William Dalrymple is one of the finest historians and authors who has previously brought to life the story of East India Company and how it led to British India.
In this book, he focuses on, among other things, how Buddhism spread all over.
It's a fascinating topic. He also has a very successful podcast [1][2] -- with his co-host Anita Anand -- The Empire Podcast, which chronicles the rise and fall of empires. Highly educational, not to mention entertaining.
I second the podcast recommendation. Empire pod has been truly worth every bit of the time. I am currently on the episode about History of coffee - ethiopia, ottoman turks etc.
In the West the numerals we use are known as "Arabic numerals" because it came from the middle east. But in the middle east the same numerals are known as "Indian numerals" because that's where they got it from.
Similarly, algebra came from the middle east where it was called "al-Jabr" [1], but they in turn got it from India [2].
The invention of calculus is traditionally attributed to Isaac Newton and Gottfried Leibniz. But this book [3] says calculus was brought into being in India in the 14th century by a mathematician named Madhava.
Years ago I worked on a site which was translated into Arabic. Some of the early partners were adamant that we needed to use the Indic numerals everywhere, so I had localized all of our UI, including some JavaScript widgets and worked with a third-party to localize their embed. A few years later, they had a larger conference with a wider swath of the Arabic-fluent world represented and pretty much everyone said we should use what the English-speaking world calls Arabic numerals because most people were more familiar with them than the Indic numeral variants they didn’t use, and computers were making that even more pronounced.
It's difficult to call Madhava's work calculus as we know it. He and his school certainly made huge contributions to analysis of infinite series, but they did not develop that into a unifying framework. It's important to note that no mathematics is done in a vacuum, if Newton/Leibniz "invented" Calculus, it's by building on top of hundreds of years of work done throughout the world.
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
Going to preface this by saying that it's completely uncontroversial that the Kerala school discovered a lot of foundational mathematical concepts independently and in many cases earlier than Europeans, but the evidence for any transmission of those ideas to europe is _extremely_ thin, and you can see the organic development of it mostly within Europe in the textual record after the introduction of Arabic numbers and Arabic mathematical texts to Europe (which of course themselves where _hugely_ influenced by Indian mathematics themselves).
You can of course make the argument that colonialist historians are motivated to erase the contributions of foreign mathemeticians, but the _mathematicians themselves_ were not shy about crediting the influence of arabic philosophers and mathematicians, so it would have to be explained why they were fine with crediting al-Khwārizmī but drew the line at crediting Madhava. If they did have direct access to the works of the Kerala school, they'd have developed Calculus much more quickly, that's for sure.
It's very possible that there was some extremely vague indirect transmission through word of mouth where the source was obscure even to the mathematicians themselves, but I think it's hard to make the argument that "if not for Madhava, calculus would not have been discovered in Europe." There were many scientists and mathematicans all circling around the same problems and several of them solved aspects of it independently at the same time from different directions.
And the idea that _Euler_ only discovered power series with help from Indian mathematics is ridiculous. You can see in his own books and his correspondence with others how he gradually worked them out from first principles over time. There's no mystery as to where it came from. If he had just read about them from a book, he would have used them and not spent several years trying to figure it out.
The development of math in Europe was _absolutely_ dependent on the introduction of Indian mathematical ideas through Arabic texts, though.
Not in any mathematically significant way, except for the big endian little endian swap. In Indian system the units is specified first, then the 10s then the hundreds and so on. Arabs took the same system but they wrote right to left. That's why we write the numbers the way we do.
Anyone interested in Indian history and its interaction with the world should check out the Odd Compass channel on YouTube (https://www.youtube.com/@OddCompass) - it's really well done, entertaining and just makes you appreciate the world a little more - especially how interconnected we've all been, especially through trade.
I love this channel! The videos are well made narratively while still preseving the facts and citing sources. Almost the ideal combo of being academic and entertaining
> The great mathematician Aryabhata (476-550), in his masterwork composed when he was only 23, covers square and cube roots, the properties of circles and triangles, algebra, quadratic equations and sines, and contains a decent approximation of the value of pi at 3.1416.
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.
Which suggests a long oral and|or easily destroyed "document" tradition of teachings being passed down which came to Aryabhata who compiled such things in a manner that survived.
The sophistication of Babylonian mathematics boggles my mind. I am sure a credible science fiction story could be told where the Babylonians are a sophisticated alien race making Earth their home.
However, "the same kind of mathematics" rings dismissive. Trigonometry as we know it, came to its own and flourished in the middle ages in Indian, Arab and Persian civilizations. I am not aware of Babylonic trigonometry.
The story of the name of sin is itself quite interesting. It was half a 'jyay' (meaning chord subtended by an angle) in India. Through transliteration it became 'jayb' to the Arabs. Or the Europeans who were translating the Arabic mathematical literature derived from India, transliterate it as 'jayb', a phonetically similar bonafide Arabic word, that to this day is used to mean, a pocket/wallet/cavity. So pocket becomes sinus in Latin and then it evolves into just 'sin'. I think it was Napier who gave the name that we use.
Cultivation of geometry by Indian scholars go further in the past, to about 8th century BC as recorded in Sulbasutra.
You might be interested to know that combinatorics was also a hot topic among the Indian mathematicians. What we know as Fibonacci goes back far in the past, to Pingala (250 BC +/- 50). Pingala had worked out the binary numeral system and the 'Fibonacci' series.
What I am really keen to know is the mathematics of the Indus valley civilization, they were contemporaries of the Babylonians. Scarce little is known about their mathematics.
Did not expect to see this here. But I have been looking forward to this book for a while. I’m a big fan of William Darlymple. The Anarchy is my favorite of his.
I picked it up from the title alone because I was on a medieval history kick and I unreasonably expected it to be about... "The Anarchy"... you know, the period of history that actually goes by that name... but no. I was surprised and disappointed to find it was about The East India Company which I didn't have much of a hankering after. I still feel cheated!
Wow reading the comments here, there is so much in terms of invention and contribution that I wasn’t taught (in American schools). Given the importance of some of these contributions, I feel a bit shocked at how much is left out of our education in America (and I assume Europe) about what other cultures have provided. India in particular feels like a blind spot - they’re the largest country by population but also conspicuously missing. All you hear about is Gandhi, and even that is not covered well (in terms of the politics of colonization or the partition of India). It feels to me like it is purposeful - how else do you explain skipping out on all these math inventions that are critical to the modern world?
Many things are not taught in school, because basic education is short, and a lot of it happens before people's cognitive abilities have fully developed. The average kid probably spends about a year learning about society and culture, and much of it must be devoted to topics that are relevant to daily life in their own society.
Back in Finland some decades ago, there was pretty decent coverage of India in three topics: world religions and the history of religion; European explorers, colonialism, and imperialism; and "modern" history with Gandhi, Nehru, the partition, and the wars. There were also some passing mentions in other topics. Overall, we probably spent more time on Indian history than American history.
> and a lot of it happens before people's cognitive abilities have fully developed.
This is sort of trivially true, in that there is generally literally no point at which a person's cognitive abilities are "fully developed" (cognitive abilities being non-uniform in development, and some aspects of cognitive ability tending to continue developing until very late in life, long after most of the rest have been declining for quite a long time.)
The West hardly knows anything "true" of India; almost everybody looks at it only through a narrow perspective and thus are "incorrect".
The History of India is a multi-branched tree with the main branches being; a) Oral Tradition b) Linguistic Tradition c) Philosophical/Religious Tradition d) Literature/Poetry e) Mythologies f) Historical writings g) Writings by other cultures/civilizations h) Archeology. All of them have to be studied to get an idea of what India was/is.
1) A good book to start with is A.L.Basham's The Wonder That Was India: A Survey of the Culture of the Indian Sub-Continent Before the Coming of the Muslims.
Not just that! For many hundreds of years there were over 250 ships a year trading between India and Rome, and tariffs on that trade alone accounted for 25% of the Roman Empire’s tax revenue. Among many other things, India fed the demand for transparent clothing, which was all the rage in the Roman Empire during that period.
I’ve been fascinated with Indian history (mostly for selfish reasons to understand my lineage). My oversimplified summary has been,
India has had 3 major golden ages in its history.
The earliest was during the Mauryan Period. This is the age Dalrymple is talking about, the time of Aryabhatta, the popularization of Buddhism, arguably the time when most of the Indian epics were written (dating them has become notoriously political to the point that discerning the truth is hard now). This is the age Indian nationalists stress on, and the left wing tries to ignore. This age declines because the empire got too big, lost control and slowly disintegrated.
The second age was during the Islamic golden age for India which the right tries to ignore and left stress on. My general sense is that there was a golden age of architecture, poetry and arts and probably not so much in Science. The Taj Mahal for instance was built during this time. This age declined mostly due to wars, the Mughal rulers were never successful in fully unifying India, even to the extent the Mauryans did. They fought consistent wars against pockets of resistance in the South, and towards the end began losing these wildly expensive wars (leading to a brief reign by Shivaji)
The third golden age which no one wants to admit (left or right) is the British Golden age. There really was a renaissance in Indian thought in arts, science during British rule. This was when Indian history was “rediscovered”, first by British orientalists and then by mostly Indian Bengalis. CV Raman won the Nobel prize in science, Tagore won the Nobel prize in literature, Ramanujan etc, the names are numerous. The British rule also was the largest and most stable unification of India till the modern times. After 1850s there were almost no pockets of military resistance against the British rule. The British age declined with WW1 and WW2, and ended with Indian independence.
Post independence was not a smooth going party. If you came to India during 1980s, you could argue independence had been a disaster with everything getting worse post independence. But since 1990s economic liberalization India has a new ish golden age with unique characteristics. Who knows how long this will continue and when it will inevitably end
I don't think British India can be described as a Golden Age, when it was arguably the biggest transfer of wealth in history. When the British started colonizing India, its share of the World GDP was approximately between 1/5 and 1/4. So it was a major player. When they left, GDP share was a mere 4/100. Under British rule, India experienced massive de-industrialization. For example, the UK disassembled most of the handloom industry.
By golden age I only focus on top of the line Science, Art, Architecture etc which is a what a lot of people intuitively focus on since that is what we remember a period by. What rule was best for the common population is unclear.
It might have been the Mauryas but we don’t have enough details of how the common man lived in the Mauryan empire. The arthashastra seems like our best source. Both the Mughals and the British had their share of evils against the common population but my sense is that if you were Hindu you were better off in the British empire than the Mughal Empire except for the rare tolerant Mughal ruler (eg ; Akbar).
You forgot the Gupta Empire from mid 3rd century CE to mid 6th century CE. Aryabhatta - the famous mathematician was a product of this age.
The empire unfortunately promulgated the extensive spread of Buddhism, became pacifist and passive and was easily invaded and looted by invaders from Central Asia. The late Gupta empire's army was a joke - only existed on paper.
The fall of Indian civilization from the 7th century on-wards is very painful to read. No military of any note - Islamic invaders would regularly invade and loot and burn great temples and libraries every few years with little to no trouble. Foundational records and knowledge of ancient India was utterly annihilated. So much critical informational on Aryuveda, astronomy, mathematics and the physical sciences was lost forever. Some libraries were so huge, they burned for days.
Islam was at the height of its military era at the time and easily destroyed and annexed other passive cultures. Lots of journals of invaders from that time that documented their utter contempt for peoples who couldn't defend themselves.
The West had learned this lesson early and well from the Roman era: Si vis pacem, para bellum.
> CV Raman won the Nobel prize in science, Tagore won the Nobel prize in literature, Ramanujan etc, the names are numerous.
So golden age of India was when the country with a seventh of the world's population won 2 nobel prizes over 5 decades ?
> The British rule also was the largest and most stable unification of India till the modern times. After 1850s there were almost no pockets of military resistance against the British rule.
Mughal and Gupta empires lasted over 3 centuries, Mauryan empire a little under 1.5 centuries. By comparison, east india company rule lasted a century and the British crown's rule less than that. So again completely incorrect.
> The third golden age which no one wants to admit (left or right) is the British Golden age.
There's your hint: if people on both sides of the aisle don't "want to admit" something, maybe it doesn't make sense. Not to mention a slap in the face of billions of Indians.
> The British age declined with WW1 and WW2, and ended with Indian independence.
Thank god for that decline, otherwise Indian taxpayers would have been funding Brexit and the crumbling British economy right now.
> My oversimplified summary has been
This is not a summary, it's a lazy opinion backed by little research.
Have you read any books at all by Indians who lived through the British empire. Maybe start with “My Experiments with Truth” by Gandhi. The caricature that some modern Indians have made of the British empire would make even Gandhi turn in his grave.
But if you want to read something really heretic, maybe try reading An autobiography of an unknown Indian by Niraj Choudhary. Choudhary was a British raj supporter, as in an Indian who opposed Indian independence. Does that shock you? There were actually quite a lot of them, more than you’d expect.
Then if you want to get really metal, read in his own words, by Subedar Sitaram Pande. Sitaram Pande, was a soldier for the bengal army, for the British empire from 1812 to 1860, it’s one of the rare first author accounts we get of an Indian in that era. It will give you a glimpse of how an Indian at that time thought generally (hint: it was far more dominated by caste than you’d expect), how he viewed the empire and his relation to it. At that point I would say you are ready to try to understand Indian history that is not an avengers movie plot.
I would follow it with CK Majumdars history of modern India, one of the best historians so committed to the truth that Nehru had to throw him out of the government and try to prevent him from writing his book. Don’t worry, he’s not an heretic, he was an Indian freedom fighter, but you will find that he was far more honest about his life under Britain, under Indian national Congress and the state of the country in different periods of time (he also has a 12 volume set covering India for over 2000 years that I never had a chance to complete).
>Mauryan empire a little under 1.5 centuries. By comparison, east india company rule lasted a century and the British crown's rule less than that
This is a very dishonest way to obscure the actual facts.
Direct rule from Britain lasted for almost 90 years: 1858 to 1947. Even by your numbers then, that's 190 years: longer than the Mauryan empire's whole lifespan, and much closer to that of the Mughals. From there the question remains whether it's the longest "unification", and this mostly comes down to exactly when each of the aforementioned empires could be considered to have "unified" India.
By any definition the Mughals united the subcontinent by 1707AD at the latest: but by 1751, less than fifty years on, their effective domain had declined to a few pockets in Rajputana and Bengal.
The Guptan Empire on the other hand, while certainly a key predecessor to later Indian states and a major unifying force in the northern half of the subcontinent, never conquered the southern half -- what is today Karnakata, Kerala, and Tamil Nadu never entered their control. The closest they got was ~420AD after the south-eastern conquests of Chandragupta II, but again within fifty years they again lost control of today's Orissa, and even lost large swathes of north+western India to invasions from the steppe.
You call GP's post "a lazy opinion backed by little research", but when you dig into the facts I can't see how you could argue that his claim is incorrect. The British Raj alone seems to qualify as the longest-lasting unification of India before the modern Indian state, and if you include any part of the EIC's rule then it's indisputably so.
As an Indian, it is extremely naive and childish to dismiss any consideration of the British rule as a "slap in the face". The British introduced electricity, railways, capitalism and a thousand other things we take for granted behind those saffron-tinted glasses.
Hell, the British Raj was what unified India into a single national identity. It was more fractured than the European continent otherwise.
Good summary, even if I disagree with the 3rd piece being called a golden age. But my question is why ignore the cholas, pallavas and pandyas? Only because they didn’t capture the whole subcontinent? Our schools history books were Delhi centric but we don’t have to be going forward. Those 3 empires had a lot of arts and culture too - and in terms of impact, they spread to parts of SE Asia too.
That's the specialty of a centralized state. There is a heavy push towards homogenizing Indian culture often at the cost of regional subcultures since almost 1947.
I hear what you’re saying, but this idea of a golden age under rule by some outside force is often the positive take about many historical periods where someone who committed significant crimes (invasions, genocides, etc) is then praised hundreds of years later because of the positive effect they had for the economy or whatever. Julius Caesar is one such example that comes to mind.
But I feel that approach sometimes discounts what the country and its people could have achieved on their own if they were not invaded. For example, the Taj Mahal is just one building in India. There are lots of other structures that exist that were not built under rule from outside invaders, but they never get attention (at least I’ve not heard of them). With Britain - you mentioned that they unified India and brought stability but how true is that if you compare it to the period of India much earlier? India was rocked by Islamic invasions (which led to the Mughal period) and colonized by those Islamic rulers for hundreds of years. That’s what the European colonizers took over. Is that really a fair period to compare against? India basically spent a millennium ruled by one outside genocidal invader or the other. I am guessing the period before that was more peaceful and not in need of some outside unifying force to feel ‘stable’.
They never reached the highs reached by the Mauryan Empire, but were still a period of relatively prosperity and stability compared to what immediately followed (until the Mughals came)
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It's a fascinating topic. He also has a very successful podcast [1][2] -- with his co-host Anita Anand -- The Empire Podcast, which chronicles the rise and fall of empires. Highly educational, not to mention entertaining.
Recommended!!
[1] Empire Podcast - Apple - https://podcasts.apple.com/gb/podcast/empire/id1639561921 [2] On Spotify - https://open.spotify.com/show/0sBh58hSTReUQiK4axYUVx?si=45f5...
Dead Comment
Similarly, algebra came from the middle east where it was called "al-Jabr" [1], but they in turn got it from India [2].
The invention of calculus is traditionally attributed to Isaac Newton and Gottfried Leibniz. But this book [3] says calculus was brought into being in India in the 14th century by a mathematician named Madhava.
[1] https://en.wikipedia.org/wiki/Al-Jabr
[2] https://www.theguardian.com/world/article/2024/sep/01/hidden...
[3] https://www.nytimes.com/2024/07/24/books/review/the-secret-l...
https://persianlanguageonline.com/wp-content/uploads/2023/07...
Years ago I worked on a site which was translated into Arabic. Some of the early partners were adamant that we needed to use the Indic numerals everywhere, so I had localized all of our UI, including some JavaScript widgets and worked with a third-party to localize their embed. A few years later, they had a larger conference with a wider swath of the Arabic-fluent world represented and pretty much everyone said we should use what the English-speaking world calls Arabic numerals because most people were more familiar with them than the Indic numeral variants they didn’t use, and computers were making that even more pronounced.
@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
Citations: [1] https://www.linkedin.com/pulse/madhava-man-who-taught-trigon... [2] https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama [3] https://mathshistory.st-andrews.ac.uk/Biographies/Madhava/ [4] https://en.wikipedia.org/wiki/Madhava_series [5] https://indicmandala.com/the-kerala-school-european-mathemat...
You can of course make the argument that colonialist historians are motivated to erase the contributions of foreign mathemeticians, but the _mathematicians themselves_ were not shy about crediting the influence of arabic philosophers and mathematicians, so it would have to be explained why they were fine with crediting al-Khwārizmī but drew the line at crediting Madhava. If they did have direct access to the works of the Kerala school, they'd have developed Calculus much more quickly, that's for sure.
It's very possible that there was some extremely vague indirect transmission through word of mouth where the source was obscure even to the mathematicians themselves, but I think it's hard to make the argument that "if not for Madhava, calculus would not have been discovered in Europe." There were many scientists and mathematicans all circling around the same problems and several of them solved aspects of it independently at the same time from different directions.
And the idea that _Euler_ only discovered power series with help from Indian mathematics is ridiculous. You can see in his own books and his correspondence with others how he gradually worked them out from first principles over time. There's no mystery as to where it came from. If he had just read about them from a book, he would have used them and not spent several years trying to figure it out.
The development of math in Europe was _absolutely_ dependent on the introduction of Indian mathematical ideas through Arabic texts, though.
Not in any mathematically significant way, except for the big endian little endian swap. In Indian system the units is specified first, then the 10s then the hundreds and so on. Arabs took the same system but they wrote right to left. That's why we write the numbers the way we do.
Dead Comment
TIL...
- Invention of Zero - Decimal Place-Value System - Astronomical Calculations - Understanding of Negative Numbers
here is a good YouTube video on this subject https://www.youtube.com/watch?v=jgjcy04PDRM
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.
https://en.wikipedia.org/wiki/Babylonian_mathematics
Which suggests a long oral and|or easily destroyed "document" tradition of teachings being passed down which came to Aryabhata who compiled such things in a manner that survived.
https://en.wikipedia.org/wiki/Babylonian_mathematics
However, "the same kind of mathematics" rings dismissive. Trigonometry as we know it, came to its own and flourished in the middle ages in Indian, Arab and Persian civilizations. I am not aware of Babylonic trigonometry.
The story of the name of sin is itself quite interesting. It was half a 'jyay' (meaning chord subtended by an angle) in India. Through transliteration it became 'jayb' to the Arabs. Or the Europeans who were translating the Arabic mathematical literature derived from India, transliterate it as 'jayb', a phonetically similar bonafide Arabic word, that to this day is used to mean, a pocket/wallet/cavity. So pocket becomes sinus in Latin and then it evolves into just 'sin'. I think it was Napier who gave the name that we use.
Cultivation of geometry by Indian scholars go further in the past, to about 8th century BC as recorded in Sulbasutra.
https://personal.math.ubc.ca/~cass/courses/m309-01a/kong/sul...
You might be interested to know that combinatorics was also a hot topic among the Indian mathematicians. What we know as Fibonacci goes back far in the past, to Pingala (250 BC +/- 50). Pingala had worked out the binary numeral system and the 'Fibonacci' series.
https://en.wikipedia.org/wiki/Pingala
What I am really keen to know is the mathematics of the Indus valley civilization, they were contemporaries of the Babylonians. Scarce little is known about their mathematics.
I read this somewhere earlier, in some article about the history of mathematics, maybe Indian, Chinese, or both:
Take the number 113355 (easy to remember). Split it down the middle to get 113 and 355. Divide the latter by the former. E.g. in the Python shell:
> print (355/113)
Result:
3.1415929203539825
which is a slightly closer approximation to pi than 3.1416.
https://en.wikipedia.org/wiki/The_Anarchy
https://en.m.wikipedia.org/wiki/Wu_Zetian
Back in Finland some decades ago, there was pretty decent coverage of India in three topics: world religions and the history of religion; European explorers, colonialism, and imperialism; and "modern" history with Gandhi, Nehru, the partition, and the wars. There were also some passing mentions in other topics. Overall, we probably spent more time on Indian history than American history.
This is sort of trivially true, in that there is generally literally no point at which a person's cognitive abilities are "fully developed" (cognitive abilities being non-uniform in development, and some aspects of cognitive ability tending to continue developing until very late in life, long after most of the rest have been declining for quite a long time.)
The History of India is a multi-branched tree with the main branches being; a) Oral Tradition b) Linguistic Tradition c) Philosophical/Religious Tradition d) Literature/Poetry e) Mythologies f) Historical writings g) Writings by other cultures/civilizations h) Archeology. All of them have to be studied to get an idea of what India was/is.
1) A good book to start with is A.L.Basham's The Wonder That Was India: A Survey of the Culture of the Indian Sub-Continent Before the Coming of the Muslims.
The Book: https://archive.org/details/TheWonderThatWasIndiaByALBasham_...
The Author : https://en.wikipedia.org/wiki/Arthur_Llewellyn_Basham
2) For a modern idiosyncratic and funny take, see Sam Miller's book A Strange Kind of Paradise: India through Foreign Eyes.
Dead Comment
The earliest was during the Mauryan Period. This is the age Dalrymple is talking about, the time of Aryabhatta, the popularization of Buddhism, arguably the time when most of the Indian epics were written (dating them has become notoriously political to the point that discerning the truth is hard now). This is the age Indian nationalists stress on, and the left wing tries to ignore. This age declines because the empire got too big, lost control and slowly disintegrated.
The second age was during the Islamic golden age for India which the right tries to ignore and left stress on. My general sense is that there was a golden age of architecture, poetry and arts and probably not so much in Science. The Taj Mahal for instance was built during this time. This age declined mostly due to wars, the Mughal rulers were never successful in fully unifying India, even to the extent the Mauryans did. They fought consistent wars against pockets of resistance in the South, and towards the end began losing these wildly expensive wars (leading to a brief reign by Shivaji)
The third golden age which no one wants to admit (left or right) is the British Golden age. There really was a renaissance in Indian thought in arts, science during British rule. This was when Indian history was “rediscovered”, first by British orientalists and then by mostly Indian Bengalis. CV Raman won the Nobel prize in science, Tagore won the Nobel prize in literature, Ramanujan etc, the names are numerous. The British rule also was the largest and most stable unification of India till the modern times. After 1850s there were almost no pockets of military resistance against the British rule. The British age declined with WW1 and WW2, and ended with Indian independence.
Post independence was not a smooth going party. If you came to India during 1980s, you could argue independence had been a disaster with everything getting worse post independence. But since 1990s economic liberalization India has a new ish golden age with unique characteristics. Who knows how long this will continue and when it will inevitably end
It _was_ the Golden Age ... for the Brits! :-D
It might have been the Mauryas but we don’t have enough details of how the common man lived in the Mauryan empire. The arthashastra seems like our best source. Both the Mughals and the British had their share of evils against the common population but my sense is that if you were Hindu you were better off in the British empire than the Mughal Empire except for the rare tolerant Mughal ruler (eg ; Akbar).
The empire unfortunately promulgated the extensive spread of Buddhism, became pacifist and passive and was easily invaded and looted by invaders from Central Asia. The late Gupta empire's army was a joke - only existed on paper.
The fall of Indian civilization from the 7th century on-wards is very painful to read. No military of any note - Islamic invaders would regularly invade and loot and burn great temples and libraries every few years with little to no trouble. Foundational records and knowledge of ancient India was utterly annihilated. So much critical informational on Aryuveda, astronomy, mathematics and the physical sciences was lost forever. Some libraries were so huge, they burned for days.
Islam was at the height of its military era at the time and easily destroyed and annexed other passive cultures. Lots of journals of invaders from that time that documented their utter contempt for peoples who couldn't defend themselves.
The West had learned this lesson early and well from the Roman era: Si vis pacem, para bellum.
So golden age of India was when the country with a seventh of the world's population won 2 nobel prizes over 5 decades ?
> The British rule also was the largest and most stable unification of India till the modern times. After 1850s there were almost no pockets of military resistance against the British rule.
Mughal and Gupta empires lasted over 3 centuries, Mauryan empire a little under 1.5 centuries. By comparison, east india company rule lasted a century and the British crown's rule less than that. So again completely incorrect.
> The third golden age which no one wants to admit (left or right) is the British Golden age.
There's your hint: if people on both sides of the aisle don't "want to admit" something, maybe it doesn't make sense. Not to mention a slap in the face of billions of Indians.
> The British age declined with WW1 and WW2, and ended with Indian independence.
Thank god for that decline, otherwise Indian taxpayers would have been funding Brexit and the crumbling British economy right now.
> My oversimplified summary has been
This is not a summary, it's a lazy opinion backed by little research.
But if you want to read something really heretic, maybe try reading An autobiography of an unknown Indian by Niraj Choudhary. Choudhary was a British raj supporter, as in an Indian who opposed Indian independence. Does that shock you? There were actually quite a lot of them, more than you’d expect.
Then if you want to get really metal, read in his own words, by Subedar Sitaram Pande. Sitaram Pande, was a soldier for the bengal army, for the British empire from 1812 to 1860, it’s one of the rare first author accounts we get of an Indian in that era. It will give you a glimpse of how an Indian at that time thought generally (hint: it was far more dominated by caste than you’d expect), how he viewed the empire and his relation to it. At that point I would say you are ready to try to understand Indian history that is not an avengers movie plot.
I would follow it with CK Majumdars history of modern India, one of the best historians so committed to the truth that Nehru had to throw him out of the government and try to prevent him from writing his book. Don’t worry, he’s not an heretic, he was an Indian freedom fighter, but you will find that he was far more honest about his life under Britain, under Indian national Congress and the state of the country in different periods of time (he also has a 12 volume set covering India for over 2000 years that I never had a chance to complete).
Direct rule from Britain lasted for almost 90 years: 1858 to 1947. Even by your numbers then, that's 190 years: longer than the Mauryan empire's whole lifespan, and much closer to that of the Mughals. From there the question remains whether it's the longest "unification", and this mostly comes down to exactly when each of the aforementioned empires could be considered to have "unified" India.
By any definition the Mughals united the subcontinent by 1707AD at the latest: but by 1751, less than fifty years on, their effective domain had declined to a few pockets in Rajputana and Bengal.
The Guptan Empire on the other hand, while certainly a key predecessor to later Indian states and a major unifying force in the northern half of the subcontinent, never conquered the southern half -- what is today Karnakata, Kerala, and Tamil Nadu never entered their control. The closest they got was ~420AD after the south-eastern conquests of Chandragupta II, but again within fifty years they again lost control of today's Orissa, and even lost large swathes of north+western India to invasions from the steppe.
You call GP's post "a lazy opinion backed by little research", but when you dig into the facts I can't see how you could argue that his claim is incorrect. The British Raj alone seems to qualify as the longest-lasting unification of India before the modern Indian state, and if you include any part of the EIC's rule then it's indisputably so.
Hell, the British Raj was what unified India into a single national identity. It was more fractured than the European continent otherwise.
There hasn't been a single famine in india since independence. The last one was the bengal famine of 1943. India became independent in 1947.
But I feel that approach sometimes discounts what the country and its people could have achieved on their own if they were not invaded. For example, the Taj Mahal is just one building in India. There are lots of other structures that exist that were not built under rule from outside invaders, but they never get attention (at least I’ve not heard of them). With Britain - you mentioned that they unified India and brought stability but how true is that if you compare it to the period of India much earlier? India was rocked by Islamic invasions (which led to the Mughal period) and colonized by those Islamic rulers for hundreds of years. That’s what the European colonizers took over. Is that really a fair period to compare against? India basically spent a millennium ruled by one outside genocidal invader or the other. I am guessing the period before that was more peaceful and not in need of some outside unifying force to feel ‘stable’.
https://en.m.wikipedia.org/wiki/Gupta_Empire