Biman Nath, a professor of astrophysics at Raman Research Institute, Bengaluru, has a great article in *Frontline* about the development of proto-calculus ideas by ancient Indian mathematicians -- specifically, Aryabhatta, Brahmagupta, Bhaskara II (all in Sanskrit over 1000 years ago), and Madhava (in Malaalam, some 600 years ago).

Along the way, he makes a good point about the state of the 'debate' in the public sphere on history of Indian scientific ideas:

... [M]ost of us are not historians, and such discussions become pointless after a while. One has come across the names Aryabhata or Varahamihira in school textbooks, but one would be hard pressed to name any specific achievement of these ancient Indian scientists. School students are taught about the discovery of the zero but they never learn what it means to have “discovered” it.

How does one suddenly invoke a number and bring it down to the realm of reality? We teach our children that our ancestors must have had a remarkable knowledge of metallurgy and give the example of the iron pillar in Delhi, but we never spell out what exactly they knew.

This superficiality in our collective knowledge often leads to meaningless, rhetorical debates on the achievements of ancient Indian scientists. [...]

Nath peppers his article with snippets that give us a richer picture of how these mathematicians presented their results. For example, here's Aryabhatta:

... Aryabhata’s sine table was considered the most accurate ... He had tabulated the sines of 24 angles, equally spaced between zero and 90 degrees (with a difference of 3.75° between them). The table was in the form of a verse, and he had come up with an ingenious way of coding the table in the words of the poem. The introduction to his book Aryabhatia explained the code. [...]

And, Brahmagupta and Madhava:

It is not clear how Brahmagupta derived the formula—he never explained it anywhere—but historians think he used geometry.

[...]

Like his predecessor Brahmagupta, Madhava never explained how he came up with the formula. Historians of mathematics again suspect that he obtained it through geometrical methods.

Finally, all this leads to question of what happened after Madhava's work, and why it did not blossom into calculus as we know it today? I look forward to a follow-up article by Nath.

## 9 Comments:

- Archimedes had a more sophisticated anticipation of calculus, as the world learned from the famous Archimedes Palimpsest. He used Reimann sums to prove theorems and calculate physical quantities such as area enclosed by a parabola.

- One feature of Greek math was that proof was see as an important part of mathematics. Indians, on the other hand were more numerical and algebraic in their thinking. The Greeks lacked sophistication in algebra. The world had to wait till the globalization brought about by Islam in the 8th - 12th century CE for these two schools to be unified. In North Africa and Spain, Islamic scholars rediscovered the importance of the Greek canon. At the other end of the world, they learned Indian and Chinese mathematics.

- The Islamic commentaries and discoveries made their way to Europe via Fibonacci and other traders.

- As the Rennaissance unfolded, Fransesca Vieta and Rene Descartes understood the importance of mathematical notation. The importance of the development of modern math notation is often unrecognized. Then, Descartes also invented the notion of coordinate geometry to translate hitherto geometrical concepts into algebra.

- By the time Newton and Leibnitz discovered calculus, Descartes, etc. we're already well known across Europe, thanks to the printing press (lack of a printing press would have handicapped Madhava and co.).

- it is only with all these intermediate developments, many of which fortuitously hsppened in Newton and Leibnitz's world, that we get to their discovery of calculus. It is also the 'reason' why, Newton and Leibnitz did not happen in the subcontinent.

For more, see for example, ET Bell's "Development of Mathematics," Dover. It is a flawed book, but is a great overview of the history of mathematical ideas. He doesn't talk about Archimedes and calculus, IIRC, since the palimpsest discovery is fairly recent.

This is a very useful resource: http://nptel.ac.in/courses/111101080/2

A few comments:

0. Based on my earlier readings of the subject matter (and I have never been too interested in the topic, and so, it was generally pretty much a casual reading), I used to believe that the Indians had not reached that certain abstract point which would allow us to say that they had got to calculus. They had something of a pre-calculus, I thought.

Based (purely) on Prof. Nath's article, I have now changed my opinion.

Here are a few points to note:

1. How "jyaa" turned to "sine" makes for a fascinating story. Thanks for its inclusion, Prof. Nath.

2. Aryabhat didn't have calculus. Neither did Bramhagupta [my spelling is correct]. But if you wonder why the latter might have laid such an emphasis on the zero about the same time that he tried taking Aryabhat's invention further, chances are, there might have been some churning in Bramhagupta's mind regarding the abstraction of the infinitesimal, though, with the evidence available, he didn't reach it.

3. Bhaskara II, if the evidence in the article is correct, clearly did reach calculus. No doubt about it.

He did not only reach a more abstract level, he even finished the concept by giving it a name: "taatkaalik." Epistemologically speaking, the concept formation was over.

I wonder why Prof. Nath, writing for the Hindu, didn't allocate a separate section to Bhaskara II. The "giant leap" richly deserved it.

And, he even got to the max-min problem by setting the derivative to zero. IMO, this is a second giant leap. Conceptually, it is so distinctive to calculus that even just a fleeting mention of it is enough to permanently settle the issue.

You can say that Aryabhata and Bramhagupta had some definite anticipation of calculus. And you cannot say anything more about Archimedes' method of exhaustion either. But, as a sum total, I think, they still missed calculus, per say.

But with this double whammy (or, more accurately, the one-two punch), Bhaskara II clearly had got the calculus.

Yes, it would have been nice if he could have left for the posterity a mention of the limit. But writing down the process of reaching the invention has always been so unlike the ancient Indians. Philsophically, the atmosphere would generally be antithetical to such an idea; the scientist, esp. the mathematician, can then be excused.

But then, if mathematicians had already been playing with infinite series with ease, and were already performing calculus of finite differences in the context of these infinite series, I explicitly composing verses about the results, then they can be excused for not having conceptualized limits.

After all, even Newton initially worked only with fluxion, and Leibnitz with the infinitesimal. The modern epsilon-delta definition still was some two centuries (in three-four centuries of modern science) in the coming.

But when you explicitly say "instantaneous," (i.e. after spelling out the correct thought process leading to it), there is no way one can say that some distance had yet to be travelled to reach calculus. The destination was already there.

And as if to remove any doubt still lingering, when it comes to the min-max condition, no amount of merely geometric thinking would get you there. Reaching of that conclusion means that the train had not already left the first station after entering the calculus territory, but had in fact gone past the second or the third as well. Complete with an application from astronomy---the first branch of physics.

I would like to know if there are any counter-arguments to my new view spelt above.

[continued in my next comment]

[continued from my comment above]

4. Madhava missed it. The 1/4 vs. 1/6 is not hair-splitting. It is a very direct indication of the fact that either Madhava did a "typo" (not at all possible, considering that these were verses to be by-hearted by repetition by the student body), or, obviously, he missed the idea of the repeated integration (which in turn requires considering a progressively greater domain even if only infinitesimally). Now this latter idea is at the very basis of the modern Taylor series. If Madhava were to perform that repeated integration (and he would be a capable mathematical technician to be able to do that should the idea have struck him), then he would surely get 1/6. He would get that number, even if he were not to know anything about the factorial idea. And, if he could not get to 1/6, it's impossible that he would get the idea of the entire infinite series i.e. the Taylor series, right.

5. Going by the content of the article, Prof. Nath's conclusion in the last paragraph is, as indicated above, in part, non-sequitur.

6. But yes, I, too, very eagerly look forward to what Prof. Nath has to say subsequently on this and related issues.

But as far as the issues such as the existence of progress only in fits here and there, and indeed the absence of a generally monotonously increasing build-up of knowledge (observe the partial regression in Bramhagupta from Aryabhat, or in Madhav from Bhaskar II), I think that philosophy as the fundamental factor in human condition, is relevant.

7. And, oh, BTW, is "Matteo Ricci" a corrupt form of the original "Mahadeva Rishi" or some such a thing? ... May Internet battles ensue!

--Ajit

[E&OE]

Nice resource

Actually, it was not Aryabhata (nor Brahmagupta) who first imputed to zeros all the properties attributed to it in modern place value notation. This paper by D H Bailley shows that the first known evidence occurs in a anonymous Sanskrit text on astronomical observations (the observations contained therein fortuitously allow him to accurately date it, or so he claims) to a few decades before Aryabhata.

Abi, Rahul Siddharthan linked to this article which has some detailed analysis http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol47_4_11_PPDivakaran.pdf

Here is a wonderful book about the birth of Indian Calculus,dealt in a philosophical manner--

"Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16th C. CE" by C.K.Raju,an Indian Mathematician.(BTW his articles on the same topic--all of them can be read in Internet--are quite thought-provoking and excellent.)

AND ALSO--

” Finally, a few words about Indian mathematics, which I consider has not yet gained the global or domestic recognition it deserves. Apart from the well-known numeral system, and the algorithmic/computational revolution it sparked, the number of advances made in India long before they were (re-)discovered in Europe keeps increasing as we learn more of our own history. Look at these examples: a large part of algebra, first solutions to linear and quadratic indeterminate equations (Aryabhata, Brahmagupta); the binomial theorem, the combinatorial formula and Pascal’s triangle (Pingala 3rd century); second order interpolation formulas and the Newton–Raphson method (Brahmagupta), the Fibonacci numbers (Virahanka 700 CE, Hemachandra ~1150 CE); the basics of differentials, maxima of functions, mean value theorem, etc. (Bhaskara ~12th century, Munjala ~ 800 CE); infinite series, and a precursor of what later came to be known as calculus and analysis (Madhava 14th century): so the list goes on. These contributions are not just ‘little’ mathematics but the ‘Big picture’ “--Roddam Narasimha,a well known Indian Scientist.

In my understanding,I have concluded by asking a question to myself--

Can anyone guess why Euler--a great Mathematician-- has written a lot about Indian Mathematics? To sum it up, I add--

"If Mathematics is the Queen of Sciences" then, "India is the Queen of Mathematics”, albeit—an unknown Queen..

Thank You.

Archimedes used primitive exhaustion techniques like his other contemporaries. He did not know about Riemann sums.

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