Firstly, your claim about Stillwell's "conclusion" is a misrepresentation. Stillwell makes no such conclusion about the lack of transmission. In fact, he explicitly states that the Kerala school knew these mathematical series before 1540. This selective reading and distortion of Stillwell's work is intellectually dishonest and undermines genuine historical inquiry.
The Jesuits sent to India weren't not your typical bible thumpers; they were highly trained mathematicians and astronomers with a specific mission to study and acquire Indian mathematical and astronomical knowledge. The primary motivation for Europeans to import knowledge from India wasn't mere academic curiosity - it was a matter of practical necessity, particularly in navigation. By the mid-16th century, Europeans were grappling with significant errors in their calendar calculations. The true solar year was about 11.25 minutes shorter than the assumed 365.25 days, an error that had compounded over centuries, leading to serious discrepancies in timekeeping and navigation. Matteo Ricci, the Jesuit astronomer and mathematician in a letter from India to Giovanni Battista Maffei (Italian mathematician) he states that he requires the assistance of an “intelligent Brahmin or an honest Moor” to help him understand the local ways of recording and measuring time.
If one wants a smoking gun—a direct admission of knowledge transfer—is either naïve or deliberately obtuse. Do you also believe that tea plants magically teleported from China to India? The British East India Company's industrial espionage in China's tea industry parallels the Jesuits' activities in India perfectly. Both were covert operations aimed at acquiring valuable "know-how" for economic and strategic gain. Do we have a signed confession from Robert Fortune or his kin admitting to tea espionage?
The cumulating circumstantial evidence isn't just substantial—it's overwhelming. We have documented records of Jesuits studying Indian texts, teaching Indian concepts, and corresponding with European mathematicians (see my other comment for examples). The methodological similarities between Kerala mathematics and later European work, like the striking parallels between Wallis and the Yuktibhasa (15th century), where Wallis (in 17th Century) is using the exact expression and reasoning as given in the Yuktibhasa, aren't coincidences—they're smoking guns.
Your dismissal of this substantial body of evidence goes beyond healthy skepticism. It appears to disregard the complex realities of historical knowledge transfer risks coming off as a deliberate attempt to erase non-European contributions to mathematical history.
The claim of "extremely thin evidence" for the transmission of Kerala mathematics to Europe by Jesuits is far from accurate. In fact, there's a wealth of circumstantial evidence supporting this possibility. Jesuits were present in Kerala from 1540-1670, with many, like Matteo Ricci, being highly trained mathematicians tasked with studying Indian sciences.
We have clear documentation of their interest in local mathematics, astronomy, and timekeeping, even incorporating subjects like jyotisa into their curricula. Numerous examples show Jesuits actively gathering and transmitting knowledge, from Ricci's inquiries about Indian calendars to Schreck's astronomical observations sent to Kepler.
Their close relationships with the Court of Cochin provided access to valuable mathematical manuscripts, and there's evidence of collaboration with Brahmins in translating Sanskrit works.
The Jesuits were strongly motivated by practical needs in navigation and calendar reform. Moreover, Marin Mersenne's extensive correspondence network demonstrates that awareness of Indian mathematical knowledge was circulating in Europe. Intriguingly, there are methodological similarities between Kerala mathematics and later European developments, such as parallels between methods used by Wallis and those in the Yuktibhasa.
I believe it's crucial to consider the historical context of knowledge transmission between cultures, which often involved clandestine methods. A prime example is the case of Robert Fortune, a Scottish botanist, who in 1848 undertook a covert mission for the British East India Company. Fortune, disguised as a Chinese merchant from a distant province, infiltrated China's heavily guarded tea-growing regions. His objective was to acquire tea plants and seeds, along with the closely guarded secrets of tea production. Fortune's mission was successful; he managed to remove thousands of tea plants and seeds from China, effectively ending the Chinese monopoly on tea production.
This act of industrial espionage had far-reaching consequences, leading to the establishment of vast tea plantations in India and Ceylon (now Sri Lanka), and fundamentally altering the global tea trade. While this example pertains to botany rather than mathematics, it illustrates the lengths to which nations went extract knowledge.
(Source: Joseph, G. G. (2011). The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition). Princeton University Press.)
I think it’s important to clarify that describing the Babylonian method of estimating the area under a curve using trapezoids as "proto-integration" or "pre-calculus" might be a bit misleading. While their approach demonstrates an impressive grasp of geometry and an early method for approximating areas, it doesn't quite align with the formal development of calculus that emerged centuries later.
Madhava of Sangamagrama, a 14th-century Indian mathematician, made groundbreaking contributions to calculus that were far more advanced. He is known for discovering infinite series expansions for trigonometric functions such as sine, cosine, and arctangent, as well as deriving power series for π. His work included innovative methods for numerically approximating π to remarkable precision. In comparison, Madhava's achievements represent a significant evolution in mathematical thought. While the Babylonians were certainly ahead of their time, their techniques were still relatively basic when juxtaposed with the sophisticated concepts introduced by Madhava. His work laid critical groundwork for the later development of calculus by figures like Newton and Leibniz.
The Babylonians, while advanced for their time, were still operating in a more primitive mathematical framework. So while the Babylonians showed an inkling of ideas that would later blossom into calculus, it's an overstatement to equate their methods directly with calculus. Madhava's work represents a much more mature and developed understanding of these concepts. The Babylonians were pioneers, but Madhava was a revolutionary in comparison. Let's give credit where it's due!
I encountered this topic a while, back and had a deep look into it, I will be sharing my insights and formed opinion based on the facts that I encountered.
@Dx51Q
I appreciate your perspective, but I'd like to clarify a few points regarding Madhava's contributions to calculus. While it's true that Madhava and his school may not have created a unifying framework like Newton and Leibniz did, their work laid crucial groundwork for what we now consider calculus. Madhava is credited with developing infinite series for trigonometric functions such as sine and cosine, which are equivalent to the Taylor series we use today.
For example, his series for sin(x) and cos(x) predate those discovered in Europe by over 200 years[1][2].
His followers, like Jyeṣṭhadeva, further elaborated on these concepts in texts like the Yuktibhāṣā, providing proofs and demonstrating their applications[3][5]. Moreover, Madhava's methods for approximating pi were remarkably accurate, achieving values correct to 11 decimal places, showcasing his advanced understanding of numerical analysis[2][4].
This indicates that he was indeed engaging with concepts foundational to calculus, such as limits and convergence. Thus, while Madhava's work may not fit neatly into the modern definition of calculus, it represents a significant and sophisticated mathematical tradition that deserves recognition as a precursor to later developments in the field.
While we are on this topic, we can stop for a second and ponder on why the ancient Indians needed these mathematical formulation. The answer is astronomy, and thus needing a language/framework to understand the cosmos, i.e. mathematics.
Additionally, I wanted to share some interesting insights about the Jesuit transmission of both calculus and the Gregorian calendar from Kerala to Europe. The Jesuit missionaries were not only spreading Christianity through their work, but they were scholars in their own right and could see the value of the advance mathematics they encountered by the Kerala (India) school of mathematics by madhava and the advanced calendar, more accurate than the julian calendar used in Europe at the time.
Jesuit missionaries, especially Matteo Ricci, were really fascinated by the advanced mathematical knowledge coming from the Kerala school. They connected with local scholars, like Brahmins and Kshatriyas, to learn about their mathematical concepts, including those found in texts like the Yuktibhāṣā and Tantrasangraha [5]. This collaboration was part of the Jesuits' efforts to understand local cultures and improve their missionary work. It’s fascinating to think that this exchange not only contributed to the Gregorian calendar reform in 1582 but also helped introduce key calculus concepts into European mathematics.
As to why this is not common knowledge, it’s partly the British colonial policies that muddied the waters and/or suppressed the source of information. But If one looks at it with time, the evidence is there
While we praise Aryabhatta man, i would like to shed some lights on Madhava of Sangamagrama c. 1340 - c. 1425 CE from India who less well known
Key Contributions
Infinite Series and Trigonometry
Discovered power series expansions for trigonometric functions:
Madhava's Sine Series: Infinite series representation for the sine function.
Madhava's Cosine Series: Infinite series representation for the cosine function.
Madhava–Gregory Series: Series for the arctangent function, predating James Gregory by over 200 years.
Calculus and Mathematical Analysis
Laid early foundations of calculus through: 200 years before Newton or leibniz
Methods of term-by-term integration and iterative techniques for solving transcendental equations.
Concepts related to the area under curves, similar to integral calculus.
Introduction of convergence tests for infinite series.
Creation of trigonometric tables with accurate sine and cosine values.
The Jesuit missionaries in India played a crucial role in the transmission of advanced Indian mathematical and astronomical knowledge to Europe by learning local languages, collaborating with local scholars, and documenting key works, thereby significantly influencing the development of mathematics in the West.
The Jesuits sent to India weren't not your typical bible thumpers; they were highly trained mathematicians and astronomers with a specific mission to study and acquire Indian mathematical and astronomical knowledge. The primary motivation for Europeans to import knowledge from India wasn't mere academic curiosity - it was a matter of practical necessity, particularly in navigation. By the mid-16th century, Europeans were grappling with significant errors in their calendar calculations. The true solar year was about 11.25 minutes shorter than the assumed 365.25 days, an error that had compounded over centuries, leading to serious discrepancies in timekeeping and navigation. Matteo Ricci, the Jesuit astronomer and mathematician in a letter from India to Giovanni Battista Maffei (Italian mathematician) he states that he requires the assistance of an “intelligent Brahmin or an honest Moor” to help him understand the local ways of recording and measuring time.
If one wants a smoking gun—a direct admission of knowledge transfer—is either naïve or deliberately obtuse. Do you also believe that tea plants magically teleported from China to India? The British East India Company's industrial espionage in China's tea industry parallels the Jesuits' activities in India perfectly. Both were covert operations aimed at acquiring valuable "know-how" for economic and strategic gain. Do we have a signed confession from Robert Fortune or his kin admitting to tea espionage?
The cumulating circumstantial evidence isn't just substantial—it's overwhelming. We have documented records of Jesuits studying Indian texts, teaching Indian concepts, and corresponding with European mathematicians (see my other comment for examples). The methodological similarities between Kerala mathematics and later European work, like the striking parallels between Wallis and the Yuktibhasa (15th century), where Wallis (in 17th Century) is using the exact expression and reasoning as given in the Yuktibhasa, aren't coincidences—they're smoking guns.
Your dismissal of this substantial body of evidence goes beyond healthy skepticism. It appears to disregard the complex realities of historical knowledge transfer risks coming off as a deliberate attempt to erase non-European contributions to mathematical history.