What Indian Mathematicians Knew in Calculus

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.

Jayant Haritsa: Why Indian students should be goats, and not sheep

1. This year's Infosys Prize winner from our Institute, Prof. Jayant Haritsa, has a blog post at the Infosys Science Foundation website: Why Indian students should be goats, and not sheep.

2. The Infosys Prize ceremony happened a few days ago, and the videos have been posted on a YouTube channel devoted to the Prize. Jayant's acceptance speech is here, and the other highlight of the event, Prof. Amartya Sen's speech, is here.

   Here's a video profile of Jayant (you can find profiles of the other winners at the Infosys Prize channel). Knowing Jayant's sharp wit, I'm not surprised that he identifies "a very good sense of humour" as the first item when asked about "attributes of a researcher."

Links: Vedic Aviation Edition

Sidebar

See also:

Jayant Narlikar: Going Back in Time -- Science and Technology in Epics (The Times of India 1995).

Mayank Vahia (TIFR) in DNA: Cry my beloved science, cry my beloved country.

* * *

1. Hartosh Singh Bal in The Caravan: Why the Indian Scientific Community is to Blame for the Lack of Science at the 102nd Indian Science Congress.

   A copy of the proceedings that would include Bodas’ draft will only serve to confirm what the Islamic scholar Abu'l Raihan al-Biruni, writing almost a thousand years ago, had said about the state of Indian scientific knowledge: “I can only compare their mathematical and astronomical literature, as far as I know it, to a mixture of pearl shells and sour dates, or of pearls and dung, or of costly crystals and common pebbles. Both kinds of things are equal in their eyes, since they cannot raise themselves to the methods of a strictly scientific deduction.”

   But even these harsh words would constitute a charitable view. The reason the proceedings may still end up carrying Yonath and Ashtekar’s pearls along with Hindutva’s dung is not because our scientists cannot distinguish between them, but because they choose to look away in the face of a new political dispensation.

2. At one point, Bal refers to the Indian Science Congress as "India's premier scientific gathering." It looks like he hasn't got the memo that the ISC turned into some kind of a sad joke quite a while ago -- most Indian scientists avoid going to it if they can. In case they do go there, it's more likely because of an invitation from some hapless friend who got saddled with the responsibility of organizing a session on something or the other.

3. In any event, we now have some info on how the session on "Ancient Science through Sanskrit" got into the program, and how the organizers vetted the "contributions":

   Organisers defend session Organisers said the idea to conduct a session on ancient sciences emerged following a meeting of various university vice-chancellors with the Governor of Maharashtra. It was the vice-chancellor of Kavikulaguru Kalidas Sanskrit University in Nagpur, Uma Vaidya, who proposed the idea for such a session, said S B Nimse, chairman of the 102nd Indian Science Congress. Nimse said the sessions had been decided by a committee he headed and which had seven members, including TIFR Director Mustansir Barma, IIT Bombay Director Devang Khakhar, scientist Anil Kakodkar, Professor Kothari and two local secretaries. He too agreed that the controversy had “impacted other sessions”.

   Associate prof and head of Sanskrit department at Mumbai University, Gauri Mahulikar, who gave an overview on ancient Indian sciences through Sanskrit at the Science Congress, said that between August to December this year, there were several meetings between Vaidya and Sanskrit department of Mumbai University. “In one of the meetings, which included Vaidya, myself and teachers from Mumbai University’s Sanskrit department, it was decided to have such a topic at the Indian Science Congress. Thereafter, we called for papers and received five abstracts or presentations, which were jointly reviewed by Vaidya, myself and our teachers,” said Mahulikar.

4. It also turns out that neither the author of the Vedic aeronautics paper nor the organizer of that session is keen to part with the paper:

   When asked the reasons for not sharing a public presentation, Gauri Mahulikar, co-host of the event, associate professor and Head of Sanskrit Department, Mumbai University, said, "There are copyright issues. We fear that others who have nothing to do with this research, will claim it as their finding," she said. She said that a few students of Sanskrit had already taken some papers from them, and claimed to be their own works.

Links

1. Relevant to the goings on at Science Congress: H.S. Mukunda et al (1974): A critical study of the work “Vymanika Shastra”. [Hat tip to V. Vinay on Twitter]

   A study of the work “Vymanika Shastra” is presented. First, the historical aspects and authenticity of the work are discussed. Subsequently, the work is critically reviewed in respect of its technical content. It appears that his work cannot be dated earlier than 1904 and contains details which, on the basis of our present knowledge, force us to conclude the non feasibility of heavier‐than craft of earlier times. Some peripheral questions concerning dimensions have also been touched upon. [Bold emphasis added]

2. Also relevant: an engaging BBC Radio 4 podcast on Indian mathematics (45 minutes, mp3, via Sidin Vadukut). See also this review of Mathermatics in India and Square Roots in the Sulbasutra by Cornell mathematician David Henderson. [Hat tip to Siddharth Varadarajan]

3. Dalmeet Singh Chawla in Science Insider: India’s major science funders join open-access push

4. My colleage Prof. E. Arunan has started blogging. Here are the first three posts:

   • Indian Science: Catch up with India, then worry about China.

   • The more zeroes you have to your name, the better scientist you are!

   • Is there a leadership crisis in Indian science?

On the Indian Science Congress Agenda Today ...

... is a session on Ancient Sciences through Sanskrit -- see page 20 of the program. Some of the talks in the session have attracted media commentary (justifiably) filled with ridicule and scorn.

For many, this is like a bad dream that just won't go away -- Vedic astrology, Vedic nanotech, and now this. All trying to gain public legitimacy by being seen in the company of real science.

Here's just a sample of news reports on this session:

1. Stop Scientific Distortion (Mumbai Mirror). "NASA scientist Ram Prasad Gandhiraman, who started an online petition against an ISC lecture on ancient aviation, on why we must fight pseudo-science. [...] He has launched an online petition demanding that Capt Bodas's lecture scheduled for January 4 be cancelled as it brings into question the 'integrity of the scientific process'."

2. Hindu scientific temper: Elephant and cow urine to fuel aeroplanes? (Dinesh Sharma, Daily O). "Hindu organisations are on a mission to portray mythological and Sanskrit texts as the origins of modern science."

3. At Science Congress, Vedic aeroplanes and virus-proof suits - See more at: http://indianexpress.com/article/india/india-others/at-science-congress-vedic-aeroplanes-and-virus-proof-suits/99/#sthash.sUaAtlHU.dpuf (Mithika Basu, The Indian Express).

4. Science meet to discuss ancient Indian aviation (Kalyan Ray, Deccan Herald).

Shevgaonkar's Resignation: Update #3

ToI reports that while MHRD is yet to forward Shevgaonkar's resignation letter to the President, it is still keen on nailing him for the Mauritius initiative:

Meanwhile, the ministry is readying itself by preparing a case against Shevgaonkar on the Mauritius issue. Sources said IIT-Delhi has furnished all the details that the ministry wanted. "An attempt is being made to make Shevgaonkar personally responsible for the memorandum of understanding with the Mauritius institute. "The ministry has prepared a note against Shevgaonkar and is debating how to initiate action against him," one HRD source said. Technically, the ministry will have to seek permission of the President who is the visitor and appointing authority.

It is still unclear what exactly MHRD is talking about here, and no newspaper has managed to unearth the specific illegality it clearly wishes to pin on Shevgaonkar. This is an asymmetric war, since politicians, their followers on Twitter, and their un-named underlings in government can spew allegations (using intemperate, ominous, or even vile language), but the supporters of IIT-D (especially those within IIT-D) have to couch their rebuttal in a calm, dignified tone which does not play well on news media.

In any case, whatever MHRD has thrown at IIT-D and its Director has been rebutted convincingly. The latest example is this wonderful op-ed from Prof. M. Balakrishnan from IIT-D -- Why IIT can't fly. After laying all the facts out in his op-ed, Balakrishnan expresses his deep disappointment at all the meddling that he has seen:

A key aspect of the final MoU was that, contrary to the original proposal, administrative and financial control of IITRA would not rest with IIT Delhi, making it that much harder to bring up a quality institution. At that stage, we were deeply disappointed with the HRD ministry for clipping our wings even before we could attempt to fly. This was around the time that newspapers were full of stories about an amendment to the higher education act, permitting foreign institutions to open campuses in India. In hindsight, I am profoundly thankful that the bureaucrats at the HRD ministry knew the rules — no one is allowed to fly in this country, especially if you are funded by the government.

If that was about the previous government, this is about the present one:

... The only disappointment is that this country believed in the prime minister’s “minimum government, maximum governance” call and thought that he would devote his energies to enable people to fly and not to try to enforce 60 years of “no-fly” regulation. [...]