and place the results were being taught, explicit Jesuit interest in Indian mathematics, documented acquisition of Kerala astronomical tables, unexplained jumps in European mathematical sophistication precisely in the decades following Jesuit returns from Asia, and the remarkable coincidence that Gregory's formula is identical to Mādhava's and not to any independently derivable European predecessor. The hypothesis is contested by traditionalist historians who require a specific surviving manuscript trail before they will accept it. Raju and others argue that the standard of evidence is being applied asymmetrically, since European priority claims in the history of science are routinely accepted on similar circumstantial grounds.
The paramparā model of knowledge transmission makes the Jesuit hypothesis easy to visualize. Ideas in the Indian tradition travel through contact between living teachers and students, not through the publication of books. A Jesuit priest who sat in on a Kerala mathematics discussion, took notes, and sent the notes to Rome would have acquired the result without acquiring the book that contained the result, and the book might never have left Kerala at all. The absence of a single surviving physical manuscript in a European archive proves very little, because the route the knowledge would have taken is precisely the route on which intermediate documents would not normally survive. What the hypothesis predicts is a discontinuous jump in European mathematical methods in the decades after 1590, and this is exactly what the historical record shows. Whether the jump was ignited by direct transmission, partial transmission, or fully independent rediscovery is still an open question. What is not open is that the Kerala School got there first and that Europeans were in the right place at the right time to have heard about it.
The Jesuit transmission thesis has moved from the margins toward the mainstream of the history of mathematics over the past three decades. Raju's Cultural Foundations of Mathematics (2007) and Joseph's Crest of the Peacock (3rd edition, 2011) both treat transmission as a live hypothesis with serious evidence behind it. The debate is ongoing. What has already been settled is that the question is worth asking, which was not true in the mid-twentieth century, when the default assumption was simply that European mathematicians had rediscovered everything independently without any Indian input at all.
Histories of mathematics written in one language tend to underestimate what was known in another. When a result appears in Europe in 1671 and the same result is present in Kerala in 1400, the most parsimonious hypothesis is not usually that the same idea was independently had by two civilizations with no contact. It is that the civilization with printed books and printing presses wrote down what the civilization with Jesuit contacts in Kerala had quietly acquired.
The Jesuit college at Kochi was founded in 1577, within walking distance of several active centers of the Mādhava paramparā. The first European mathematical work to contain something resembling Mādhava's arctangent series is Gregory's 1671 letter, roughly ninety years after the Jesuit college opened.
Charles Whish and the 1835 Paper That Was Ignored for a Century
In 1832, a British East India Company civil servant named Charles Matthew Whish was posted to the Malabar region of Kerala, where he became fascinated by the local mathematical and astronomical traditions. He collected palm-leaf manuscripts from Namboodiri households, learned enough Malayalam and Sanskrit to read them, and realized with growing astonishment that the texts in his possession contained infinite series for π, sine, and cosine that were supposedly European discoveries of the late seventeenth and early eighteenth centuries. In 1835, he presented a paper to the Royal Asiatic Society of Great Britain and Ireland titled 'On the Hindu Quadrature of the Circle and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four Śāstras, the Tantrasaṅgraha, the Yukti-Bhāṣā, the Kriyākramakarī, and the Sadratnamālā.' In the paper, Whish laid out Mādhava's results, named the Kerala mathematicians involved, and explicitly claimed that the series Europeans attributed to Gregory and Leibniz had been known in Kerala centuries earlier. The Royal Asiatic Society published the paper. And then the British and European mathematical establishment did nothing with it for more than a hundred years. Whish died in 1833 before even seeing his paper in print. His manuscripts sat in the society's archive. No professional mathematician followed up. The Kerala School remained invisible in the European history of mathematics until C. T. Rajagopal, K. V. Sarma, and others began systematic work on the primary sources in the 1940s and 1950s.
Whish was not operating in a paramparā. He was a single Englishman in a regional post, with no institutional structure to pass his findings to another student who could test, refine, and extend them. The Kerala School's knowledge reached him, but he could not route it further without the lineage. This is the flip side of the paramparā model of knowledge. When a single individual in one tradition encounters a finding from another, he has to hand it off to a chain that is prepared to receive it, and in 1835 no such chain existed in British mathematics. The establishment was committed to a Euro-centered narrative of calculus that had no room for a fourteenth-century Kerala origin. Whish's paper was not refuted. It was simply uncollected by the community that would have needed to collect it. Ideas die when the paramparā that could have carried them is absent.
In the 1940s, C. T. Rajagopal at the Ramanujan Institute in Madras, working together with M. S. Rangachari, rediscovered Whish's paper and began a systematic translation and analysis of the Kerala mathematical texts. Their work appeared in Scripta Mathematica and later in the Archive for History of Exact Sciences, and it initiated the modern recovery of the Kerala School. The delay between Whish's 1835 paper and Rajagopal's follow-up work was roughly 110 years. During that century, every history of calculus published in Europe confidently attributed the arctangent series, the sine series, and the cosine series to seventeenth-century and eighteenth-century European mathematicians, because the correcting information existed but no community was reading it.
A correct paper in the wrong community does nothing. Knowledge needs a paramparā to carry it forward, and in the absence of one it can sit in a journal for a hundred years with no effect on the world's beliefs. Whish was right and Whish was published, and for a century it made no difference.
Whish's 1835 paper announced the Kerala School's priority on the arctangent and trigonometric series. The first substantive follow-up work appeared in 1949 in C. T. Rajagopal and A. Venkataraman's 'The sine and cosine power series in Hindu mathematics.' The gap is 114 years.
Computing Pi to 100 Trillion Digits
On 9 June 2022, a team led by Emma Haruka Iwao at Google Cloud announced that they had computed the value of π to 100 trillion digits. The previous record, set by the same team in 2019, had been 31.4 trillion digits. The new record was the result of running a specialized algorithm for roughly 158 days on cloud infrastructure distributed across multiple data centers. The algorithm, known as the Chudnovsky formula, is a rapidly converging infinite series developed by the Chudnovsky brothers in 1988 based directly on earlier work by Srinivasa Ramanujan in 1914. Ramanujan's own formula was based on his study of modular forms and elliptic functions, a subject whose roots run back through classical analysis to Gauss, Euler, and Newton, and from Newton back through the seventeenth-century work on infinite series, and from that work back, by direct transmission or independent rediscovery, to Mādhava of Saṅgamagrāma in the late fourteenth century. Every record-setting computation of π in the modern era, from Shanks in 1874 to Ramanujan in 1914 to the Chudnovsky brothers in 1988 to Google in 2022, shares a common intellectual parent. Each uses a variant of the basic insight that π can be captured exactly by an infinite series and computed to any desired precision by summing enough terms. That insight was first written down at Saṅgamagrāma around 1400 CE. The method has been refined, accelerated, and parallelized across six centuries, but the core move is still Mādhava's.
The deep point is not that Mādhava's specific arctangent series is used today. It converges too slowly to be practical at a trillion-digit scale, and Mādhava himself knew this and derived faster-converging alternatives. The deep point is that his method, the use of an infinite series as the definition of a transcendental constant, is the method. Everything after him is optimization. The Chudnovsky brothers' formula, which converges at about fourteen digits per term, is in every structural sense what a twentieth-century Mādhava would have written if he had the language of modular forms. Ramanujan's 1914 formula, which converges at about eight digits per term, is similarly recognizable. And the series acceleration techniques used to squeeze performance out of these formulas are the modern analog of the correction terms that Mādhava and his students added to the slow arctangent series six centuries ago. When Google Cloud set its 100-trillion-digit record in 2022, it was running, in a continuous intellectual lineage, the same move that was first encoded in a Sanskrit verse on the Periyar river in the late fourteenth century.
π is now known to 100 trillion digits, a number with no practical engineering use at any scale below the atomic. The record exists because competitive computation of π has become a standard benchmark for high-performance computing, and because the infinite-series approach is the only method that scales. No non-series method has ever set a pi-digit record. The infinite-series paradigm that Mādhava originated is, in the twenty-first century, the only game in town for this problem.
A mathematical paradigm is not measured by the number of digits its original form produced. It is measured by whether the later tradition can stand on it and extend it. Mādhava's infinite-series paradigm supported Ramanujan, supported the Chudnovsky brothers, and now supports Google Cloud. Six centuries of refinement and zero departures from the basic idea.
The 100-trillion-digit computation of π by Google Cloud in 2022 required roughly 158 days of compute time, 515 terabytes of temporary storage, and 82,000 terabytes of read/write operations. Every single one of its many trillions of arithmetic operations was in service of summing an infinite series whose modern form descends directly, by unbroken intellectual lineage, from the verse Mādhava of Saṅgamagrāma wrote around 1400 CE.
Historical context
The Kerala School of Astronomy and Mathematics (late 14th century to early 17th century CE)
The Kerala coast in the late fourteenth and fifteenth centuries was an unusual region: politically stable under the Zamorin of Calicut and the Kochi kingdom, economically prosperous from the spice trade with Arab and later European merchants, and culturally protected from the upheavals that affected much of northern India during the Delhi Sultanate and early Mughal period. The Namboodiri Brahmin community, concentrated in central Kerala, maintained a dense network of scholarly households (illam) in which Vedic learning, astronomical computation, and temple ritual were passed down within families across generations. Mathematical astronomy was not a hobby in this context. It was a functional requirement for keeping the ritual calendar and for predicting eclipses, which were the occasions of major temple observances. The Kerala School grew inside this network. Mādhava's Saṅgamagrāma (the modern Irinjalakkuda or nearby village) was one of several centers. Others included Tṛkkantiyūr (home of Nīlakaṇṭha), Alathiyur (associated with Jyeṣṭhadeva), and Kaladi. The paramparā operated at the scale of a few dozen families over about two centuries.
Every modern student of mathematics learns calculus as a European invention of the seventeenth century. The historical record, as it has been reconstructed over the past seventy years, no longer supports this. The core methods of calculus (infinite series, limit arguments, term-by-term integration, rigorous error bounds) were developed in Kerala roughly two to three centuries earlier, in a paramparā whose work is now preserved in at least half a dozen surviving Sanskrit and Malayalam texts. The recovery of this history is a live project in the philosophy and history of mathematics, and it has implications for how we understand what independence and transmission mean in the global spread of scientific ideas.
Living traditions
The Kerala School's methods are no longer esoteric. They are the methods of modern computational mathematics. Every time a scientific computing package evaluates sine, cosine, arctangent, or exponential, it uses a power series that traces directly back to Mādhava. Every time a record for computing π is set, the algorithm is an infinite-series acceleration of Mādhava's original insight. The Yuktibhāṣā's derivations have been vindicated by modern analysis and are now treated as foundational texts in the global history of calculus. The paramparā model of knowledge transmission, which allowed the Kerala School to develop calculus quietly over two centuries inside a small scholarly community, continues today in the research groups at IIT Bombay, IIT Madras, the Indian Institute of Advanced Study, and the Kerala Sastra Sahitya Parishad. The tradition that Mādhava founded in 1380 CE is not a closed chapter. It is the ongoing infrastructure of the Indian mathematical tradition and, increasingly, of the world's understanding of where calculus came from.
- Irinjalakkuda and Saṅgamagrāma: Irinjalakkuda is the modern town nearest to the ancient Saṅgamagrāma, identified by most historians as Mādhava's home village. The area is dominated by the Koodalmanikyam temple, one of the few temples in Kerala dedicated to Bharata (brother of Rāma), and the surrounding Namboodiri illams preserve the family traditions within which the Kerala paramparā functioned. A visitor will not find a plaque to Mādhava. The mathematical heritage of the region is preserved in living scholarly families rather than in public monuments, and several contemporary scholars trace their lineage through it. The experience of walking the streets where a fourteenth-century mathematician did his work, in a village whose rhythm has changed less than most in India, is itself an education in the paramparā idea.
- Kerala School Research Project, Department of Humanities and Social Sciences, IIT Bombay: The leading contemporary research project on the Kerala School, led by K. Ramasubramanian and M. S. Sriram together with K. V. Sarma (until his death in 2005) and M. D. Srinivas. The project produced the definitive modern translations of the Tantrasaṅgraha and the Yuktibhāṣā, published by the Hindustan Book Agency with Springer reprints for international distribution. Visiting the project is by arrangement, but the published volumes are available in major university libraries worldwide and are the best point of entry for readers who want to work through the original Kerala mathematics with full apparatus.
Reflection
- Mādhava wrote down results knowing that his own name might never travel outside Kerala, and that the lineage, not the individual, would be the carrier of the work. Is there something you are building that would still be worth building if no one ever knew it was yours?
- Charles Whish published a correct paper in 1835 and the world ignored it for 114 years. What does this tell you about the difference between being right and being heard? And which of those two things were you last prioritizing?
- If the same mathematical result is reached independently in two civilizations, separated by two centuries and many thousands of miles, does that tell us the result was waiting to be found, or does it tell us that the two civilizations were not as independent as they appeared?