Mādhava of Saṅgamagrāma, born around 1340 CE, led a flourishing school of astronomy and mathematics that lasted a couple of centuries. From Mādhava to Acyuta Piṣāraṭi, who died around 1620 CE, six or seven generations of an uninterrupted line of teacher-student mathematical community, with a majority of them living on the banks of the river Nila in Kerala, produced such signal contributions as Tantrasaṅgraha and Yuktibhāṣā. Containing the seeds of calculus in mathematics and a planetary model that foreshadowed the later Keplerian model in astronomy, the unique contributions of this school—that now stand convincingly established on the modern intellectual firmament—came to sharp focus essentially in the latter half of the 20th century. Coming together sometime in 1991, M.D. Srinivas, M.S. Sriram and K. Ramasubramanian—the three protagonists interviewed here—undertook an intense and discerning study of the Mādhava school, publishing authentic English translations of their works including that of Tantrasaṅgraha and Yuktibhāṣā. In a marathon effort lasting nearly ten hours, spanning over five long sessions, the trio shared their travails on the Saṅgamagrāma trail, as well as the several component threads of the Indian tradition, with much excitement and focus, and the present article tries to capture some of the exhilarating moments that the Bhāvanā team spent in their virtual company.
Greetings from Team Bhāvanā. This interaction is part of the History of Mathematics in India (HoMI) initiative based out of Gandhinagar, with which Bhāvanā is happy to be associated. This interaction is also the very first occasion when we are engaging with three minds simultaneously, to feature them together in a single theme-based interview. Considering the visible and continuous thread of ideas that runs through each of your individual works, and binds you all together, thematically, this is not surprising at all.
Starting right from the beginning, could each of you individually summarise your journeys starting from early childhood, and going perhaps all the way up to your doctoral studies?
M.D. Srinivas: Thank you for inviting the three of us to this joint session. Hopefully, we will be able to convey some highlights of our joint work.
I was born in Bengaluru (erstwhile Bangalore) on 2 October 1950. I studied up to class eight in a Kannada medium school. Then I joined a well-known school called National High School in 1962, from which time onwards, I had the privilege of having Sriram as my classmate. I completed my pre-university course at the same institution. This institution was set up during the freedom movement in 1917, under the idea of having desiya vidyashalas. The originators were all inspired by a high level of svadeśi and nationalist spirit, many of whom had participated in the freedom movement.
Our teachers were inspiring in several ways. Our mathematics teacher taught us with a lot of enthusiasm. Our Sanskrit teacher was also good, and I had the privilege of studying Sanskrit with another professor from what is today called Sanskrit University in Bengaluru, which was then called Chamarajendra Sanskrit College, located quite close to my home. The professor was an expert in alankāra śāstra, but he was happy to teach elementary things too. He made me memorise a number of śabda-rūpas and dhātu-rūpas, and tried to make me memorise some verses from the Amarakośa too. He also taught some simple texts, which was a very good experience.
In school, we once heard a brilliant lecture on the Līlāvatī of Bhaskaracharya by K.S. Nagarajan. That lecture was interesting, of course; beyond that, we had not heard anything more of Indian traditions in mathematics, astronomy or science. In 1961, Nagarajan had published a nice translation of the Līlāvatī in Kannada, explaining its mathematical content.1
When we entered university, Sriram and I were drawn to theoretical physics, due to an enthusiastic teacher, R.S. Srinivasa Murthy, who taught us at National College in Bengaluru. He used to lecture on a wide variety of topics in mathematical physics. He himself was inspired by a renowned professor at Central College, D.S. Subbaramaiya. For close to two years, we interacted regularly with Srinivasa Murthy, and consequently became seriously interested in various aspects of theoretical physics. Another teacher who inspired us greatly at National College was the Sanskrit professor H.S. Varada Desikacharya, who later became well known as Rangapriya Swamiji.
Around this time, we came in contact with T.S. Shankara, who later on took up sanyāsa, and became well-known as Paramananda Bharati Swamiji. He was a student of Alladi Ramakrishnan in MatScience,2 Chennai, and had earned his PhD in mathematical physics under his guidance. Sriram and I started working on some aspects of the `phase space representations of quantum mechanics’ with him, and even had a couple of joint publications by the time we finished our BSc honours course. Sriram moved to IIT Kanpur in 1969, and I continued at Bangalore University for my Master’s degree.
In 1972, I joined the University of Rochester in the USA to do my PhD with Emil Wolf, the famous co-author of Max Born’s Principles of Optics. The Indian physicist Girish Saran Agarwal, had worked on the phase space representations of quantum mechanics in the context of quantum optics with Wolf. Wolf was originally a Czech national, who moved to England and then to the United States to set up the famous Quantum Optics Group in the physics department at the University of Rochester. I also had the good fortune of learning mathematical physics from Gerard Emch, who was a mathematical physicist from Switzerland. A few years earlier, Kalyan Bidhan Sinha had worked for his PhD with him. Rochester was also the alma mater of E.C.G. Sudarshan, which he visited a few times during my stay there. My serious interest in subjects like quantum optics and the foundations of quantum mechanics got a boost due to the influence of scholars such as Wolf, Emch and Sudarshan.
In 1976, I finished my PhD and returned to India. I hadn’t had an opportunity to pursue the subject of Indian traditions of science in Rochester. However, in the Rochester library, I found a book written in 1967 by Frits Staal, called Word Order in Sanskrit and Universal Grammar, which I read with great interest. It was a small monograph of roughly 110 pages on Indian logic, linguistics, and their development from the Vedic period. It discussed many aspects of Paninian grammar, from which I came to realise that Indians had achieved a very high level of sophistication in the study of languages.
After returning in 1976, I joined the Department of Theoretical Physics at the University of Madras, where I spent the next 20 years. Although it was a small department, headed by P.M. Mathews, both the faculty and the research scholars were serious about theoretical physics. Sriram and Jatinder Bajaj joined the same department in 1978 as postdoctoral fellows with G. Rajasekaran, who was earlier at TIFR, and joined the department as a professor in 1976. He later moved to the Institute of Mathematical Sciences (IMSc). Several other friends of Sriram from IIT Kanpur would meet us frequently, and we would engage in serious discussions on the quality and relevance of the work that we were doing, how it could be of some use for our country and our people, scientific progress in India, and so on. This was also because these very topics were being debated and discussed in many places in India from different perspectives.
In July 1981, P.N. Haksar, Raja Ramanna and P.M. Bhargava and others, released the famous statement on scientific temper3 that provided a context for our own discussions on the subject. While in the 1970s, India was talked about as one of the leading Third World countries in science, with a high number of scientific publications, it was becoming clear by the 1980s that the efforts in science and technology did not quite have a commensurately significant impact either on the world of science or in meeting our national challenges and the needs of our people.
The main argument in this discussion had been that scientists in India were doing excellent work, but it was Indian society that lagged significantly behind in the so-called Modern Era. It suffered from a lack of scientific temper, and therefore the need of the hour, it was felt, was a large-scale reform of Indian society. Superstitious practices prevalent in Indian society were cited as being among the main reasons contributing to the country’s failure to capitalise on our endeavours in science and technology.
We took this discussion fairly seriously. One of the things that came up in all our discussions was that contrary to the prevalent view among scientists, as well as the general public, all available evidence seemed to show that India had a flourishing tradition of science and technology from ancient times. As is evident from the work of Dharampal,4 our sciences and technologies were functional till the late 18th century and attracted considerable attention and appreciation from European observers.
Over the course of several long and intense discussions, we came to the conclusion that science and technology in India lacked vitality because of two reasons. Firstly, modern scientists and technologists in India made little or no effort to connect their pursuits with the Indian traditions in their fields. Efforts to even record those traditions and the history of Indian sciences and technologies were scarce. Secondly, the problems that were being pursued had almost no relation to our national concerns and the needs and aspirations of the larger populace of India.
With this understanding, we constituted ourselves into a group dedicated to working towards comprehending and ameliorating the problems of Indian science and technology, and gave ourselves a name: the Patriotic and People-oriented Science and Technology (PPST) Group. Some of us went around various institutions in the country to initiate local chapters of the PPST Group. We also began studying the history and epistemology of the earlier Indian sciences and technologies on the one hand, and the current state of our science and technology efforts in various fields, and launched the PPST Bulletin, a forum for sharing our studies, analysis and concerns.
In 1991, a `Multi-Institution Project’, funded by the Department of Science of Technology (DST), was initiated by one of our friends (late) Navjyoti Singh who was then in Delhi. Under this project, a participating group from IIT Kanpur was looking at linguistics; Sriram was looking at astronomy, and Ramasubramanian joined as a research scholar. I was to collaborate with K.V. Sarma on the study of proofs in Yuktibhāṣā. A conference was held under this project in 1992 where several scholars from all over India participated and discussed different aspects of Indian mathematics and astronomy.
I opted for voluntary retirement from the university in 1996. A few years later, in 2001, I published a monograph summarising much of the work I had done at the University of Madras on the conceptual and mathematical foundations of quantum mechanics.5 Since 1996, I have been working at the Centre for Policy Studies. Here, apart from Indian science and technology, I work on polity, society and economy of both the eighteenth century and contemporary India.
Thank you for your reply. Can we move to Sriram for his part of the reply?
M.S. Sriram: I was born in a lower-middle-class family in Bangalore on 4 November 1950 and had all my early education up to BSc in Bangalore. Our family’s then financial situation was not great, but sufficient to provide reasonable education to all six children. It was implicitly understood that we had to strive and do well in our studies to move up in life. Initially, I carried out my early studies in Acharya Pathashala. As a matter of fact, C.N.R. Rao and Roddam Narasimha also had studied earlier in Acharya Pathashala. It is interesting to note that we had no science classes till class 8. We studied two languages, English and Kannada, where the latter was also the medium of instruction. Then funnily, there was a choice between Sanskrit and hygiene, and I had chosen Sanskrit. We also studied mathematics, history and geography.
Then we went to pre-university in the same institution. It was around this time that Srinivas and I made up our minds to study science, though it was expected in the family that I better study Engineering, get a decent job and earn well. But I was adamant and enrolled in the BSc Honours course at Central College, Bangalore. At that time Central college was the postgraduate college and research centre of Bangalore University, and the library there was good and well maintained. We enthusiastically got familiar with many classics, especially on relativity and quantum mechanics, studying them together.
After completing my BSc (Hons) in Physics in 1969, I went to IIT Kanpur, which was known to be very good for physics. In my class, there were 20 people, of which, ten were toppers from various Indian universities. So, the competition was somewhat severe. After my MSc, I continued there, working in theoretical high-energy physics for my PhD. It took me seven years, but such things were fashionable in those days. My time at IIT Kanpur coincided with a lot of interesting developments happening in the areas of particle physics and high-energy physics. I finished my PhD in 1978, and in the same year, I came to the University of Madras as a postdoctoral fellow. Srinivas was already a faculty member there. I was a postdoctoral fellow for three years, and I had to find myself a full-time job.
We had practical classes which were rare among high schools at that time in Karnataka
In 1981, I got a job at the University of Allahabad and was there for five years. It was a good centre, though the research activities had declined over the years. Also, for personal reasons, I had to come back to the south. I got married in 1983, and in 1986, I returned to Madras as a lecturer. My interest in Indian science took root only in 1990. From 1986 to 2011, I was at the University of Madras, joining as a lecturer and retiring in 2011 as a professor. I worked in both physics, and Indian astronomy and mathematics, but essentially gave up on research in physics after around 2001. Since 2011, I have been associated with the K.V. Sarma Foundation, a private organisation devoted to Indian civilisational studies.
Can we now move to Ramasubramanian for his take on the first question?
K. Ramasubramanian: Thank you. I hail from a highly traditional family and was born in 1969 in a village called Mullaikudi near Tirukkattupalli in Thanjavur district.
My grandfather’s name is Brahmasri Rajagopala Ghanapāṭhigal. The pāṭhaśālā which he used to run was described as “mobile pāṭhaśālā” by Kanchi Paramacharya, since my grandfather had to move from village to village along with the śiṣyas in order to earn some money by way of sambhāvanā for doing pravacanam. I heard from my father that he spent about three to six months in a place, doing Gītā or Rāmāyaṇa pravacanam. Once that is over, some sambhāvanā would be given by the villages. With that, he would move to a different village, and so on. When he was beyond 55 years of age, the Paramacharya asked a senior official of a company in Mettur to arrange for a permanent setup for him, and then Sri Shankara Gurukulam was established there, since the mode of migration from place to place was not tenable any more due to advancing age.
My father Krishnamurthi Sastrigal learned from my grandfather and had his education in Yajurveda. My grandfather was very keen not to send my father to school, and so soon after his Vedic education, he entered a Sanskrit college. In a similar manner, my grandfather also did not want me enrolled in a school and insisted on regular Vedic education. Somehow, my father was keen that I should pursue both Vedic as well as modern education. I used to go to school as usual, but at the age of seven, we had an initiation ceremony, where my grandfather initiated my formal Vedic adhyayana. There is an adventurous event which I seem to have been involved in when I was just four years old. My father had taken me to a school named Vidya Bharati, which was about 500 metres from my house and left me there. Within two hours, I was back home, and my parents were really surprised. They went to the school asking how I had gone back home. They were all puzzled, and later found a gaping hole in the gate, which explained the whole episode.
I also studied in Tirumala for a year during my second standard. From classes three to five, I studied in Madras in a nearby school, where many classes would happen simultaneously. From class six, I went to Oriental School, located inside Madras Sanskrit College, which was a real blessing. Oriental schools were established perhaps in the 1960s, primarily for introducing Sanskrit in schools, a move that was strongly opposed then in Tamil Nadu. These schools had an additional subject, Sanskrit, besides the regular state board curriculum. So from classes six to ten, I had the opportunity to learn Sanskrit formally in school. Some teachers took a special interest in me, possibly because I was the son of a professor teaching at Madras Sanskrit College, also located on the same campus. My mother was also interested in making me learn Hindi and Sanskrit, and insisted that I take up exams conducted by Bharatiya Vidya Bhavan, and Hindi Prachar Sabha. So I learnt Sanskrit, as well as Vedic traditions simultaneously till the 10th standard. After 10th, I switched over to study science in PS High School. By then, I had developed a very serious inclination to adopt the traditional style of clothing and attire, typical of Brahmin communities of India. Therefore, I requested the school to allow me to come attired in a dhoti. I even had to meet the authorities to obtain the necessary permissions.
After class 12, I was in a dilemma, debating a choice of either engineering or science. Even during my school days, I had faced this dilemma at various stages, when I used to attend Vedic sabhas with my father, where the gathering of scholars routinely conversed with each other in Sanskrit. I had always felt in such situations that I should quickly excel in Sanskrit, and go on to become a scholar just like my father. But then my father himself was insistent that I pursue science, and opined that I would anyway learn the śāstras growing up. So, after class 12, I joined Vivekananda College, which was just a few minutes away from home, to earn a BSc in physics. An interesting option available for BSc students was to be able to pursue an evening course in a chosen branch of engineering from another college, even while pursuing a full-time BSc course there, during the day. I took this up, essentially pushed by a friend to do so, who himself did not take it up! I was juggling a hectic schedule that started with an hour of Vedic classes in the morning, followed by my BSc course till late afternoon. This was quickly followed by three hours of evening classes in engineering, and all this had me completely occupied. In addition, my weekends were filled with workshops and practical classes from the engineering course. With a little extra effort, along with my BSc Honours in Physics, I also earned a Bachelor’s degree in Engineering.
By then, I was already enrolled in the first year of an MSc course in Physics. Before I completed my MSc second-year examinations, I was fortunate to attend a few lectures by Paramananda Bharati Swamiji. He used to give lectures at Madras Sanskrit College and I attended a couple of them. One of the faculty members from the Department of Physics, IIT Madras, M.V. Satyanarayana, was a devotee of the Swamiji. Through my interactions with Satyanarayana, I got introduced to Sriram and Srinivas. In April 1991, Satyanarayana came to my house, and said, `Ram, your help is needed.’ Satyanarayana mentioned a project taken up by Sriram, Srinivas and their group, and that they were looking for someone who had a background in both Sanskrit and science. He put me in touch with Sriram and Srinivas, even though I had zero exposure to astronomy. The project was titled “An analysis of the accuracy and optimality of algorithms in Indian astronomy”. I didn’t even know the names of Āryabhaṭa and Bhāskara back then. I had only studied a little bit of śāstras during my school days, picking up small bits of logic and vyākaraṇa. During the first interaction with Sriram and Srinivas, they gave me William Marshall Smart’s book Spherical Astronomy; and gave me four to five days to study the first two chapters of the text, and come prepared for an interview before taking me on board. This completely providential association has been extremely fruitful for the past 30 years.
Regarding this project, I enjoyed being a part of it and it went on till 1994. In between, we also jointly authored a very interesting paper. That is another interesting story which we’ll come to later, but an attempt was made around 1993 to meet the Registrar of the University of Madras to see if some part of the research which I had done till then, could be converted to earning a degree. This was apparently not possible, and I completed the project in 1994, following which I was in a dilemma whether to continue studying in the same discipline towards a PhD or do something else. Many people felt that if I continued along these lines, I would be left in a triśaṅku, suspended in a limbo that was neither history nor science, and urged me to drop it to take up some other contemporary and fashionable topic in physics. I then decided to become a regular research scholar and cleared the requisite UGC-JRF requirement. After that, I entered a PhD program at the University of Madras. Initially, for my PhD, we started with studying WKB approximations for semiclassical quantisation. I obtained a Mathematica programming manual through one of my friends in the US and learned how to use that tool extensively. Along with some programming techniques, I was almost ready to even publish a paper. At that point, I realised that the work had already been published by someone in Russia, and therefore we had to completely abandon it, with the work of a year and a half gone in vain. Then I found out that Sriram was working on nonlinear dynamics and chaos, which was a glamorous topic in those days. He suggested that we could work together, and eventually, we also had some interesting papers. I remember Sriram saying that by the time I completed my PhD, I should have at least three to four papers, and I was thankfully able to do that.
Since my inclinations were in the study of the śāstras, I was lucky to find an opportunity to study śāstra in a regular way, even as I was pursuing my PhD work. In 1994, a scheme was launched under the tutelage of the Sringeri Shankaracharya, after an industrialist expressed interest in funding and promoting sastric studies by offering fellowships of Rs. 1000 per month to students, and Rs. 3000 to teachers. There were four disciplines available for study—Vedānta, Mīmāṃsā, Nyāya, and Vyākaraṇa—and I enrolled in Vedānta. Sriram was kind enough to let me take off for a small duration during my research, to prepare for the examinations, every semester. By 2001, I had both a PhD and completed a more or less rigorous study in Advaita Vedānta śāstra by taking 13 semesters of studies and examinations, which, in a couple of years, fetched me the coveted title `Vidvat Pravara’. But then, I had to decide what to do next. There were suggestions to go abroad for postdoctoral options, but my family and I were both not inclined. So, Sriram and I decided to take up a project on Tantrasaṅgraha, with the Indian National Science Academy (INSA). From 2001 to 2004, we were working on this project and also published a couple of papers, covering almost the entire text.
In 2004, it was sheer providence that both Sriram and I attended a seminar organised at IIT Bombay. We delivered a couple of lectures there, and on one of the evenings of the three-day workshop, a few faculty members came over and asked me if I would be interested in joining IIT Bombay as a member of its faculty, if such a position was offered. I was not very keen, but then in a month, I again received a fairly longish letter, strongly urging me to join the institute. Both Sriram and Srinivas encouraged me to join, and that is how I joined IIT Bombay in 2004. I would like to take this opportunity to thank Professors Sriram and Srinivas, who provided the necessary mentoring all these years.
 |
 |
 K. Ramasubramanian |
Thank you, Ramasubramanian, for the happy reminiscences. We now move on to the next question, directed to Srinivas. You started your research career with a strong interest in quantum physics, focusing on the mathematical aspects of quantum physics. How did your interest eventually shift to matters related to the history of Indian contributions to science, and was it a specific event that triggered the change, or was it all, so to speak, slowly building up inside you for some time?
MDS: I have already explained the general atmosphere in India when this transformation occurred from my perspective, specifically between 1978 and 1980. It was very clear to me that in India, we needed to do some serious work on our own traditions of science, with at least the same kind of seriousness and rigour with which we pursue our modern studies. At least in the area of Indian studies, my fond hope was, and still is, that we should be the world leaders in the study of Indian knowledge and Indian civilisation. I was clear in my mind about the goal, but still not ready to undertake serious work in the area.
my fond hope was, and still is, that we should be the world leaders in the study of Indian knowledge and Indian civilisation.
The way I evolved was with the PPST movement, where I took the responsibility of bringing out the PPST Bulletin fairly seriously. I remember that I worked on all aspects of its production, apart from authoring a number of articles. There were several topics and articles that I worked on then. One of the early articles I wrote in that period was a preliminary bibliography of source works in Indian mathematics and astronomy, especially those that had been edited and translated. This was a list I first prepared back in 1981, and after 38 years, I came back to the same subject and wrote a more detailed article on it in 2019.6 I also wrote on agriculture as it was practised in 19th century India, based on various reports authored by British agricultural scientists, to show that Indian farmers were fairly conversant with efficient techniques of agriculture. I studied some aspects of Indian logic and linguistics. The framework of Indian logic is very different from that of the Greek and modern Western logic in that it’s not a logic of propositions; but a logic of jñāna, awareness, and cognition. It had developed sophisticated linguistic techniques to do what modern mathematical logic does with modern mathematical techniques. They could surely lead to many interesting consequences.
In 1985, a NASA scientist by the name of Rick Briggs wrote an article about the Śābda-bodha technique of Sanskrit logicians and grammarians. People working in AI at that time were studying techniques of `knowledge representation’, and Briggs thought that Śābda-bodha could solve certain problems in knowledge representation. This article got widely circulated, and Paramananda Bharati Swamiji was fairly excited to read about it. At his behest, the Sringeri Shankara Matha and the Computer Society of India organised an international conference in Bengaluru in December 1986 on “Knowledge Representation and Samskritam”, where Briggs and a few other international scholars such as B.K. Matilal, S.N. Srihari participated; and for three days, we had very interesting talks and interactions involving several eminent Indian scientists including E.C.G. Sudarshan, Roddam Narasimha, H.N. Mahabala, Vineet Chaitanya and others. Well-known Sanskrit pundits from all over India such as Peri Suryanarayana Sastri, N.S. Ramanuja Tatacharya, Goda Subrahmanya Sastri, Dinesh Chandra Guha, including Vidwans Ranganatha Sharma, N.T. Srinivasa Iyengar and other scholars from Bangalore, also spoke at the conference. Unfortunately, we could not bring out the proceedings of this conference.
In 1987, I wrote a preliminary study of proofs in Indian mathematics.7 I found that there was a large corpus of upapattis, or proofs, that was already available in published literature. This was amazing because all textbooks on histories of mathematics kept on saying Indians had no proofs in their mathematical source-works, and that logical rigour was unknown to them.
The first efforts at a translation of Yuktibhāṣā into English by K.V. Sarma was initiated in 1991. Around the same time, we also started looking at the work of the Kerala School on planetary theory which had not been systematically studied. We were able to see that Nīlakaṇṭha in his Tantrasaṅgraha was really proposing a new planetary model. It took a couple of years to get that paper published. Sometime later, around 2000, we started working seriously on Yuktibhāṣā, and soon Sriram and Ram also started their translation of Tantrasaṅgraha.
Thank you. Sriram and Ram may please feel free to add to what Srinivas has already spoken about.
MSS: In 1971, I started my PhD program at IIT Kanpur, and decided that I would work on high energy physics, looking at the relevant theoretical aspects of Quantum Field Theory (QFT). Any physically meaningful QFT is plagued with the problem of the appearance of infinite quantities, when one starts calculating physical quantities in higher orders of the coupling constant, or in other words, higher and higher accuracy. There is a strategy known as `renormalisation theory’ to handle this problem, and extract meaningful finite answers in the final steps. Though accurate and verifiable results are possible using this approach, the situation is not entirely satisfactory due to infinities cropping up in the intermediate steps.
At that time, under the guidance of Tulsi Dass in IIT Kanpur, I studied some papers that were based on the work of Abdus Salam, the famous Pakistani physicist, who was also the Director of the International Center for Theoretical Physics (ICTP), in Trieste, Italy. Salam attempted to resolve the problem of infinities in a quantised electromagnetic theory by coupling the gravitational fields also to the system, and using a certain special technique to remove the infinities even in the intermediate steps. My thesis work was on generalising this to other quantised field theories of physical interest. Later, I worked on what are called `soliton models’ of strongly interacting particles. This led me to the study of solitons in general, and the associated ideas of integrability and non-integrability of nonlinear systems. All of these converged in a study of the chaotic aspects of nonlinear systems, on which Ramasubramanian’s PhD thesis is also based.
From 1990 to 2000, I was seriously involved in the study of ancient Indian texts on astronomy, even while I continued to do particle physics and nonlinear dynamics. In the beginning, I was more interested in the calculational aspects of Indian astronomy, and I got into its more theoretical and conceptual study, a little later.
Thank you very much. Ramasubramanian, your take on the same question. I’ll just paraphrase the question on strange attractors of chaotic dynamical systems and Lyapunov exponents that Sriram was just now talking about, and which you also studied for your own doctoral dissertation. Were these `strange attractors’ not sufficiently attractive enough to resist your transition away from nonlinear dynamics, and towards a study of the historical aspects of Indian mathematics?
KR: That’s an interesting question. In fact, I would say that I moved from one strange attractor to another strange attractor. The first strange attractor was the astronomy project about which I had no initial clue whatsoever, but which quickly began to excite me. This is what happened in 1991. As I mentioned earlier, I was faced with a dilemma regarding the path ahead for me at the end of my secondary school, and much later in life too, after I earned my PhD in nonlinear dynamics. From close observations of the academic situation prevailing in the country, especially related to a rigorous study and practice of sastric studies, it became very clear that both the quality and the quantity of serious scholarship was lacking. There were not many competent people trained to both study and teach even standard texts such as Āryabhaṭīya and Tantrasaṅgraha, in Indian academic circles. This observation was a strong motivation for me to take up the Tantrasaṅgraha project in 2001.
The project itself was quite demanding, because this work was voluminous, with almost 430 verses that we had to interpret, understand thoroughly before putting them all in modern mathematical language. This was not only a demanding piece of work, but also posed some challenges in clearly deciphering the content of the text in some places. By the time I completed the project, an opportunity to join IIT Bombay came up wholly unexpectedly. They had started off a small unit called `Cell for Indian Science and Technology in Sanskrit’, and that is how they address it even now. This unit was started at IIT Bombay after receiving a note that was sent to all the IITs to start programmes of teaching and research in Sanskrit knowledge, by the Ministry of Human Resource Development, then headed by Murli Manohar Joshi. Apparently, a few members of IIT Bombay’s faculty, though not in favour of Sanskrit but were oriented towards Indian science and technology, felt that one could start a unit that focusses on studies in these areas which are accessible via Sanskrit.
This unit had started off in 2002, but none of us were even aware of it. However, when Sriram and I were there to deliver lectures, we were briefed about the Cell. There was a retired mathematics professor S.M. Bhave from Pune, who was offering a couple of elective courses pertaining to Indian mathematics in the newly established centre. Then, I landed up there completely unexpectedly, and they offered a position to me without even asking me to undergo the process of presentation, and regular interviews. There is another faculty member with me as part of this Cell, Malhar Kulkarni. Kulkarni works on Sanskrit grammar and computational linguistics.
We would now like to know from each of you as to what kind of work did you first undertake in your studies in the history of mathematics? When did it begin, what motivated you into such a study, and what were your initial contributions?
MDS: As mentioned before, I started off on this path in the early ’80s. The first serious work on the history of Indian mathematics was a study of the upapattis8 in the Indian source-works of mathematics and astronomy. That the Sanskrit source-works do contain upapattis was known even to Henry Thomas Colebrooke in 1817 when he translated Līlāvatī and Bījagaṇita into English, taking note of these “demonstrations”. That’s the word Colebrooke uses, when he says that the Indians are seen to be employing both geometric and algebraic demonstrations. Colebrooke used the commentaries Buddhivilāsinī on Līlāvatī (c 1545) by Gaṇeśa Daivajña, and the commentary Bījapallava on Bījagaṇita (c 1601) by Kṛṣṇa Daivajña, which give detailed upapattis. So the very first translation of Indian source-works of mathematics, clearly talked about upapattis.
An impression that Indians do not discuss proofs began to appear from the 19th century itself. And that’s because Indian commentaries do not start in the typical Euclidean fashion listing definitions, then axioms or general notions, then stating each theorem, and establishing the result with a proof. This style is not followed in Indian works. So the world of European indology thought that Indian studies of mathematics were essentially non-rigorous. Now if the Indians were not proving things rigorously, then there should have been lots of errors; if they were lax in their logic, there should have been errors galore in many Indian works on mathematics. The only error that the critics have found from the ancient texts are in two sūtras of Āryabhaṭa—concerning the volumes of the pyramid and the sphere—where the interpretation given by the commentators in the standard English translation shows that Āryabhaṭa’s formulae are not correct. But K. Elfering9 has given an explanation to the effect that Āryabhaṭa’s words should be interpreted differently, and when seen in this new light, they are free of error. The only other confusion which has been cited is that Brahmagupta did not explicitly mention that the quadrilateral, for whose area he has his famous formula, has to be cyclic. There have been recent efforts at rectifying this confusion as well.
This has a striking resemblance to the case of Ramanujan, who wrote down upwards of 3000 results in his notebooks, about which it is said that he did not have the necessary proofs. By that measure, there should have been at least hundreds, if not thousands of errors in his results. But Bruce Berndt, an authority on Ramanujan, says that there are hardly two or three results that need some correction; and that for the rest, proofs could be supplied. So, there is no question of Indians lacking rigour, or logical thinking ability, but the formalism or the foundational methodology that they employed was different from what many of us are presently accustomed to. We have to understand the methodology of proof in the Indian mathematical and astronomical traditions, and my paper of 1987 was an initial exploration in that direction. A later version of this study is in a volume published in 2005, edited by R. Sridharan, Gerard Emch, and myself.10
if the Indians were not proving things rigorously, then there should have been lots of errors
Back in 1987, I had noted that there already were 17 published works of mathematics and astronomy which contained upapattis, yuktis, or vāsanās (the Sanskrit words for proofs). Right now I think there are about 28 published works which contain proofs. Very few of them have been translated, and in fact, the best example of translation of a work containing proofs is Yuktibhāṣā, the english translation of which appeared in 2008. But many more such works need to be translated in order for us to make a definitive statement, and to provide a complete picture of how Indian mathematical methodology differs from that of the Greco-European tradition.
MSS: My serious personal involvement with Indian astronomy and mathematics took place around 1990. This was regarding a textbook project. The PPST group took up a project on preparing textbooks suitable for our school students. One of the projects involved preparing a textbook for school students, not on Indian astronomy per se, but on school-level astronomy from an Indian perspective. It was meant for students from class six or seven, to about class 11 or 12. In that connection, I had to read some amount of serious Indian astronomy, especially on how calculations had been done. Some popular books on astronomy were available, such as the popular book Nakshatra Veekshane in Kannada by G.T. Narayana Rao. There was also an earlier book by R.L. Narasimhayya, father of Roddam Narasimha, titled Nakshatra Darshana. In this book, the author gives various accounts of stars, where he uses Indian names for many, and also coined new Indian names for many others. It also dealt with modern astronomy, albeit in a very limited way. It was essentially about astronomy from an Indian perspective with many stories from Indian mythology too. That was the first involvement. My unpublished book explains how Indians viewed astronomy, discusses the Indian calendar in some detail, and gives accounts of some astronomical instruments developed by Indians, including those in Jantar Mantar in Jaipur, Delhi and other places.
Later, in 1991, I participated in a coordinated project funded by DST on Indian mathematics, astronomy, nyāya, law, linguistics, and Paninian grammar; and I took up the astronomy part. The aim was not to prove anything specifically, but to demonstrate how accurate Indian calculations were, as reported in the Pañcāṅgas; what were the stated algorithms for calculating the positions of planets, and eclipses, and the theoretical basis for them in the Indian framework; and how the computed results compared with those computed using modern positional astronomy. In essence, to know how accurate, simple, and optimal were the methods used by the ancient Indians. This project lasted from 1991 to 1994, with Ramasubramanian as a Research Fellow, and me as the Principal Investigator. In the same coordinated project, but looking at the mathematics part, was the team comprising Srinivas and K.V. Sarma. Sarma worked on translations of Yuktibhāṣā, for both its astronomy and mathematics content, looking mainly at the notion of proofs. The work on Yuktibhāṣā which started then was completed much later, only around 2008.
Along the way, we found that in Tantrasaṅgraha, Nīlakaṇṭha was talking of the revolution numbers of interior planets Mercury (Budha) and Venus (Śukra) in a very different way. In earlier texts, the number of revolutions of Śīghrocca of Budha and Śukra in a Mahāyuga would be stated. But here, Nīlakaṇṭha says that they are actually the planets’ own revolution numbers. The corrections to be applied to the interior planets to obtain their true positions were also done in a different manner here. Actually, it was the first time in the history of astronomy that the interior planets were handled correctly. I do not want to go into the technical details here, but it can be said that he carried out a major revision of the traditional Indian planetary model. The geometrical model underlying this revision is elaborated by Nīlakaṇṭha in his other works like Siddhānta-Darpaṇa, and Golasāra. Our work led us later on to a detailed study of Tantrasaṅgraha. One specific output of the project was a paper in Current Science that the three of us published on the planetary model of Nīlakaṇṭha.11 The project report itself contained a detailed analysis of the accuracy of the calculational procedures (for planetary positions and eclipses) of Indian astronomy texts. In the course of time, we studied Indian astronomy a lot more and were prepared for the next phase, which started a little later.
KR: My first exposure to Indian science per se was only through this above-mentioned project. Before this, I didn’t have any idea about mathematics or astronomy, or what may be generally referred to as science in India as presented in Sanskrit texts. This project, which I joined around 1991, was intended to analyse the accuracy and optimality of algorithms in Indian astronomy. Soon after joining the project, there was an opportunity to give a talk at a conference on Indian sciences that was held in 1992 at Madras Institute of Technology, Chromepet.
Another thing, as a fresh recruit in the project, that I was expected to do was to systematically study a few topics in spherical astronomy. Every two weeks or so, we used to have a presentation where I would study some chapters on spherical astronomy, and my guides Srinivas and Sriram would sit and listen to me speak. That is how they trained me as a researcher, and ensured that I gained a good grounding in modern spherical astronomy. They felt that before getting into Indian astronomical texts, one should have a proper understanding of these foundational aspects. Then we slowly took up various texts that Sriram was mentioning, in order to look into different aspects of astronomy that are discussed in the Indian astronomical works starting from Āryabhaṭa.
One such text that was taken up for study in the initial stages was a book edited by Prabodh Chandra Sengupta titled Khaṇḍakhādyaka: An Astronomical Treatise, by Brahmagupta. This is a karaṇa text,12 and the reason for making this choice was twofold. One, P.C. Sengupta was a well-known scholar in the field, and two, he presents relevant numerical examples that substantiate the underpinning algorithmic and theoretical framework. I did enjoy learning new things, and day by day my interest in the subject only grew – unsaturated to date. What also motivated me additionally was the realisation that I found myself doing something which not many around me were doing. So, there was a certain excitement in the novelty of my work.
This feeling got further confirmed when I interacted with a neighbour, who was then a member of the faculty of Jyotiṣa in Madras Sanskrit College. I used to discuss with him certain aspects where there was an overlap of interest. I would quiz him on something that I thought he should know, but the answer invariably was that he did not know how it related to the mathematical aspects of the subject. Generally, such experts deal with astrology only. So, I asked him about others who may be knowledgeable about these matters, and it became quickly clear to me that not many scholars had explored and analysed these texts through a mathematical prism. This was a big early motivation for me to embark on this study.
MSS: I just want to add one thing. If one is using Āryabhaṭa’s values, the sought values for the longitudes are almost correct, and off by only about two or three degrees on an average. We noted that for the Sun and the Moon, Tantrasaṅgraha gives a substantial increase in accuracy, because Nīlakaṇṭha understood the theoretical ideas better, and the parameters also were therefore adjusted better. So, the calculations are fairly good. The second point is this: Even with the parameters of the older texts, the predicted eclipse time, that is the middle of the eclipse, will be correct within a few hours. Sometimes, it may be off by an hour or two, and that is not good. But if one struggles and twiddles the parameters around a little bit, one can actually get very good results. In Indian astronomy, it is never said that one has to stick to these parameters rigidly. One can always update the parameters for achieving dṛg-gaṇitaikya. If one wants an accuracy of one second of an arc, then that may not be possible using Indian methods. But if a celestial event is to be predicted with a tolerance of a few minutes, Indian methods, with some parameters modified, can do the job.
MDS: Just a comment on the discovery of the revised planetary model of Nīlakaṇṭha. The work Tantrasaṅgraha was published in 1957. Āryabhaṭīya-bhāsya of Nīlakaṇṭha, for the Gaṇita pāda and the Kālakriyā pāda, the first two parts, were published in 1930–31. And Nīlakaṇṭha’s bhāṣya on the Golapāda, the third part, was published in 1955. So, these source-works were available for more than 30 years when we started studying them. This shows that the publication of a work doesn’t necessarily imply that the academic community is studying it, discussing it, and evaluating it technically. This is one such example where people had to wait for more than 30 years after its publication, before getting to know what was in the very first chapter of Tantrasaṅgraha. In the earlier part of Āryabhaṭīya-bhāṣya on Kālakriyā pāda too, Nīlakaṇṭha is seen explaining this planetary model; and in his bhāṣya on Golapāda, he clearly declares that Mercury and Venus go around the Sun, and the Earth is not circumscribed by their orbits.
Such a delay would be inconceivable in a functioning society which looks at its past carefully and meticulously. If a text is published, then there should ideally be hundreds of people looking at it, trying to see what new ideas and information they can get out of it; but alas, we were operating in a frame of mind where we were victims of the view that nothing new had anyway been done by Indians; and that they mostly kept repeating whatever had already been said in the earlier books. But, like unbiased beginners, here we were trying to learn the whole thing from first principles, and found that Nīlakaṇṭha was doing something very new; and that’s how this discovery could actually be chanced upon.
 |
 |  |
There is one other point to be made about the accuracy of Indian astronomy. It was noted already back in the 18th Century by a Frenchman Guillaume Le Gentil who came to witness the `Transit of Venus’ in Pondicherry. It is recorded by Le Gentil that the Indian eclipse calculations were fairly good, even up to seconds. And now for the pañcāṅga, we need the positions of the Sun and the Moon, and Indian values for them have been accurate since ancient times. After Muñjāla (c.930 CE), even the eviction correction for the Moon was properly taken care of. And so, our computations would have been fairly good for the tithi, nakshatra, and the motion of the Sun and the Moon.
So, was it in 1991 that the three of you first came together to start working on Tantrasaṅgraha?
MSS: Yes! After the first project, things were a little slow because Ramasubramanian’s PhD program was going on. But we were still maintaining some amount of activity on Indian astronomy. Meanwhile, the year 2000 was the 500th anniversary of Tantrasaṅgraha. So, we thought, why don’t we look into the full details of Tantrasaṅgraha in connection with the planetary model. We had the firm belief that it was indeed a very major revision of the traditional Indian planetary model. Nīlakaṇṭha was an important figure, and we had talked about him, and his revised planetary model, extensively.
So we felt that we should celebrate the 500th anniversary of his celebrated work Tantrasaṅgraha. Fortunately, at that time, Srinivas was associated with the Indian Institute of Advanced Studies (IIAS) at Shimla, and a grant was obtained for organising a conference on this theme. We could also invite a lot of people too. The Iranian historian Mohammad Bagheri was invited. He could not come but sent his colleagues, Negar Nader and Farid Ghassemlou. It also coincided with the 80th birthday of K.V. Sarma. Incidentally, K.V. Sarma, K.S. Shukla, and T.A. Sarasvati Amma were all born around the same time, in and around 1919. We did contact K.S. Shukla but heard that he was not keeping well. We got a response from Sarasvati Amma and she wished us all luck, but was not in a position to attend. So, it was celebrated as the 80th birthday of K.V. Sarma also, because he was one of the persons who had edited Tantrasaṅgraha based on many manuscripts. It should be mentioned that earlier K. Sambasiva Sastri had come out with an edition in 1957. There was not much difference between the two editions except that K.V. Sarma had included a detailed commentary of Yukti-Dīpikā by Śaṅkara Vāriyar. K.V. Sarma’s edition also included a portion of another commentary by the same scholar, called Laghu-Vivṛtti.
At the time this conference was organised, we had not gone into great detail about the whole of Tantrasaṅgraha, but had studied the planetary model aspects in detail. So the three of us gave a talk each on different aspects of this model, and also on the earlier or the traditional model. Many other people attended the conference. Some notable names include Balachandra Rao from Bangalore, Navjyoti Singh and Jolly K. John from New Delhi, Johny John Vembilly from Thiruvananthapuram, S. Madhavan from Kerala, V. Madhukar Mallayya from Thiruvananthapuram, L. Satpathy from Bhubaneshwar, and S.M. Razaullah Ansari from Aligarh. Many topics were covered in fairly great detail, and once it was done, IIAS agreed to publish it too. Among the many interesting topics, N.K. Sundareswaran from Calicut spoke on how Nīlakaṇṭha’s tradition is received and remembered in Kerala even to this day. It was therefore quite clear that Nīlakaṇṭha’s book is seen as an updated version of the entire tradition of Indian astronomy, not only for the planetary model, but involving many ideas related to problems of spherical astronomy and eclipses.
Incidentally, Yuktibhāṣā has been considered as a commentary of Tantrasaṅgraha. But, actually, its mathematics part is really an independent work. The second part on astronomy gives an exposition of the contents of Tantrasaṅgraha. Even at the time of the conference, we thought that we should study Tantrasaṅgraha itself in great detail, so that we understand every verse in it fully. That is how this project on Tantrasaṅgraha started around 2001. Ramasubramanian had just then completed his PhD in nonlinear dynamics, and was looking for some work. This project came in handy, and we applied to Indian National Science Academy (INSA), and immediately got the project granted.
Incidentally, I attended the conference held in Chennai in the year 2000; and like all participants was given a booklet. From the booklet, I gather that Nīlakaṇṭha composed the entire Tantrasaṅgraha in a matter of just five days, starting from 22 March 1500, to 27 March 1500. Your thoughts on this remarkable achievement.
MDS: Yes! This is mentioned by the commentator Śaṅkara Vāriyar, who is a junior contemporary of Nīlakaṇṭha. He explains that the words he viṣṇo nihitaṃ kṛtsnam indicate the starting date, and the words lakṣmīśa-nihita-dhyānaiḥ indicate the ending date. This is a historical record from those days that proves that this feat was indeed completed in just five days.
The study of source-works of the Kerala school of astronomy and mathematics, is one area in which the three of you have worked together extensively. It resulted in the publication of English translations, with mathematical notes, of Yuktibhāṣā, Tantrasaṅgraha and Karaṇapaddhati. What are the important contributions of each of these to our understanding of the work of the Kerala School?
MDS: K.V. Sarma came in fairly close contact with us during the 1980s. After he retired from Hoshiarpur in 1976, he was in Varanasi for a year. He returned to Chennai towards the end of the seventies. Initially he had a professorship at the Adyar Library. From then on, we were in contact with him. Sriram just mentioned the multi-institutional project funded by the DST, under which he had taken up a study of Indian astronomical computations. K.V. Sarma was the principal investigator for one of the projects, in which I was a co-investigator. The project involved a translation of Yuktibhāṣā, especially looking at the mathematical part, and also looking at the notion of proofs, as they were presented in the Indian tradition. K.V. Sarma also actively participated in the 1992 conference on Indian mathematics and astronomy that was mentioned earlier.
In 1993, PPST Foundation published the seminal edition and translation of Varāhamihira’s Pañca-Siddhāntikā, by T.S. Kuppanna Sastry and K.V. Sarma. Incidentally, there are three translations of Pañca-Siddhāntikā. The first of which, due to Sudhakar Dwivedi and George Thibaut, was published in 1889. Otto Neugebauer and David Pingree came up with a translation in 1970–71, published from Copenhagen. This was a much better translation than the first one, but still a lot of questions about that text remained.
There was a general review of the aforementioned DST project around 1993, where some of the proofs from Yuktibhāṣā were presented, just as they appeared in the translation, and accompanied with modern mathematical notation. R. Narasimhan, the computer scientist, B.V. Subbarayappa, and N. Mukunda were the reviewers for this project. All of them appreciated the work, and suggested that the entire work should be translated and published with detailed mathematical notes. Fortunately, the opportunity arrived in 1998, when IIAS started a new programme for the study of Indian civilisation. Under that project, K.V. Sarma completed his translation of the entire text of Yuktibhāṣā. He also edited a Sanskrit version of Yuktibhāṣā, based on a manuscript he had procured from the Government Oriental Manuscripts Library, Madras. This was published by IIAS Shimla in 2004.
Ramasubramanian had worked for a while as a post doctoral fellow in this project. Sarma asked the three of us to prepare detailed mathematical notes based on his translation of Yuktibhāṣā. The final version of the edition, translation and the notes for the mathematics part of Yuktibhāṣā were completed and sent to Shimla before Sarma passed away in January 2005. Final version of the astronomy part was also completed by 2006. However, there was a considerable delay, before the two volumes were finally published in 2008. They were released in the Chennai Mathematical Institute, by David Mumford in the last week of February 2008. Mumford was visiting India for six months, and he in fact conducted a weekly seminar on the history of mathematics in which many scholars participated. Among those present at the release function were C.S. Seshadri, S.R.S. Varadhan, Frits Staal, R. Sridharan and S. Ramanan, and several other renowned scholars.
The importance of Yuktibhāṣā is enormous. It was first talked about in 1835. C.T. Rajagopal and his collaborators wrote pioneering articles on it in the 1940s and early `50s. The mathematics part of the text had been published, along with detailed notes in Malayalam, by Ramavarma Maru Thampuran and Akhileswarayyar in 1948. Some of this material was also discussed in Sarasvati Amma’s book on geometry. But the actual wording of the entire original text along with details of its mathematical formulation became available for the first time, only with this two-volume edition and translation by K.V. Sarma and co-workers.
While several authors had discussed some of the proofs in the mathematics part of Yuktibhāṣā, as regards the astronomy part little was known to the entire scholarly world prior to this edition. In fact, regarding the astronomy part, there was just one article by K.V. Sarma, co-authored with Hariharan, which was published in the Indian Journal of History of Science, in 1991. There, Sarma and Hariharan show that Jyeṣṭhadeva is talking of a geometrical model of planetary motion, where the five planets go around the Sun, and the Sun, in turn, goes around the Earth.13 In essence, Yuktibhāṣā is actually a detailed exposition of Tantrasaṅgraha, and not a commentary, done word by word, or verse by verse. But it takes all the results and procedures given in Tantrasaṅgraha, most of which is concerned with mathematical astronomy, and explains them along with demonstrations.
MSS: The textbook nature of the exposition comes out clearly in this work. I was surprised when I started looking at the translation of the astronomy part, and tried to understand the diagrams provided there. The first 20 to 30 pages resemble what one would find in a standard spherical astronomy book.
Some of the pedagogical questions addressed are: How does one define a celestial sphere? How does one define the cardinal directions, and the north celestial pole? How does one visualise the planetary model? What is mean motion, and true motion, and why is there a difference between the two? How does one construct the various circles of interest? All these are explained in the greatest detail in the astronomy part of Yuktibhāṣā, along with proofs. In Tantrasaṅgraha, there are no proofs, because it is a `tantra’ text. It is essentially an algorithmic text, and though the algorithms are given, detailed explanations for the same are not provided. But in Yuktibhāṣā, everything is explained in detail. For instance, there is the daśapraśnāḥ (the ten problems): Considering five quantities of importance on the celestial sphere, and given any three of them, how does one determine the other two? Typically, the five quantities considered are the zenith distance, latitude of the location, declination, hour angle, and the azimuth. For instance, Nīlakaṇṭha himself has given the algorithms for systematically solving the daśapraśnāḥ.
The important thing is to understand that it is based on an underlying principle. The principle is: If there are two intersecting great circles, and if from one point on one of them, one drops a perpendicular to the plane of the other, how much is that perpendicular? That is, what is the distance between that point and the plane of the other great circle? From this one principle, all the spherical astronomy relations associated with modern spherical trigonometry become available. It is necessarily detailed because several intermediate steps are involved. I maintain this, particularly because there is a view in certain sections that Indians didn’t possess spherical trigonometry, and so they just worked with planar trigonometry and applied it to figures in the interior of a sphere. But that is not true at all. They indeed worked, quite competently, on a spherical surface. For another set of problems called pañcadaśapraśnāḥ, which do not depend upon the hour angle and the local coordinates, but are applicable globally, the answers are also given. So, everything is systematically laid out. This also surprised me greatly because many scholars, particularly David Pingree, make an allegation that Indians didn’t have any foundational method upon which subsequent work could be built on. And that, due to the absence of such a method, all they had were some stray results strewn here and there. It is as if the Indians didn’t really understand what they were doing! But here, in contrast, whether it is the planetary model, or an eclipse calculation, or a spherical trigonometry problem, everything is explained in great detail starting from the basic founding principles. They are seen posing questions such as `What are the basic quantities that are involved?’, `What in turn are the relations between them?’, and so on.
Furthermore, there are absolutely no pictures drawn, or accompanying diagrams provided in the text. Yet, completely detailed descriptions required for their constructions are given. Without any ambiguity, one can in fact draw very complicated diagrams, by just following the unambiguous instructions given there. What was equally surprising to me, and this happened later, was that this kind of treatment is also seen in Siddhānta-Śiromaṇi of Bhāskara. All in all, therefore, Yuktibhāṣā is the first completely systematic exposition from that era, and closely resembles a modern textbook on astronomy. Before I conclude, let me again add that Bhāskara’s Siddhānta-Śiromaṇi is an algorithmic text, and it is in his own commentary on it that Bhāskara provides detailed explanations and proofs.
KR: Tantrasaṅgraha has about 432 verses in all, and all of them but for one or two, are composed in the short and simplest metre anuṣṭup. These verses are arranged in eight chapters, and as in any other work in the Indian tradition, the first chapter starts with an invocation. Some of these texts which had been composed earlier, say by Bhāskara, namely the Siddhānta-Śiromaṇi, are much bigger texts, and they also have a variety of topics in them which are not directly connected with the topic of planetary motion. For instance, in Siddhānta-Śiromaṇi, one finds a beautiful description of the seasons themselves, and Bhāskara has almost devoted an exclusive chapter for that. In short, there are a few voluminous texts such as Siddhānta-Śiromaṇi, in which one would find verses that are not directly connected to the actual process of computation. But, as far as Tantrasaṅgraha is concerned, it is a fact that not even a single verse not connected with the computation that one intends to carry out, can be found. It is such a precisely written text, starting from defining what the whole purpose is, and going on to show how it is carried out, choosing the necessary parameters, and so on. All these things are done in the first chapter.
Then follows a very systematic exposition of how to find the sine values, how to approximate them, their various relations, and so on. One also finds some interesting results which led some people to believe that this text has discussed the various infinite series, but this is not supported by the textual evidence. For instance, in one of the famous books which have been written on the history of calculus, one finds people stating that Nīlakaṇṭha has provided the infinite series for the sine function. This is not quite correct. Nīlakaṇṭha gives various important results which are extremely useful in making approximations for the sine function, when the argument of the function is small in magnitude; he even treats the sine inverse both exactly and approximately.
When stating the number of revolutions made by Mercury and Venus in a mahāyuga, he uses the word svaparyayāḥ; whereas, people earlier to him would have used śīghrocca-paryayāḥ. It is this usage that immediately caught our attention as something potentially significant, and so we immediately took it up, and then went about with great enthusiasm looking at the various related things, including works of other important astronomers, just to see how they have described this concept themselves. It was a very exciting, and rigorous exercise made enjoyable by the enthusiasm of my mentors Srinivas and Sriram. Even while describing the method of finding the longitudes in the second chapter, it was very clear to us that he had a clear picture that Mercury and Venus were moving around the Sun and not around the Earth. This constitutes the major topic of the second chapter.
As we proceed further, the third chapter is the longest, and here Nīlakaṇṭha systematically describes how to find various celestial coordinates from observations, as well as the relations between them. This is essentially, in short, a way to describe the positions of planets in different coordinate systems. It can be an equatorial system, a horizontal system, or anything else. And if one wants to move from a certain system to another, and if one knows the values in that system, one can get the corresponding values in the other system; just as knowledge of a system’s cartesian coordinates, will enable one to get to its equivalent description in say, spherical coordinates. It is in this context that the daśapraśnāḥ have been discussed in great detail.
There is one other interesting thing on which we authored a paper,14 where Nīlakaṇṭha talks about the `parallax correction’. When determining the latitude of a particular location, there are two important corrections that we note. This is nicely stated by Nīlakaṇṭha. One is when the shadow of an object is measured. If the source of the light is not a point source, then one has a small correction. In particular, the angular diameter of the Sun has to be considered, so as to precisely fix the coordinates of the shadow. Also, it has to be corrected a little bit more, because as observers, we ourselves are observing not from the centre of the Earth, but from its surface. So, the radius of the Earth also makes a difference and its effect has to be accounted for. These things have to be taken care of, because even though the magnitudes may be small, theoretically speaking, they still need to be accounted for, and incorporated. This is something which has been pointed out as being important in the third chapter.
Then we come to the eclipses, and a few other things which are dealt with in great detail. These are the major features in Tantrasaṅgraha. You also asked this question about what made me take up this kind of research. I would say that partially, it was due to the enthusiasm that got generated when I attended the conference celebrating 500 years of Tantrasaṅgraha; and partly, it was also a phase in which I was not quite sure where to move to next. But inside me, there was both a natural inclination and a compulsion to take up this work. That was because I hadn’t yet completed my śāstra course in Advaita Vedanta fully, and although I had completed 13 semesters, one last semester was pending. I also had to take the final viva, so I wanted a buffer period before doing any other work. So, this was a blessing in disguise and of course, was quite exciting too.
We would now like to hear your reflections on Karaṇapaddhati.
KR: I took up a deeper study of Karaṇapaddhati, whose edition had earlier been brought out, because of a student who had then joined me for a PhD program. He is Venkateshwara Pai, currently a faculty member in IISER Pune. Pai’s joining me for a PhD is also an interesting story in itself. Sriram and I had gone to deliver lectures at a conference in Kerala, where the Mata Amritanandamayi University is located. We gave some lectures there, and one of the persons in the audience was this young man who was working there, after his MSc, as an assistant professor in physics. At the end of the day, he approached us and asked how he could go on to do research in this area, saying that he was quite excited at the prospect of doing so. Fortunately, I had a project running then, and he expressed his interest in immediately joining that project. I told him that he could come and join within a month. Just to back it up, he joined IIT Bombay first as a project student, and then within a few months, I was able to get him enrolled as a PhD student.
MSS: There is a particular type of formula which appears throughout the text, and which is related to continued fractions. There’s a 248-day period associated with the Moon and the so-called Moon’s anomaly, within which the Moon completes nine revolutions. This was known to the Babylonians, Greeks, and Indians as well. However, this induced the Indian astronomers to develop the so-called Vākya system. Karaṇapaddhati provides the theoretical basis for the Vākya system. It tells us how it can be concluded that Moon’s anomaly completes very nearly nine revolutions in two forty eight days, beginning with the revolution numbers of the Moon and its apogee (mandocca) in a Mahāyuga, which are large numbers, using the continued fraction method.
K. Narayan Kumar
What one does is try to get better and better approximations for the ratio of two given numbers, both of which are large. The theoretical framework for such calculations is given in Karaṇapaddhati. This theme pervades the whole text, and is based on techniques similar to continued fraction expansion. Putumana Somayājī describes various other things of interest, such as candravākyas, which is the Vākya system for the Moon. A similar idea, that is, a Vākya system for the Sun is also presented, and he calls it yogyādivākyas. Additionally, ideas on how the longitude of the Sun can be found out every day, using a very simple technique are also presented.
KR: There are some verses which are unique to this text Karaṇapaddhati. Even with respect to the infinite series, there are certain verses which are found in Karaṇapaddhati, presenting in beautiful metrical form some interesting fast convergent approximations that are not seen elsewhere. There is another verse in the book which makes the case why this book should not be seen merely as a karaṇa text; this is hinted at by the Sanskrit phrase karaṇa-karaṇe paddhati. This book doesn’t describe the computation of planetary positions per se. On the other hand, the primary thrust of this text is to teach one how to compose a karaṇa text.
MDS: Karaṇapaddhati is one of the four important texts that were mentioned by Whish in his article. This was first published in 1937 by Sambasiva Sastry, and then, a Malayalam translation with an explanation was published in 1953 by P.K. Koru. Two Malayalam commentaries were also edited and published in 1956. Telugu language manuscripts of Karaṇapaddhati are available in various manuscript libraries. There are a couple of Tamil commentaries also.
Still, this is one text of which both the place and the date of the author are not definitively established to date. The name Putumana Somayājī is not very illuminating either. Putumana means a new house, and Somayājī is a person who does the Somayāga. Clearly, not enough to pin down the place. Many people thought that the reference to Śivapura alluded to in the text refers to modern-day Thrissur; and we went there on an expedition to locate his house. We were led to various nearby places, but could not succeed in locating his place.
Whish says that Somayājī passed away around the 1730s. Various people quote his time from 1660 to 1730. There are several works of Putumana Somayājī which have not yet been edited, and there is one work called Veṇvārohāṣṭaka, in which he explains a way to calculate the position of the Moon. There, he uses an epoch as a base point for further calculations. Usually, when people present such algorithms, they choose epochs that run concurrently with their own time. Inferring from such plausible clues, his date is likely to be the second half of the 16th century, and he was perhaps a junior contemporary of Nīlakaṇṭha.
Now, let us look at the work Karaṇapaddhati itself. It tells you how to make karaṇa texts. In order to make karaṇa texts, we have to simplify the calculations given in the siddhānta texts which use a very large number of revolutions for the celestial bodies. Typically, the Moon performs some 5,77,53,300 number of revolutions in a mahāyuga; and the Sun performs 43,20,000 revolutions in the same period. Since such large numbers appear in various ratios, Karaṇapaddhati uses Āryabhaṭa’s technique of vallyupasaṃhāra (used in solving the kuṭṭaka problem) to arrive at the convergents of corresponding continued fractions. Later on, Dṛkkaraṇa seems to have come up with even more optimal solutions by using the nearest integer continued fraction. Since the motion of the Moon is fairly complicated, one needs to bring in many more intermediate steps known as computing the khaṇḍas and dhruvas. All in all, Karaṇapaddhati is a very important text whose translation finally appeared only in 2017.
KR: Another highlight of the text that deserves mention is that Putumana had this wonderful capacity to convey the revolution numbers of almost all planets in just a single verse, choosing the right literary metre—sragdharā, to be precise—which has 21 syllables per quarter, in contrast to the anuṣṭup, which has only eight. It may also be added that in order to achieve the objective of conveying everything that he wanted to say with precision and brevity, he often changes the metre suitably.
Another major contribution that the three of you have accomplished together has been the NPTEL course on “Development of Mathematics in India from the Vedic Period to the Modern Times”. This seems to have been widely acclaimed as a novel attempt to introduce Indian mathematics into our modern higher education curricula. How did you get initiated into this project? What have been the responses that you (individually or collectively) have received from the viewers of these courses? Do you have plans to offer more such courses?
MDS: After Ramasubramanian got appointed to IIT Bombay, he started teaching regular courses on Indian mathematics and astronomy; and they soon became very popular. So, by about 2009, he started inviting Sriram to give a series of lectures on allied topics in these courses. Then we felt that apart from conducting courses for students of IIT Bombay, why don’t we organise some general workshops meant for all those students and teachers in India, who may be interested in Indian mathematics and astronomy.
So, with help from the National Board of Higher Mathematics (NBHM), one such workshop was conducted at IIT Bombay around February 2013. Sometime later, I happened to meet Bhaskar Ramamurthy, who was then the Director of IIT Madras. He asked me about the work we were involved in, especially with reference to Indian mathematics and astronomy. I told him about the recently concluded workshop at IIT Bombay. Bhaskar then mentioned the new NPTEL courses that were being planned and suggested that we could offer a course on Indian mathematics and astronomy under the NPTEL initiative. That is how we got started on this project. Whatever material we had put together for various courses and workshops was appropriately supplemented, and turned into a detailed syllabus. The course was titled “The development of mathematics in India from the Vedic period to modern times”, and additionally, there were also two lectures towards the end, on the emergence of modern mathematics in India.
A total of 40 lectures were delivered during the entire course. It seems to have had a significant impact since, for the first time, participants could see a very systematic account of the development of mathematics in India. Another novelty was that it showed that disciplines related to Indian knowledge could be taught as a regular course in a university setup. The focus of the course was more on the delivery of the mathematical content than on the attendant historical issues. Historical issues came up only at the beginning of a lecture, or towards the end of it, for fixing the time of the person, and related issues. This approach was different from most of the available books on the history of Indian mathematics. Thus, in our NPTEL course, the main focus is on the mathematical content that is intended to be conveyed by the source-texts in Sanskrit, Malayalam etc. It was assumed that the students would possess some basic knowledge of reading Sanskrit. This is roughly how we developed this course. It is indeed very gratifying that this course has been widely used and appreciated. Some of our lectures have been seen by over 80,000 people so far. This also opens up the possibility for designing similar courses at the university or college level on other disciplines of Indian knowledge.
MSS: I don’t have too much to say except that before we signed up to offer NPTEL courses, I really had not studied some of our mathematics source-texts at a level of detail that the course demands. So, I enjoyed reading up and learning all that was required. Before we went live, there were rehearsals conducted to ensure a foolproof delivery. Over a period of time, we have also built in a certain amount of flexibility as far as the course content and rigour go. Depending on the audience, and the time available, we suitably tailor the content. We have taught such courses to students at IIT Bhubaneswar. Another innovative development has been that in 2019, we presented an instructional course on Indian sciences in the K.V. Sarma Foundation, lasting for five days, focusing on Indian astronomy, Indian mathematics, Paninian grammar, Ayurveda and Indian epistemology.
KR: Let me just add a brief historical note here. One day, a colleague from the Department of Mathematics at IIT Bombay, Jugal Kishore Verma met me and spoke to me about the then newly established National Centre for Mathematics (NCM), which is a joint venture between TIFR and IIT Bombay. Verma spoke to me about an opportunity to offer a course on Indian mathematics under the auspices of the NCM, and also to tailor the course to meet the needs of an already well-established programme called the Advanced Training in Mathematics (ATM) schools. The added attraction was to be able to shoot the session in a live classroom filled with students on campus, unlike if it had been shot in the rather sterile settings of a studio. This led to the February 2013 workshop that was mentioned earlier.
K. Narayan Kumar
MDS: Our NPTEL course on Indian mathematics was perhaps the first such experiment so to say, to design a complete course on an important discipline of Indian knowledge and record it for online delivery, and ensure that it could cater to a varied audience. We could have made it in such a way that it avoided all the original quotations from Sanskrit, and focused just on mathematics. Then that course could have been followed by any student, even by one who does not have a Sanskrit background. But then, what most people would like to know is, on what authority we lay claim to the various facts mentioned. They may even genuinely want to know if that is really what the original source-texts say. Since this whole exercise was being done for the very first time, we could not even ask them to refer to an existing chapter of a popular book. Therefore, all of it had to be integrated and presented in one place. That’s why it is perhaps a condensed and heavy course of lectures. If it is to be taught at all, it has to be taught in a graded and relaxed way. It would have been easier if we had written an accompanying book of lecture notes, that could have then served students as a learning aid to this course.
Another relevant point is the rather unfortunate situation prevailing in India today, that students entering the portals of a university have almost zero exposure during their formative school years, to the important contributions from Indians through the ages, in various subjects such as mathematics, science, arts, and other areas of human endeavour.
It’s essential that anybody who earns an MSc in mathematics in India, should know what Indians have achieved in mathematics in a systematic way. Since one has never been exposed to this body of knowledge either at BSc or at school, the NPTEL course re-designed for such a student will necessarily be heavy, compensating for a whole lot of material that is missed out on, during the earlier years. One can also think of a course without too many Sanskrit quotations, and where they do appear, doing so only in a certain appendix. Such a course can focus solely on mathematics, using the Indian technical terms. But a course that highlights relevant passages from the Sanskrit source-texts is also valued by those people who want to study this subject further, directly from the Sanskrit source-texts. Understanding the need to cater to diverse audiences has enriched our NPTEL course, and increased its ability to cater to different types of learners.
A question to Sriram. It is generally felt that many of these commentaries haven’t been studied enough by scholars, and hence there’s a mistaken notion that Indian mathematical texts are bereft of explanations, and proofs. We understand that you are currently working on a translation of the auto-commentary Vāsanā-bhāṣya of the text Siddhānta-Śiromaṇi by Bhāskara. Would you like to share with us insights that you gained during your study of this commentary?
MSS: Certainly. I have worked on the text and the Vāsanā-bhāṣya in collaboration with Sita Sundar Ram of Kuppuswamy Sastri Research Institute, Chennai, and Venketeswara Pai of IISER, Pune. First of all, the verses in Siddhānta-Śiromaṇi basically state the algorithms for the computation of various quantities of astronomical interest in a compact manner, using a particular metre. These would have been difficult to understand by the practitioners of astronomy even in those days, without a more detailed description in prose, and also explanations and proofs. Bhāskara himself wrote the Vāsanā-bhāṣya, or the commentary for his own work. I had some familiarity with commentaries on Indian astronomy texts. After all, the astronomy part of Yuktibhāṣā is essentially a commentary on Tantrasaṅgraha, which we had studied in some detail. Tantrasaṅgraha also has other commentaries, namely, Laghu-Vivṛtti and Yukti-Dīpikā, which provide explanations of what is contained in the text.
I have practically lived with this text for a few years. From the way it is written, I get the impression that Bhāskara had a premonition that his work was going to stay relevant for a long time
Bhāskara’s Siddhānta-Śiromaṇi is perhaps one of the greatest Indian works on astronomy that contains a detailed explanation of all the key astronomical concepts, algorithms for calculations, and dates back to the 12th century when it was composed. I have practically lived with this text for a few years, studying it in detail. From the way it is written, I get the impression that Bhāskara had a premonition that his work was going to stay relevant for a long time. This was evident to me as I began its study, and I found the manner in which he explains things very appealing. In the beginning itself, he says that many scholars have already authored texts on Indian astronomy, and asks if there is any need at all for one more text. Then he goes on to say that certain shortcomings seen in certain earlier books have to be removed, and newer results incorporated too, even while admitting that many contributions of his predecessors continue to be relevant and important. With this, he urges his readers to proceed with the study of his text. As a matter of fact, the entire system of Indian astronomy as practised in the twelfth century is available in the two parts of Siddhānta-Śiromaṇi. The Grahagaṇita part of the text is all about algorithms for calculations. The verses are all merely encoding algorithms.
Coming to Vāsanā-bhāṣya, for every set of verses in Siddhānta-Śiromaṇi, Vāsanā-bhāṣya provides a description in greater detail of what was condensed in that set of verses. The verses would provide formulae for finding astronomical quantities of interest, and would be mostly terse and difficult to understand at times. To address this, the corresponding explanation in Vāsanā is mostly descriptive, and written in a language that is simpler, leaving no room for ambiguity. A more important feature of Vāsanā is the presence of the so-called upapattis. These upapattis are seen wherever derivations, proofs, and demonstrations appear in the Vāsanā.
Grahagaṇita is mostly about algorithms, and a lot of spherical trigonometry is already present here. Yet, some people think that all the relevant spherical geometry is explained only in Golādhyāya. This is not true at all. Golādhyāya has more conceptual matters discussed in it. Especially questions such as “What is the situation of the Earth in the cosmos?” Further, how does it stand supportless? A beautiful explanation is given which proves via recursive reasoning that one cannot have such a support for the Earth. Suppose one does say that the Earth is actually standing on something. Now, what does this support itself stand upon? The support itself needs a support to stand up on. Stretching this further leads to an infinite regress, one will need an infinite number of supports each supporting something above it, while itself being supported by something below it. But no such thing is observed, and neither is there a preferred direction that points to an Up or a Down in empty space. So, the Earth is supportless. Such arguments which can be called physical explanations, appear in great detail in Golādhyāya.
From a purely pedagogical perspective, in physics, it’s not enough that one knows a formula, or understands some derivation. For a real understanding, one has to understand it physically. This kind of physical explanation is available in Golādhyāya. For instance, consider the rising times of various rāśis. Why do they vary? They do because of the way the ecliptic is inclined with the equator. Or, how do paths of objects moving in the sky appear when observed from various places on the surface of the Earth, say from one of the poles? Golādhyāya also has some material on many instruments used during Bhāskara’s days for astronomical observations.
What makes this `auto-commentary’ almost a unique instance is that the author himself is providing unambiguous explanations via an easy-to-read commentary, of a much more condensed and harder source-text. It is clear from this that Bhāskara himself knew that his work was going to be important, and even stand the test of time. This is how convinced he was of his own competence.
In the Indian astronomical tradition, Siddhānta-Śiromaṇi was always considered as one of the most important texts from the 12th century to even the 20th century. So, it is not surprising that even in the 19th century, when European scholars started looking at Indian works, Siddhānta-Śiromaṇi was one of the first texts to be translated. Lancelot Wilkinson edited the text with the Vāsanā and translated the verses of Golādhyāya in 1842. The whole of Siddhānta-Śiromaṇi comprising the two parts and the Vāsanā-bhāṣya was edited and published by Bapudeva Sastri in 1861, after consulting many more manuscripts and correcting the earlier editions. In Jyotiṣa departments across India, Siddhānta-Śiromaṇi is a prescribed text. The focus is not only on algorithms per se. Bhāskara also discusses how to determine the parameters associated with planetary motion such as the revolution numbers of planets, their mandoccas and śīghroccas, the length of the year and such, from careful observations. For these, he has provided explanations in great detail, though I am not sure whether all these observations can actually be performed.
Bhāskara also provides many approximation techniques that assist one in calculations. The very first chapter on mean longitudes for instance is a case in point. Even in Tantrasaṅgraha, there are only about 30 verses or so dealing with mean longitudes. Siddhānta-Śiromaṇi has nearly 120 verses. In general, approximations are very important for karaṇa texts. Bhāskara also talks of very different ways of finding the solution to the same problem, and this is so important from the perspective of pedagogy. And he himself says that correcting the works of people like Brahmagupta and Varāhamihira, whose greatness he acknowledges, was very important. He also seems to have a love-hate relationship with Lalla, whom he constantly needles, and yet whose work he cites and acknowledges.
What makes this ‘auto-commentary’ unique is that the author himself is providing unambiguous explanations of a more condensed and harder source-text
Bhāskara gives an explanation for Brahmagupta’s second-order interpolation formula, which clearly shows that he had some insights to offer. Other novelties that merit mention in the text include the use of a `principle of proportionality’ (trairāśika) in many situations to find the expressions for some physical quantities, physical explanations of how an eclipse progresses and how to visualise an eclipse, an ingenious method to find the true rates of motion of planets (involving calculus ideas) and their retrograde motions, among other things. All these make it a highly fascinating read and prods one to think deeper about the nature of the phenomena around us. Bhāskara’s text is full of juices. He invokes poetry, even while steeped deeply in an astronomical discussion. Nīlakaṇṭha’s Tantrasaṅgraha, in comparison, comes through as somewhat of a dry text. Bhāskara also offers a peek into what is an essentially Indian way of thinking about astronomical and natural phenomena.
KR: The work of Bhāskara can be summarised in a rather poetic sense using three `C’s. He is `Concise’, in his ability to condense a whole lot of ideas in a pithy manner. While being concise, he also ensures that his exposition is quite `Comprehensive’. And the third is his distinctly unique style of composing verses (filled with both śabdālaṅkāra and arthālaṅkāra),16 which is indeed very `Charming’.
The work of Bhāskara can be summarised in a rather poetic sense using three ‘C’s – Concise, Comprehensive and Charming
We would like to hear your views on the number zero, and its evolution as an important mathematical construct, especially as studied in the Indian tradition. In the same vein, could you also tell us how the notion of infinity is treated?
MDS: The word śūnya occurs in the Vedic corpus itself, in the Taittirīya Brāhmaṇa and in the Atharva Veda, and means emptiness. Of course, we all know that in the Chandaḥ-Sūtra of Piṅgala, there is this sūtra Rūpe śūnyam which uses zero as a marker. In Bhāsa’s plays Svapnavāsavadatta, and Pancarātram, there is again a reference to sūnya. Of course, the arithmetic of zero comes quite late, and as far as we know, it occurs for the first time only in Brahmagupta’s Brāhma-Sphuṭa-Siddhānta. And from then onwards, we see that zero has been accepted as a proper number.
MSS: As far as infinity is concerned, and in a particular astronomical situation, Bhāskara encounters it while calculating a quantity called the samamaṇḍala śaṅku. This is related to the geometry of a shadow of a specific kind when the Sun is on the prime vertical, and one has to therefore consider the circle joining the east and the west, passing through the zenith. Clearly, here the azimuth is zero, and the calculation of the desired quantity involves quantities containing the sine of the azimuth, which appears in both the numerator and the denominator as the divisor, through a related quantity called the digjyā. Without going into the details, it is observed that finally both the numerator and the denominator are of the form “x divided by 0”. Bhāskara doesn’t say `take the limit as the divisor goes to zero’ as we would do it now. He just takes the divisor as a zero in both the numerator and denominator, and proceeds with the calculation. The answer is correct here. Bhāskara had that kind of intuition.
Wasn’t the idea of the so-called `place-value’ already present in Āryabhaṭīya?
MDS: Āryabhaṭa did not explicitly discuss details of the place-value system of representing numbers. He mentions the sthānas, and does calculations based upon the place-value system. He gives, perhaps for the first time in the history of mathematics, the standard algorithms based on the place-value system for the computation of square roots and cube roots. So, the place-value system is seen in a fully developed form, first in the work of Āryabhaṭa. But the details of the system were explicitly written down for the first time (among the various available texts whose dates have been established) in Brahmagupta’s Brāhma-Sphuṭa-Siddhānta.
KR: Just to add to what MDS said, I would like to mention the following. This word śūnyam is generally used to refer to the absence of something. The precise statement made by Brahmagupta is: khoddhṛtam-ṛṇaṃ-dhanaṃ vā tacchedaṃ kham-ṛṇa-dhana-vibhaktaṃ vā. Bhāskarācārya uses the word khahara to refer to a number having zero as denominator. What was meant by Bhāskara by this term has been clarified in later commentaries such as Nisṛṣṭārthadhūtī which I shall try to paraphrase in a couple of minutes.
In order to help people appreciate the value that one gets by dividing a quantity by zero, the commentator Munīśvara says: harā-pacaye phalo-pacayāt parama-harāpacaya-śūnyamita-hare…. Since when the divisor decreases, the result (quotient) increases, when the denominator reaches the limit of decrement, which is equal to zero, the result reaches its pinnacle (ananta).
Viewed via the founding principles of Indian schools of philosophy, mathematics is a subset of a larger framework that seeks to understand the world around us, precisely and quantitatively. Could you please elucidate how the tenets of nyāya, tarka and anumāna apply to the world of mathematics?
MDS: It’s not just Indian mathematics, but all of the systematised disciplines of Indian knowledge seem to be rooted in a certain epistemology, i.e, methods of acquiring and validating knowledge, which is discussed in the Indian schools of philosophy, the Darśanas. Be it Indian medicine, astronomy, or mathematics, they are all largely rooted in the epistemology of the Nyāya school, which is also known as the school of Indian logic.
You just mentioned `logic’ in the context of `nyāya’. I was reminded of `tarka’, which is normally used in logical arguments. Are the two related?
MDS: Yes, they indeed are. Tarka is a technical term found in Nyāyaśāstra, and many times, the two are even used interchangeably. When speaking of a logical argument, tarka is used in the context of coming up with a good argument and settling an issue in an argument vāda. There are 16 categories defined in Nyāyaśāstra to logically settle an issue, and tarka is one of those 16.
Nyāyaśāstra is specialised in the discussion of epistemology, that is the discussion of how to acquire knowledge and, further, validate the same. This subject is called pramāṇa-śāstra. Astronomy, and mathematics, use these pramāṇas extensively. For instance, in Jyotirmīmāmsā, which is one of Nīlakaṇṭha’s important works on the epistemology of Indian astronomy, he quotes from the Taittirīya Āraṇyaka to convey the fact that the idea of a pramāṇa is not specific to only the Nyāya school of philosophy, but can be clearly traced back to the Vedic corpus.
smṛtih-pratyakṣamaitihyam {\small\vert }
anumānaścatuṣṭayam{\small \vert}
etairāditya-maṇḍalam {\small \vert}
sarvaireva vidhāsyate{\small \vert }
These pramāṇas are accepted in Indian scientific texts as essential for acquiring valid knowledge.
But now there is something special about Indian epistemology, or even, Indian logic. Let us try to see what is it that makes it special. Firstly, pratyakṣa roughly means a perception. Anumāna can be translated as an “inference” going from existing knowledge or understanding to another new knowledge. Also, this idea of anumāna in India is not merely of an inference which is bereft of perception. Anumāna, interpreted literally, splits as anu + māna, which roughly means, `a knowledge which follows [a prior knowledge]’. In order to carry out inference in Indian logic, one should have prior knowledge of a general relationship, which is called vyāpti-jñāna. One sees smoke on a mountain, and one knows already that wherever there is smoke, there is fire. And therefore one infers that there is fire on the mountain. This conclusion is the standard example of anumāna. So, in essence, anumāna is not merely a logical deduction from one proposition to another, but actual knowledge that one gains by first knowing one thing, and then going on to know the next level of knowledge. This kind of reasoning is seen in all sciences, including astronomy.
Apart from pratyakṣa, which is perception, and anumāna or inference, there is also another pramāṇa which is called āgama, or śabda. Āgama is an authoritative statement, or a collection of statements, from a text or from a reliable person. It may be an ancient text, or even an authoritative statement from somebody who is considered to be a reliable source, capable of making that statement. Most of science, and most of mathematics is based upon these three pramāṇas. Without having an established older text, one cannot arrive at a newer corpus of understanding. Āgama is this corpus of tradition on which one works upon. Be it science or mathematics, especially as they have been practised all through history, pratyakṣa, anumāna and śabda are all relevant. There has been some needless confusion about the use of āgama that it makes Indian thought dogmatic. This confusion is not of much significance. It is only after one has internalised the contents of Isaac Newton’s Principia, or Paul Dirac’s The Principles of Quantum Mechanics, that one can improvise upon them. Bhāskara too says that he is building on the older texts of Brahmagupta. It is all about building newer knowledge upon the older body of knowledge.
Since you mentioned Newton, I recall Newton himself saying that if he could see further, it was because he was standing on the shoulders of giants.
MDS: Right! There is one more point that is rather unique to Indian epistemology. That is that, pratyakṣa is considered paramount, and lower in the hierarchy lies āgama. If any statement in an āgama is contrary to pratyakṣa, then pratyakṣa prevails. Or, one has to then provide an explanation to justify the presence of such an āgama, so that pratyakṣa is not seen to be in contradiction with it. A valid anumāna, or inference, is one that is not contradicted either by pratyakṣa, or by āgama. In other words, a valid inference is one that is not in contradiction with what is already seen, or something that is coming directly from an authoritative text. There is a beautiful statement in the Vātsyāyana-bhāṣya on Nyāyasūtra that conveys the essence of the entire approach that I have spoken of. It says that an inference is faulty, if it contradicts pratyakṣa or āgama.
pratyakṣāgamābhyām īkṣitasyān-
vīkṣaṇam anvīkṣā{\small \vert } tayā pravartata
ityānvīkṣikī nyāyavidyā nyāyaśāstram{\small \vert}
yat punaranumānaṃ pratyakṣāgama-
viruddhaṃ nyāyābhāsaḥ sa iti {\small \Vert }
Thus we have a clear hierarchy among the pramāṇas, namely that the pratyakṣa can overrule āgama; and pratyakṣa as well as āgama, can both overrule anumāna. This postulated hierarchy is something highly special, and unique to India. This recognition and understanding of an underlying hierarchy of pramāṇas is something that is also seen in Indian sciences. Books of Indian medicine emphasise this point. While defining the theoretical categories through which the medicinal properties of a substance are to be determined, Suśruta-saṃhitā (40.20–21) advises that the wise physician should never raise theoretical arguments about the curative properties of a drug when they are already known and established in tradition, based on actual practice:17
An expert physician should never doubt through reasons (hetus) the efficacy of drugs which have been established in practice and whose nature is well known. Even thousands of reasons will not force the Ambaṣṭha group of drugs into purgative action. So, the wise physician should stick to established tradition and not [get lost] in reasons.
Now, mathematical knowledge in India is also based upon the three pramāṇas. There is an understanding in the West that all of mathematics is derivable from logic alone, that is, starting from sentences, or propositions. On the contrary in India, Nyāya philosophy does not even allow for such kinds of understanding. Nyāya logic may start with sentences, but they have to express some knowledge that one actually possesses. With that, one moves on to the next level of knowledge, until one reaches a conclusion. So, Indians do not talk of abstract propositions, nor do they talk about the notion of logical validity being separated from material truths. Similarly, Indian philosophies do not conceive of analytical truths which are independent of experience. Hence, mathematical knowledge is placed on the same footing as knowledge in empirical, natural sciences.
Finally, let us consider proof by contradiction, or the so-called reductio ad absurdum. This is where we encounter tarka. Here, in order to prove some result, one assumes the opposite and shows that it leads to a conclusion, which contradicts what one has already assumed or established. Indians use this kind of an argument, called tarka, to check an already known general statement. But, crucially, they do not use such arguments to prove the existence of an entity. According to Indian logicians, proof by contradiction or tarka cannot be used as an independent source of knowledge:
tarko na pramāṇasangṛhītaḥ{\small \vert}
na pramāṇāntaraṃ vā {\small\vert } pramāṇānām
anugrāhakaḥ tattvajñānāya kalpate {\small\vert }
Proof of existence via reductio ad absurdum goes as follows: If one initially assumes that a certain differential equation does not have a solution, and goes on to prove a contradiction, then one concludes that there exists a solution for this differential equation. This will not be considered a valid way of establishing the existence of a solution of this differential equation, according to those who follow what is currently known as the “constructivist school” of philosophy of mathematics or the “constructive approach” to establishing mathematical existence. In India, this position has been held consistently, and that’s why `non-constructive’ arguments are not employed. On the other hand, in Euclid’s Elements, a large majority of the proofs are based upon reductio ad absurdum.
Historically speaking, it’s been emphasised that Indian contributions to mathematics are predominantly algorithmic in nature. What specific examples enable one to draw such a conclusion and, are there recorded instances that suggest that the algorithmic approach, although prominent, was only one among other approaches explored by Indian scholars?
MDS: It’s not just in mathematics that algorithms are important. The canonical texts of different sciences—or indeed of any of the śāstras or fields of knowledge which have been systematised in the Indian tradition—are not mere compilations of propositions, or universal truths. Instead, these texts are mainly concerned with presenting a series of rules which characterise systematic procedures in order to accomplish specific goals. The procedures prescribed in a śāstra to achieve a specific task, are called variously by the names of vidhi, a parikarma, a karaṇa, a sādhanā, or even an upāya.
Every śāstra, in the Indian tradition, had to first clearly state its goals or purpose. Let’s take the example of Ayurveda, whose purpose is to keep a healthy person healthy, and restore the health of an unhealthy person. Astronomy is supposed to be the science of determining the direction, place and time (dik-deśa-kāla) on the surface of the Earth by observing the motion of celestial bodies. Similarly, mathematics is the science of calculation with numbers, geometrical objects, and such. So, every science, and even art forms, have their standard objectives and algorithms or procedures to fulfil them.
These algorithms are not just some haphazard recipes, and in order to obtain them, one should have a fairly deep understanding of the domain where they operate. The design and construction of algorithms involves the possession of a sophisticated underlying theoretical framework. But the theoretical framework itself is not the primary goal of the śāstra. The goal of the śāstra is to provide efficient and optimal procedures which fulfil the various objectives. It is the same with respect to mathematics also. So, the phrasing of the question has a little bit of a problem because Indian contributions are not restricted to the design and construction of algorithms. Indian contributions encompass the concepts, and the attendant theories, on the basis of which efficient and optimal algorithms were created to achieve various objectives. There are many texts which only supply algorithms, just as there are other texts which talk solely of the concepts and principles behind them. But one needs both, be it in medicine, mathematics or astronomy. In essence, the Indian approach is more realistic, and systematic procedures are taken to be more important than merely stating theories and universal principles.
MSS: It is repeatedly emphasised that without a comprehensive understanding of the situation, it is not possible to have a grasp of the rules. This is exemplified for instance in Nīlakaṇṭha’s planetary model. He goes on to describe how the latitudinal motion of the interior planets is calculated. In his Āryabhaṭīya-bhāṣya he says that the latitudinal motion is said to be due to the motion of the śīghrocca. The immediate question that arises is how is this possible at all? How can the latitudinal motion of a physical object be attributed to the motion of something else? Clearly, this could not be. So, he takes a leap of imagination which is well-founded, and proposes that what was considered as the śīghrocca of an interior planet in earlier textsh must be the physical planet itself.
Nīlakaṇṭha takes a leap of imagination, and proposes that what was considered as the śīghrocca of an interior planet must be the physical planet itself
This is an excellent example of making an inference that is guided by a comprehensive understanding of the entire physical situation, so as to reconcile two different kinds of formulae for latitudes and longitudes. Another example is that of Bhāskara’s detailed description of the observational method to determine the various parameters associated with planetary motion. It is after all still an algorithm that allows one to calculate the desired physical quantity; but without understanding the instrument which is used to make the measurements, and the underpinning theory, one will not be able to know what exactly is happening.
With these two examples, I want to highlight the fact that the algorithmic approach in Indian astronomy is only a part of the story. Indian astronomers did realise that this approach needed strong observational and theoretical backing.
KR: Let me just state this beautiful verse which explains the unmistakable emphasis that has been laid down in thoroughly understanding the entire gamut of ideas surrounding the problem, before employing an algorithm. This is nicely articulated using three similes, by an astronomer called Nityānanda. He says:
rātrau-sadma-vicitra-citra-racanaṃ
dīpaṃ vinā kiṃ yathā
trailokyaṃ sakalārtha-pūrṇamapi kiṃ
bhāsvatprakāśaṃ vinā\vert
vaidagdhyena-vinā-pragalbha-taruṇī
saundarya-yuktāpi kiṃ
tadvat-golavivarjito-gaṇitavit
praśnottare kiṃ vibhuḥ \Vert
A rough reading of the verse goes like this: Of what appeal is, in the darkness of the night, a beautifully built and an aesthetically decorated house? Of what use are the three worlds full of opulence, without the light of the Sun (bhāsvat)? Of what use is the young, confident maiden endowed with beauty, but without smartness?…
With these examples Nityānanda essentially wants to convey that merely knowing how to numerically compute and obtain the final result without trying to understand what goes behind, is not something that will be appreciated by scholars.
MDS: There is a very important principle that is emphasised in all Indian texts, science or even other disciplines. In all the investigations dealing with worldly matters, the procedures which are taught in the śāstras are neither unique, nor absolute. All tools, theories, techniques are merely upāyas, and they must be understood only as such. Also, only as long as they provide correct results are they accepted. If some other upāya also gives correct results, that is also accepted. None of these upāyas are constrained by any requirement other than that they should satisfy the objective that the śāstra has set for itself. This key message is clearly stated in vyākaraṇa, jyotiṣa (astronomy), Ayurveda and many more places.
An important relevant statement is by Bhartṛhari. In vyākaraṇa, which is one of the most perfect sciences of India, the sūtras of Pāṇini enable one to derive the valid śabdas (utterances) of Sanskrit language by means of around 4,000 sūtras. And yet Bhartṛhari says that these rules of Pāṇini are upāyas that are relevant as long as one uses them and gets the intended job done. Apart from this minimal requirement, there is no necessary limitation that needs to be imposed on these upāyas. The commentator explains by saying that a certain teacher could be using Pāṇini’s grammar to derive all the utterances, while another teacher could be using another grammar, to derive all the utterances; either way, there is no restriction on which grammar is to be employed.18 This amounts also to cautioning us that all procedures, as well as the concepts and theories behind them, need to be constantly revised, or continuously checked.19
MSS: I want to add something related to the idea of computational positivism being the distinguishing feature of Indian astronomy. I think there is a problem with that kind of reasoning. It is true that one can use whatever upāya is appropriate to get the desired result. However, Indian astronomers clearly understood what the main results were, what were supposed to be exact, and what were those things for which one needed some imagination in modelling. One can, on the other hand, also recast an existing problem like Nīlakaṇṭha did. The result of such a recasting may not necessarily produce the most optimal objective, computationally speaking. Being computationally optimal is a good guideline to have for certain categories of problems, but there exist certain other problems which are understood to be very physical, and do not yield to approaches founded on computational optimality. Problems like eclipses are good relevant examples.
KR: I would just like to add briefly to this discussion. Even in Vedānta, they have this very interesting, important and oft-quoted verse, that is attributed to Sureśvarācārya:
yayā yayā bhavet puṃsāṃ
vyutpattiḥ pratyagātmani \vert
sā saiva prakriyā jñeyā
sādhvī sā cānavasthitā \Vert
In short, what it says is that one may choose whatever means is convenient at their disposal to facilitate the other person to understand what the truth or principle behind the phenomenon being studied is. All those means that serve the purpose are to be considered equally valid, and there are no limitations to such means.
Similarly, if one has to solve a problem, one can choose any method which is available, but the ultimate aim is to get the problem solved. Also, it may not be appropriate to call any one method the most effective method or as being superior to the rest.
There is another emphatic statement which is akin to what Srinivas was mentioning earlier. Here, I may also cite Vācaspati Miśra’s words. He notes:
na hyāgamāḥ sahasramapi ghaṭaṃ paṭayitumīśate \vert
In order to convey the primacy of pratyakṣa over āgama, the above verse emphasises that even if there are a hundred āgamas which try to convince us that a pot is in reality only a piece of cloth, all the hundred āgamas should be rejected.
How fundamentally different is the western approach to mathematics compared with the Indian approach, in an overall sense?
MDS: Let me just cite from an article of David Mumford that he wrote for the March 2010 issue of the Notices of the American Mathematical Society reviewing the book Mathematics in India by Kim Plofker:20
It is important to recognise two essential differences here between the Indian approach and that of the Greeks. First of all, whereas Eudoxus, Euclid and many other Greek mathematicians created pure mathematics devoid of any actual numbers and based especially on their invention of indirect reductio ad absurdum arguments, the Indians were primarily applied mathematicians focused on finding algorithms for astronomical predictions, and philosophically predisposed to reject indirect arguments.… The Indian mathematics tradition consistently looks for constructive arguments, and justifications and numerous numerical algorithms.
So, he is identifying two characteristics: One, that we prefer objects which are concrete, and this is evident when we invoke numerical algorithms for calculating various objects of interest associated even with geometrical figures. Two, that we do not use indirect arguments, preferring constructive arguments. So, these are some of the major differences as Mumford expresses it. He further goes on to say:
It is high time that the full story of Indian mathematics from Vedic times through 1600 becomes fully known. I am not minimising the genius of the Greeks and their wonderful invention of pure mathematics, but other people have been doing mathematics in different ways and they have often attained the same goals independently. Rigorous mathematics in the Greek style should not be seen as the only way to gain mathematical knowledge. In India, where concrete applications were never far from theory, justifications were more informal and mostly verbal rather than written. One should also recall that the European Enlightenment was an orgy of correct and important but semi-rigorous mathematics in which Greek ideals were forgotten.
So, the basic point he is emphasising is that in the history of mathematics, progress has been achieved by different methods, with India and Greece as typical examples. The predominant characteristics of Indian mathematics are the focus on constructive algorithms, rather than abstract proofs of existence; and a focus on applied, numerical algorithms. I think this is a very important statement coming from Mumford. Although he repeatedly emphasises that Indians were applied mathematicians, there was nothing much applied, in the sense of use in astronomy or any other practical discipline, about Vargaprakṛti, or Cakravāla, at least at the time of Brahmagupta. So, it’s not necessarily true that everything arose from applied mathematics. It is just that the Indians always had a concrete situation in mind, and that mathematics was always about the real world.
The question that arises now is what do the Indian mathematical texts reveal about their methodology so that we can understand how it differs from the Greco-European tradition? The Greco-European tradition, as Mumford has explained, is of two types. For a brief period around the time of Archimedes, Apollonius and Euclid, there is evidence of rigorous mathematics, which is again seen during the recent period, i.e., from the past 200 years, starting from Karl Weierstrass, Richard Dedekind and others up to the present day. But in between these two periods, there’s a whole lot of mathematics which does not conform to that kind of rigour. Until the 19th century, even the idea of a real number, and the idea of an irrational number could not be expressed in an axiomatic system. The Euclidean framework could not be applied to it, and therefore all kinds of “semi-rigorous” arguments were being used.
In comparison, it is useful to see what the corresponding situation in India was, during the same time. What were they up to? Indians differ from Europe as I said earlier, in their very understanding and formulation of the essential nature of mathematical knowledge. Firstly, it’s not particularly different from the knowledge available in the natural sciences. So, both results in natural sciences and mathematics are constrained by similar conditions under which a particular result is held to be true.
Secondly, whether all of mathematics can be derived purely from “abstract reasoning” with no empirical reference. This abstraction founded approach has not been the approach in India. In the Indian approach to mathematics, at least both pratyakṣa and anumāna are involved. And even pure anumāna, or inference in the Indian approach, is crucially dependent on pratyakṣa. Nīlakaṇṭha analyses this question in detail, and says that the common mathematical knowledge or sāmānya-gaṇita, which comprises arithmetic and geometry, can be arrived at without the use of any text or āgama. He says that competent mathematicians can always obtain or demonstrate these results for themselves, without taking recourse to any text, by using yukti21 which, like anumāna, always subsumes pratyakṣa. Nīlakaṇṭha also notes that this feature of sāmānya-gaṇita is in marked contrast with grahagaṇita or the mathematical astronomy of planetary motion where one considers observations carried out over a long time; and hence grahagaṇita or mathematical astronomy is clearly based on crucial use of textual evidence or āgama.
The other thing that the Indians emphasised was that, in mathematics, one had to necessarily provide proofs. Merely providing a number of instances where a result is valid will not do. An argument to show that the result is valid in a more general sense is demanded. Even if one observes something empirical, that empirical observation should also belong to a typical case of a much more general situation, and cannot just be representative of something that is narrow, or particular. Proofs should lead one to state general conclusions.
Bhāskara has a practical way of characterising what constitutes an upapatti or proof. It is that which allows one to become a respected member of the mathematics community, and which allows one to have no further doubts about a particular result. The purpose of proof was to ensure that the result that one presented was accepted by other mathematicians, and which also enabled one to explain the result without any confusion. There was no ideal mentioned in Indian mathematical works that a proof should show the absolute or eternal validity of a statement.
Finally, apart from their constructivist approach to the question of demonstrating the existence of mathematical objects, another crucial way in which the Indian approach differs from the Greco-European approach is the following. The Greeks thought that systematisation of knowledge was best achieved by listing at the outset a set of postulates, a certain set of general assumptions which are common to all subjects, and a set of assumptions specific to the domain of interest, which then came to be called axioms. Though the Indian texts such as the Yuktibhāṣā proceed systematically from known results to new results, there is no attempt made to present all these results in an axiomatic framework. There are many philosophical issues discussed in Nyāyaśāstra concerning the tenability of the very notion of an axiom which, at least till the 19th century, was thought of as a self-evident truth, and so on.
Pāṇini is famed for his canonical approach to establishing the rules of grammar and that which now underlies many Indian languages spoken uninterruptedly to this day for many millennia. What mathematical ideas did Pāṇini dip into, and employ?
MDS: The status that Pāṇini’s grammar seems to have enjoyed in India is similar to what Euclid’s Elements seems to have enjoyed in the western tradition. In recent times, R.G. Bhandarkar was one of the first scholars to have pointed this out. What is so special about Pāṇini’s grammar is that it deals with algorithms and procedures to derive valid utterances with the aid of around 4,000 rules. The techniques that Pāṇini uses to characterise or formulate a sequence of procedures have been utilised in all other śāstras, later. That is, if there are two rules given to achieve a certain objective, which among them takes precedence? When are the rules ordered? If there is a rule with reference to an earlier one, or if there is a repetition of a rule as in a recursion, then how does one go about these situations? Importantly, algorithms developed in Indian mathematics and astronomy, including basic techniques of stating an algorithm and formulating them seem to owe a great deal to how Pāṇini had formulated his procedures (prarkriyā) for the derivation of valid utterances in Sanskrit grammar.
Specifically, on mathematical topics, there are two ideas in Pāṇini which are of significance. One is the use of symbolic representations for a group of sounds, using the śivasūtras. A collection of sounds of the Sanskrit varṇamālā (involving vowels, consonants and such) is given a symbolic representation and this is also extended to a chosen collection of subsets of sounds, which are also treated symbolically. The notion of a `set’ was not used in India in the abstract sense of the word today.
But Indians were using and formulating symbols to denote a collection of sounds, a collection of sub-collection of sounds, and so on. Apart from this, some precursor of the notion of zero is present in Pāṇini’s works in the notion of a lopa.
The status that Pāṇini’s grammar seems to have enjoyed in India is similar to what Euclid’s Elements enjoyed in the western tradition
MSS: To me, it probably looks as though it was the other way around. Instead of thinking of what mathematical ideas Pāṇini dipped into, especially as must have been prevalent at that time, my feeling is that the mathematics that evolved in India was itself patterned in the Pāṇini mould. It is well known that Pāṇini gave various rules to generate words and sentences. In the trigonometry section of Siddhānta-Śiromaṇi by Bhāskara, is an idea named jyotpatti—which splits as jyā+utpatti—referring to a method for the generation of `sines’. This is patterned after Pāṇini’s sūtra format. For many important ideas in Indian mathematics and astronomy, Pāṇini is the proper paradigm, just as in the west, it is the Euclidean paradigm.
KR: The Paninian paradigm has deeply influenced the study of all branches. Even earlier, certain attempts were made to explain the formation of the right usages of words in vogue. How does one decide that a certain expression is a valid expression, while a certain other is invalid? This is primarily decided by structure. On the idea of using sets and allied ideas in grammar, I would say that the idea of sequences of operations is also very important in the works of Pāṇini. A sequence defines the order in which various operations have to be carried out. In addition to sets and sequences, we also have something akin to the notion of equivalence classes fairly well established in the Paninian structure. An interesting idea called apavāda appears in Pāṇini’s works referring to those contexts where a certain operation on a set does not hold, albeit for certain elements inside the set. In essence, a variety of mathematical structures, operations that are carried out on them, and the result of such sequences of operations—all of which are fundamental in our understanding of mathematics as understood and practised today—are easily and evidently visible in Paninian grammar.
MDS: There is another interesting statement, again from Mumford’s article, that seems appropriate here:
Pāṇini… introduced abstract symbols to denote various subsets of letters and words that would be treated in some common way in some rules; and he produced rewrite rules that were to be applied recursively in a precise order. One could say without exaggeration he (Pāṇini) anticipated the basic ideas of modern computer science.
Music with its lilting rhythms and charming variations lends itself to a mathematical study. Please elaborate on an internal link between music and mathematics, especially focusing on an Indian contribution that forges the link strongly.
MDS: Mathematics is seen frequently in texts of music. In the older texts, such as Bharata’s Nāṭya-Śāstra and Śārṅgadeva’s Saṅgīta-Ratnākara, it occurs when one tries to understand śrutis and svaras, or sound intervals and notes. That study continued well up to modern times with many important developments being discussed in several texts of Saṅgīta-śāstra during the 16th–18th centuries.22 Both the mathematics and the physics of sound are studied. The Veena is taken as the quintessential musical instrument to define and study these aspects.
The other areas where many important mathematical developments are seen is in the study of combinatorial considerations that arise naturally in music. We have the authoritative text Saṅgīta-Ratnākara of Śārṅgadeva, dealing with various permutations of svaras, in a section on the tāna-prastāra. Just as in Piṅgala’s Chandaḥ-Sūtra (c.300 BCE) where all binary sequences of a given length are ordered in an array or prastāra, here the 4! permutations of `sa’, `re’, `ga’, and `ma’ are ordered in a prastāra. All the usual questions that Piṅgala deals with, such as saṅkhyā, naṣṭa, and uddiṣṭa have been discussed in this context also.
Similarly, there is another combinatorial exercise involving various tālas, where one enumerates all the rhythmic possibilities of a given duration. This theory is also discussed very elaborately in Saṅgīta-Ratnākara, where Śārṅgadeva has developed the relevant mathematics.
In tāla, we encounter a situation that is similar to the mātra-vṛitta (moric metre). One has some basic constituents of tāla which are of durations one, two, four, and six. These four units are taken, and one has to now make all possible sequences out of them, to arrive at say a total of 20 durations. In how many possible ways can these four basic units come together to finally add up to 20? The basic units are ordered partitions, and a study of all the complex ways in which the sequences behave, involves fourth-order recurrence relations, instead of the second-order recurrence relation defining the Fibonacci sequence.23This subject is further developed in some of the books on music written during the 17th–19th centuries, which take into account some more kinds of tāla units also. Altogether, this is a very complex subject, and if one goes into aspects involving naṣṭa and uddiṣṭa in this prastāra, it becomes a fairly involved combinatorial problem. Śārṅgadeva’s profound contribution was in initiating this entire subject in the 13th century. This study went on, adding lots of musical and combinatorial ideas to its rich body of knowledge till the early 20th century. I am told that in the early 20th century, the Tacchur Singracharya brothers wrote some treatises in Telugu, where more intricate tāla-prastāra techniques are dealt with. The study of musical texts in India has not been systematically undertaken in recent times, though they used to be studied seriously until perhaps a 100 years ago. There is a need to evolve a new set of scholars, who also have an understanding of the mathematical nuances involved.
Trade and commerce, languages, music, architecture are various facets from everyday life that contributed to, and enriched Indian approaches to mathematics. Vedic rituals, observation of religious and cultural events relied on a precise estimation of celestial events. In what concrete terms was the so-called Siddhāntic era a significant break from the ideas of its past, and why is the study of this era important to understand Indian contributions to the sciences?
MSS: In Vedānga-Jyotiṣa (c. 1200 BCE), which is a compact text on astronomy, we find the influence of observations in evolving a calendar. A type of arithmetical astronomy is seen in this text. One sees a systematisation, albeit of a limited nature, because they considered only the Sun and the Moon; and even there, they considered their motions as being uniform. It is only during the Siddhāntic period that one encounters astronomy that was based on sophisticated procedures involving mathematics; trigonometry and spherical trigonometry, to be specific. An example is Āryabhaṭīya, which is a complete system towards understanding astronomical phenomena. However, the text is written in a cryptic manner, in only 121 verses. But it prescribes a definitive and clear approach to the study of astronomical phenomena. Mean and true positions of planets, the celestial sphere, the three problems of dik-deśa-kāla (direction, location and time), the possibility of eclipses, and many more are presented in a systematic manner, effectively leading the reader from a query to the desired answer.
This achievement became possible only during the Siddhāntic era. Incidentally, though Āryabhaṭīya is the first work to have been discovered and known, it is recorded that there were a total of 18 earlier siddhāntas. Five of them have been summarised in the Pañca-Siddhāntikā of Varāhamihira. Post Āryabhaṭīya, the same pattern of scientific inquiry is followed, with the only change being the addition of a few more topics over time, with elaborations of individual ideas. Chapters on mathematics are found in the beginning, with trigonometry being the strong focus. Since the Āryabhaṭīya was composed around 499 CE, and since it is known that there were 18 earlier siddhāntas, the generally accepted start date for the Siddhāntic era is at least the first century, with the era itself lasting up to the nineteenth century.
MDS: Varāhamihira, writing around 550 CE summarises five of the siddhāntas that existed earlier to him in his Pañca-Siddhāntikā. But what was the original form of those siddhāntas that Varāhamihira is summarising? Varāhamihira was clearly aware of what Āryabhaṭa had written around 500 CE. But, did he include insights from Āryabhaṭīya while making these summaries of the five siddhāntas? These are issues that are yet open, and not settled.
KR: I would not call the Siddhāntic period a complete break in terms of the fundamental approach to the subject of astronomy as such, but it indeed marks a major transition in terms of specifying parameters, devising new mathematical techniques and algorithms etc. It seems to me that the word is perhaps used to describe the nature and content of the text better. For instance, one of the connotations of siddhānta is that it presents a more refined picture of what had happened earlier. The second is that it is more comprehensive in its treatment of the important topics, and the third is that here one also finds certain reflections of past works in the present time. These three features are usually seen in a Siddhantic work. Here is a verse from the Sarvasiddhāntarāja of Nityānanda that gives an idea of the kind of topics that a siddhānta text would explicitly deal with, or implicitly contain:
pāṭī-kuṭṭaka-bīja-gola-nipuṇaḥ
vedāṅga-pāraṅgamaḥ \vert
kāvyālaṅkṛti śilpaśāstra-kuśalaḥ
yo nyāyaśāstrādivit \Vert
siddhānte\acute{\;}tra-samasta-vastu-sahite
tasyādhikāro bhavet \vert
………
Precise astronomical calculations required (to pun) a celestial mix of mechanics, analysis and trigonometry. What is the most authoritative Indian commentary yet from historical times on the accurate prediction of eclipses and other celestial phenomena of interest? Also, have there been instances of debates between competing Indian schools regarding the heliocentric versus the geocentric explanations of solar system dynamics?
MSS: First of all, there is no `mechanics’ here. These are all kinematical descriptions in astronomy. There is no concept of force, acceleration, or dynamics in ancient astronomy. What is understood to be mechanics in astronomy, are mostly kinematical descriptions, which provide an accurate description of the motion, and the geometry of orbits. This is true of the Indian, as well as the Greeko-European and Islamic traditions up to, and including, Kepler in the early 17th century.
On what the most authoritative Indian commentary on the accurate prediction of eclipses is, I really don’t know. But, people have attempted to compare various texts to find an answer. In the process, our overall understanding has improved. We ourselves started with the calculation of eclipses, and how its knowledge had evolved from an earlier text, to a later text. I guess around the beginning of the 16th century, Tantrasaṅgraha was more accurate than the rest of the texts, and certainly so for the motions of the Sun and the Moon. Corrections to the parameters associated with the Sun and the Moon are better in Tantrasaṅgraha. So, I would say that Tantrasaṅgraha comes close. And to it, also add Siddhānta-Darpaṇa by Samanta Chandrashekhara, composed in 1869 CE. Chandrashekhara considers more corrections to the Moon and has more accurate parameters, but he may have been influenced by modern astronomy. In fact, he acknowledges that he had studied modern astronomy, but his own work is completely in the Siddhāntic framework. So, Chandrashekhara’s Siddhānta-Darpaṇa is probably the most accurate. But, certain things are not typically discussed in Indian astronomy, like the path of the solar eclipse, though the times and the places where they will be visible are announced.
Regarding the question on Indian schools debating heliocentricity versus geocentricity, they don’t talk about that really. They talk about observations made from the centre of the Earth, and include corrections to the calculation to account for the fact that the observations are actually made from a point on the surface of the Earth, which has a specified radius. The centre of a celestial sphere and related ideas are considered, but they don’t talk about heliocentric or geocentric approaches at all. Since the geometry of the orbits are of interest to Indian astronomers, Nīlakaṇṭha went on to say that the planets go around the Sun. Now, since the Sun itself goes around the Earth in that scheme, the whole picture effectively becomes somewhat of a quasi-heliocentric model. Nevertheless, I do not see any debate happening on heliocentricity versus geocentricity.
KR: In terms of debate, we hear of something like Bhabhramaṇa-vicāra, tantraparīkṣā etc. As noted by Sriram, debate on heliocentric versus geocentric is certainly not present. However, an inquiry whether one can conceive of the Earth as a rotating frame, or should it be treated only as a non-rotating frame is definitely seen in ancient Siddhāntic texts.
As far as the other part of the question is concerned regarding an authoritative and reliable commentary of historical vintage, it’s a little difficult because we have not looked into many of these commentaries. For instance, some of us have been talking about this commentary called “the Vāsanā-bhāṣya of Pṛthūdakasvāmi on Brāhma-Sphuṭa-Siddhānta”. We do not even have a complete manuscript here, and even the portion which is available has been only partly studied. This is how things are, as regards the study of even important texts. Among the commentaries which have been looked into, Vāsanā-bhāṣya on Siddhānta-Śiromaṇi that Sriram is now working on, can be considered an authoritative text. There are also certain other commentaries which are extensive, but they may not be as precise as Bhāskarācārya’s works. So, to provide a definitive answer saying that such and such text(s) is/are the most authoritative ones is a little tricky. Currently, we don’t have the resources to make a convincing comment on such matters.
MDS: On the first question of mechanics. Mechanics is a 17th century discipline, and historians of science clearly say that mechanics arose out of developments in astronomy. The main test for mechanics was the motion of the Sun, the Moon and the planets. The Indian approach to astronomy was not geometrical, but what may be called analytical, in that Indians used formulae involving the trigonometric sine and cosine functions to describe the orbits of a planet, that is a description of their positions expressed as a function of time.
In contrast, the Greeks were exclusively using a geometrical picture of planetary motion, from which one could of course calculate the numerical values of the position of the planet at any particular time. But the Greeks did not have this idea, or the mathematical apparatus, for describing motion in terms of mathematical formulae. The Indian formulation of astronomy was not only analytical, but also physical.
The second key aspect is that the Greeks assumed that celestial bodies had to move with a uniform velocity along a circle, and that they were fixed to some `ethereal crystalline’ spheres. From a time even prior to Ptolemy and right up to the time of Kepler, these ethereal spheres held their sway. Essentially, these were the sort of ideas that they considered in their models.
In contrast, in the Indian models, these bodies could move non-uniformly. There was no restriction that they move with uniform velocity. They had these famous functions `jyā (Rsine)’ and `koṭijyā (Rcosine)’, which were functions of the arc of a circle. It must be noted that these were not functions of angles, but were carefully defined as functions of arcs of a circle. Using `jyā’ as a function of the arc, they could calculate the position of a planet, as a function of time. Further, the velocity of the planet, taken as a function of time, was calculated. Crucially, Indians came up with the `trigonometric cosine’ function as the derivative of the `trigonometric sine’, as early as in the tenth century, as seen in the works of Muñjāla and Āryabhaṭa II. Much later, this is seen also in Bhāskara’s Siddhānta-Śiromaṇi where he explicitly discusses the concept of `instantaneous velocity’. He is able to explain how instantaneous velocity is different from looking at the change of planetary position from day to day, and that the correct way to look at it was as something that changed from an instant to the very next instant. This was certainly a sophisticated understanding of the mechanics.
In many Indian texts from that era, there is a discussion of the three spatial coordinates that are needed to fix the position of the Moon, clearly demonstrating the familiarity of Indian astronomers with the conception of three-dimensional coordinate systems. This way they could even calculate the distance between the centres of the lunar and solar spheres. In essence, they possessed a sophisticated armoury of physical and mathematical concepts.
They also had an impetus theory of motion through the notion of samskāra which is discussed in the texts of the Vaiśeṣika School. This was in response to the often discussed problem as to how the motion of an arrow continues well after it is shot, that is, what causes its further motion? They did not have Newton’s laws for sure, but they had a fairly evolved understanding of how to go about discussing motion.
There was no debate in India as to whether the Sun or the Earth ought to be the centre of the universe. The importance of the Sun, in its role of sustaining all forms of life on Earth, was very well recognised. Its role in influencing and regulating the dynamics of all the planets was known too. They even had the idea that the description of planetary motion varies with respect to the so-called `frame of reference’ from the time of Āryabhaṭa himself. Their calculations were always explicitly done with respect to a frame of reference, which itself was usually the location of the observer. Corrections for parallax were also available those days, for calculations that involved different points of reference on the surface of the Earth. Texts like Yuktibhāṣā even talk about a hypothetical situation where an observer finds oneself on the Moon and is now looking at the Earth. The fact that the frame of reference has now changed from the Earth to the Moon, and therefore the observed motion of the planets will necessarily look different from how they are seen from the Earth, comes out very clearly in this example.
So when Nīlakaṇṭha says that as seen from the Earth, all the planets go around the Sun, with the Sun itself seen going around the Earth, he is talking about an Earth-based frame of reference for discussing planetary motion. Nīlakaṇṭha did not arrive at this conclusion based on any philosophical debate involving heliocentricity versus geocentricity. His statements were borne out of simple, yet crucial observations that he considered, namely the following: The latitudinal motions of Mercury and Venus, that is, their deflection from the ecliptic happens over a period of 88 and 234 days, respectively. In contrast, the period of their longitudinal motion, going around all the stars, occurs in 365 days. This phenomenon of two different periods is seen only for the interior planets. In the case of the exterior planets, both the latitudinal and the longitudinal motion occur over the same period of 365 days. It is on the basis of this crucial observation that Nīlakaṇṭha concludes that the orbits of Mercury and Venus go around the Sun with a period which is the same as the period of their latitudinal motion. The Sun carries both of them around the stars in 365 days, which is the period of their longitudinal motion. All this happens because the Earth is not enclosed in the orbits of Mercury and Venus around the Sun. This is the heliocentric model since, as far as the exterior planets are concerned, their orbits go around the Earth as well as the Sun. Nīlakaṇṭha’s argument is very sophisticated, and not at all speculative. It is not based upon some philosophical argument about heliocentricity versus geocentricity, but based purely on planetary theory and observations.
How would you capture the contributions of Mādhava, who pioneered the flourishing Kerala school that lasted for over a couple of centuries?
MSS: We will start with the infinite series, which incidentally are described in all later works, but are always attributed to Mādhava. Formulae in the form of a series exist for the sine and the cosine functions, the arc-tangent function and \pi. Mādhava also realised the slow convergence of his infinite series for \pi. To address this, he first provided a truncated series, and then simply approximated the rest of the terms with a remainder term. In Yuktibhāṣā, there is a very systematic procedure to obtain these remainder terms using which one can achieve higher and higher levels of approximation. I think this is one of Mādhava’s major contributions to the world of mathematics.
Of course, he has significant works in astronomy too. Veṇvāroha and Sphuṭa-Candrāpti are his works. A novel contribution here is how to calculate the position and the exact longitude of the Moon. Here, he refined existing methods to make them more accurate. Secondly, he has provided an ingenious procedure to calculate the longitude of the Moon accurately, at nine different instances on any given day. Tracking the motion of the Moon is important primarily because, with respect to the stellar background, it is moving at a much faster rate; and close to about 13 times that of the Sun. Another example of his contribution is for finding out mandakarṇa, which is related to the true distance of a planet from the centre of the Earth. This comes up in astronomical calculations. Here, he has a formulation which is typical of Indian planetary models, and somewhat different from the planetary model of the Greeks. These are some of his major contributions.
KR: As was pointed out by Sriram, Mādhava’s idea to expedite the observed slow convergence of a series is brilliant. One can look up the article by Jolly John in the volume celebrating 500 years of Tantrasaṅgraha.24 It’s an interesting study of the formula by Mādhava that incorporates antya-saṃskāra, and is also derived in Yuktibhāṣā as well as in Śaṅkara Vāriyar’s commentary on Līlāvatī. It is relevant to note here that Mādhava’s capacity for producing such brilliant results, is known only through citations of the same by other people. There are several verses which describe these infinite series, and even Karaṇapaddhati presents many of these verses.
There are a couple of other works of Mādhava which merit attention. Specifically, one of them is a short work Mahājyānayana-Prakāra (it is possible that it was authored by Mādhava) which has been studied by Sriram and his student, Rajarajeswari.25 This gives a derivation of the infinite series for sine and cosine functions which is essentially the same as in Yuktibhāṣā, but briefer. The other work is Lagna-Prakaraṇa, which has been studied by Aditya Kolachana, a former student of mine, who is currently a faculty in IIT Madras. This text is an extremely difficult one to decipher, and only half of it was done by Aditya for his PhD thesis. It is a very interesting and valuable text because it explains various techniques by which one can precisely calculate the important quantity called lagna in astronomy. It is a mathematically sophisticated text, and took us an enormous amount of time, just to appreciate how Mādhava had visualised the various geometric projections inside the celestial sphere, before arriving at results that are quite complicated. A careful study also informs us that this whole exercise seems to have been done for the sheer joy of arriving at important results, by employing wholly different approaches. The sign of a truly accomplished mathematician! I am convinced that he must have been a mathematician of the highest order, to have undertaken this problem. A total of four chapters have been studied by us, with four more to go. In short, as regards Mādhava’s contributions to the world of mathematics, certain things are known, but many more things have to be carefully explored.
I am convinced that Mādhava must have been a mathematician of the highest order
MDS: One more work of Mādhava, called Agaṇita-Grahacāra has not yet been published. Mādhava’s work on the vākya system also includes a higher order correction for Vararuci’s vākyas. While the ancient vākyas of Vararuci gave the true longitude of the Moon (over a cycle of 248 days) correct to minutes, Mādhava’s work provides the true longitudes which are correct to seconds. He not only provided proofs, and discovered general results, but also gave beautiful computational methods to approximate them to any desired order. So, he is both an accomplished innovator in theory as well as in computations.
KR: Regarding Mādhava, I want to make one more point. I have had occasions to look into the historical aspects surrounding the discovery of ideas related to differential and integral calculus, and their evolution. During this study, I noted the sheer volume of praise that has been showered on Gottfried Leibniz for his contributions. Adjectives such as glorious, and brilliant are routinely attributed to Leibniz, especially with regards to his work on infinite series. Now that we recognise the high status accorded to Leibniz in the world of mathematics, the following words by Mumford, for Mādhava, while reviewing Kim Plofker’s book, speak volumes about the calibre of the man.
The discovery of the finite difference equation for sine led Indian mathematicians eventually to the full theory of calculus for polynomials and for sine, cosine, arcsine, and arctangent functions, that is, for everything connected to the circle and sphere that might be motivated by the applications to astronomy. This work matured over the thousand year period in which the West slumbered, reaching its climax in the work of the Kerala school in the fourteenth to sixteenth centuries.…
Kerala is a narrow fertile strip between the mountains and the Arabian Sea along the southwest coast of India. Here, in a number of small villages, supported by the Maharaja of Calicut, an amazing dynasty of mathematicians and astronomers lived and thrived. A large proportion of their results were attributed by later writers to the founder of this school, Mādhava of Saṅgamagrāma, who lived from approximately 1350 to 1425. It seems fair to me to compare him with Newton and Leibniz. The high points of their mathematical work were the discoveries of the power series expansions of arctangent, sine, and cosine. By a marvellous and unique happenstance, there survives an informal exposition of these results with full derivations, written in Malayalam, the vernacular of Kerala, by Jyeṣṭhadeva perhaps about 1540. This book, the Gaṇita-Yukti-bhāṣā, has only very recently been translated into English with an extensive commentary. As a result, this book gives a unique insight into Indian methods. Simply put, these are recursion, induction, and careful passage to the limit.
It is only a small portion of Indian manuscripts from ancient times that are known to have been preserved and catalogued; of them an even smaller number have been published. Since a large portion of such manuscripts have even been barely studied, and their content yet to be understood and assimilated, how are modern-day scholars like yourselves addressing this huge challenge?
MDS: Firstly, the magnitude of this problem is very large. Fortunately, due to David Pingree and K.V. Sarma, we now have exhaustive lists of manuscripts of source works of Indian mathematics and astronomy, and their location. Using these resources, we could estimate that there are roughly about 30,000 manuscripts on mathematics and astronomy (excluding the works on astrology). These manuscripts pertaining to mathematics and mathematical astronomy here, amount to roughly about 9000 source-texts. In the early period of Indology, between 1800 and 1875, only 15 of these were edited, and ten among them translated, mostly in parts. Colebrooke stands out here with his translation of the mathematical sections of Brāhma-Sphuṭa-Siddhānta and the Līlāvatī and Bījagaṇita of Bhāskarācārya. From 1875 till Independence, about 135 texts were edited, and 20 among them were translated. Only in the post-independence era have some 300 texts been edited, with 65 of them being translated. So, out of these 9000 texts, only about five percent, that is, 450 texts have been edited, and among them, only about 90 works have been translated. An even smaller number of works have really been explained fully as regards their technical content, in terms of mathematical equations, diagrams, etc.
The situation as regards most of the other disciplines of Indian knowledge is much worse than what we find in the case of astronomy and mathematics. These two subjects happened to be among those in which the western scholars also got interested in the 19th century. Consequently, some texts were edited, and a few of them were studied carefully. In many other fields of Indian knowledge, there is very little by way of modern scholarship based on an intensive study of the source-works.
Unfortunately, this is the present situation, and this situation is not going to change dramatically unless we make research and teaching of all disciplines of Indian knowledge—not just Indian mathematics and astronomy—a part of the subjects taught and researched, in our institutions of higher learning. Only if that is done, will there be a sufficient number of scholars who will be competent to study the source-works of Indian knowledge. If disciplines of Indian knowledge are incorporated into the scheme of studies and research pursued in our institutions of higher learning, there will be a number of scholars who can be expected to carry out doctoral research in these areas. And soon enough, we would have created a strong ecosystem of scholars and students, who together will systematically study the available corpus of manuscripts, and explain their technical content.
The poor state of maintenance of not only manuscripts, but even other valuable sources like archaeological sites, metallurgical objects, etc., leave much to be desired. In order to safeguard their preservation, one has to train scholars to preserve as well as catalogue, and digitise available assets. For working with manuscripts, one has to train scholars to read and understand the subject, the script, and the content. This is indeed a tall order. We have progressed significantly in the past few decades, compared to how we fared in the 150 years prior to independence. But a serious, and large-scale effort has to be initiated for us to make measurable progress. The large number of institutions of higher learning in India have to seriously get involved in the rediscovery of the great knowledge heritage of India.
An unfortunate trend seen in our school and college curricula is the tendency to not credit many pioneering Indian scholars from the past, and even worse, transfer pioneering Indian contributions to non-Indian sources. What is the way forward to remedy this? Does the recently formulated New Education Policy (NEP) address these foundational issues?
MSS: I am not involved in activities regarding the new education policy, and its framing. But I want to make a few observations. In India, there is a lot of status-quoism, and it is not easy to change that. University administration and educational bureaucracy sometimes can slow things adversely. It has been observed that if one introduces Indian knowledge systems at any level of study, it is immediately remarked by some that the country is being led back to medieval times. This is a barrier that has to be overcome. The quantum of effort that is presently required to sensitise and educate our youth about our rich cultural and academic heritage, and do it systematically and relentlessly for the coming generations, is huge. Efforts like NPTEL that we ourselves were involved in, are excellent starts; but small, in scale and reach. Only when the change that we seek begins to enter school and college curricula across the country, will it acquire the desired scale and momentum.
I think NEP is a good start that needs to be sustained and nurtured. It is also high time we broke from a past that has shackled us intellectually for far too long. As far as the school-level mathematics curriculum in India is concerned, a very significant part of it can be presented through the works of Indian mathematicians done through the ages. That being the scenario, we have two options: change the syllabus completely, and create a new syllabus from an Indian perspective; or, retain the present framework, suitably incorporating Indian contributions into its existing body. Whatever option one chooses, at some level the modified pedagogy has to merge smoothly with what is required for the higher classes. The Central Board for Secondary Education (CBSE), in consultation with experts, had prepared some books based on Indian Knowledge Systems a few years ago, as material for elective courses for CBSE classes 11 and 12.26 But, it has not been implemented, and there’s a lot of resistance to introduce them even as an elective. This incident alone is symptomatic of the challenges ahead of us.
KR: We have to find a way forward to remedy this, as Sriram was just mentioning. It has to be taken to schools and colleges, and there are certainly interesting events that are happening currently in which I am personally involved. NEP of course has this provision to revise entire curricula that are being designed by various groups to incorporate Indian knowledge into its body. Inside the working groups of NEP’s various committees, there are entities called `cross-cutting groups’. One such cross-cutting group is called “Knowledge of India”. There are about eight members in that group, and I too, am a part of it. We function in an advisory role regarding incorporation of relevant content.
In a different context and outside of NEP, there are initiatives led by individual citizens and private bodies, which have more freedom to design a curriculum based on Indian texts. One such initiative called “Agastya Gurukulam” is shaping up well. Here, teaching happens only in Sanskrit right from class one onwards, and the entire school syllabus, to the extent possible, is primarily based on Indian texts. In parallel, there is also a board which is presently coming into existence, after receiving the necessary governmental sanctions. It is called “Bharatiya Shiksha Board”, and is along the lines of the Central Board of Secondary Education (CBSE). The board has started designing and framing the curricula, and necessary textbooks are presently being authored. I am involved in the mathematics part of this initiative. This board has mandated that Sanskrit has to be taught right from class one onwards, and all Indian contributions be incorporated in the textbook to the extent possible. There is also work going on to ensure a smooth transition from these curricula, to existing college level courses.
MDS: Coming to NEP, we may recall the following broad prescription it makes right at the beginning of the policy statement (NEP 2020, p.2.):
The rich heritage of ancient and eternal Indian knowledge and thought has been a guiding light for this Policy. The pursuit of knowledge (Jñāna), wisdom (Prajña), and truth (Satya) was always considered in Indian thought and philosophy as the highest human goal. The aim of education in ancient India was not just the acquisition of knowledge as preparation for life in this world, or life beyond schooling, but for the complete realisation and liberation of the self. World-class institutions of ancient India such as Takshashila, Nalanda, Vikramshila, Vallabhi, set the highest standards of multidisciplinary teaching and research and hosted scholars and students from across backgrounds and countries. The Indian education system produced great scholars such as Charaka, Susruta, Āryabhaṭa, Varāhamihira, Bhāskarācārya, Brahmagupta, Chanakya, Chakrapani Datta, Mādhava, Pāṇini, Patañjali, Nāgārjuna, Gautama, Piṅgala, Sankardev, Maitreyi, Gārgī and Thiruvalluvar, among numerous others, who made seminal contributions to world knowledge in diverse fields such as mathematics, astronomy, metallurgy, medical science and surgery, civil engineering, architecture, shipbuilding and navigation, yoga, fine arts, chess, and more. Indian culture and philosophy have had a strong influence on the world. These rich legacies to world heritage must not only be nurtured and preserved for posterity but also researched, enhanced, and put to new uses through our education system.
So, there is an all purpose mandate to include all Indian knowledge traditions into our education system, be it at school, college or research level. The problem is clearly in implementation. Out of the budget of the Department of Education, `Indian Knowledge Systems’ as a collective entity, have recently been allocated a budget of about 50 crore rupees over three years. Clearly, this allocation is inadequate, but it must be noted that even this small allocation did not exist earlier. Recently, a committee has been constituted by the UGC to frame guidelines for including Indian knowledge in higher education curricula. I happen to be its Chairman, and Ramasubramanian and many other eminent scholars are members of this Committee. While these are only some preliminary steps, it is possible that over time there may be some progress towards introducing teaching and research in various disciplines of Indian knowledge in the portals of our institutions of higher learning.
Anyone who is reasonably acquainted with the development of mathematics in India would know that a large part of mathematics we teach in schools either originated in India, or was developed further here. Sadly, our students cannot speak even a few meaningful sentences about the works of giants like Āryabhaṭa, Brahmagupta, Bhāskara or Mādhava. So, the first task that needs to be addressed is that of content planning and content generation, which will lead to preparation of books on Indian mathematics that can be read by our school students at different levels.27 In the same way, it is essential that scholars with expertise in other disciplines of Indian knowledge devote a part of their time and energy towards preparing books that are meant for study by our school students.
Mathematics as we know it now, along with the real number line is unthinkable without the idea of negative numbers. When did these numbers begin to appear in Indian works?
MDS: Negative quantities are seen in Arthaśāstra itself, while the arithmetic of negative numbers was formulated by Brahmagupta. These ideas were developed further in Bhāskara’s works. For instance, while discussing the intercepts (abādha) of the base of a triangle by the altitude drawn from the vertex in Līlāvatī, Bhāskara shows that an intercept will be negative when the foot of the altitude falls outside the base.
Even though Brahmagupta’s works were partially translated into Arabic, in the medieval Islamic and European traditions, negative numbers appeared only occasionally; they were in general abhorred. Al-Khwarizmi’s al-jabr w’al-muqabala does not even contain the notion of negative numbers. It was common to consider six different types of quadratic equations, with the coefficients being restricted to be positive on both sides. It is revealing to learn that the product of minus one and minus one, is still minus one, in some texts of this period. In essence, it seems that what the Indians knew then to be correct, and considered perfectly normal as regards negative numbers, was not known or acceptable to the rest of the world till the seventeenth century.
What is the very first explicit Indian source that clearly shows that the Indian masters were adept at handling irrational numbers. For instance, is there any literature available that suggests a knowledge of continued fractions in ancient or medieval Indian mathematics.
MDS: Approximations to square-roots of a non-square number (called karaṇi) and for \pi are known from the time of Śulbasūtras. The fact that \pi is irrational, and that it cannot be represented as the ratio of two integers was stated by Nīlakaṇṭha, in a couple of places. The reason cited is the appearance of square roots in the computation of \pi preventing the computation of its exact value.
The idea of continued fractions associated with a rational number, and the basic properties of the convergents of such continued fractions are very much implicit in the Kuṭṭaka algorithm of Āryabhaṭa. Further, the fact that these convergents provide an optimal approximation to the originally chosen rational number, is also implicit in the works of ancient Indian mathematicians and astronomers. Non-terminating continued fractions do arise in relation to the cakravāla algorithm. The cakravāla algorithm was reinterpreted by Krishnaswamy Ayyangar in 1929, as being associated with a particular kind of semi-regular continued fraction representation of a quadratic surd.
Mādhava gives three successive correction terms for the end-correction term for the \pi / 4 series; each of which is successively more accurate than the one before. From the derivation of these end-correction terms, as explained in Yuktibhāṣā, we can see that the successive correction-terms are actually convergents of a continued fraction.28 Using this, one can even express \pi as a continued fraction. William Brouncker’s continued fraction comes to mind here; although Mādhava’s continued fraction turns out to be quite different.
MSS: In astronomy, continued fractions appear as reasonable approximations to quantities of physical importance in planetary motion, as seen in the works of Bhāskara, and Nīlakaṇṭha and in later Kerala works.
Among the crowning achievements of Indian mathematics is a very accurate formulation of the trigonometric sine function; a function which in itself is both a conceptual, and a computational mainstay for astronomical calculations. Please elaborate on its original Indian name, its acceptance and use, and on the principal Indian scholars who contributed to its long history, culminating in the Kerala School.
MDS: There’s something interesting about the way the sine function seems to arise in Indian astronomy. A good reference text from where we can see origins of this function is the ancient Vasiṣṭha-Siddhānta which has been summarised in the Pañca-Siddhāntikā. It is clear that this version of Vasiṣṭha-Siddhānta dates much prior to Āryabhaṭa’s own time, 499 CE. Physically speaking, the sine or cosine functions are used to represent the acceleration or velocity of a body. The Vasiṣṭha-Siddhānta of Pañca-Siddhāntikā, discusses the velocity of the Moon in terms of what modern historians of astronomy call a linear zig-zag function, which is a periodic wedge function, highlighting constant rise and fall of a chosen quantity. This wedge function occurs in Babylonian mathematics also, but in a numerical manner. But in the Indian text, it is presented as a function which alternately consists of an increasing and a decreasing arithmetic progression. This periodic zig-zag function is a first approximation to the sine or cosine functions. Physically, it describes a body moving at constant acceleration, whereas the sine function arises when the acceleration of the body is proportional to its distance from a reference point. This is the generalisation from the Vasiṣṭha-Siddhānta which the Indians seem to have achieved at least by the time of Āryabhaṭa.
So, Indians were not obtaining the sine function by looking at Ptolemy’s chord, and bisecting it. On the other hand, its Indian origin perhaps lay in the fact, noticed first by Āryabhaṭa, that second order sine-differences are proportional to the function itself. Āryabhaṭa not only defines the sine function as a function of the arc and gives the formula for the second order sine-differences, he also seems to have been aware of the rational approximation of the sine function given in the works of Bhāskara I, which is accurate up to one percent.
Thus, by the time Āryabhaṭa used the sine function and gave us a table of its values, Indians had already understood many subtle aspects of the sine function. It is strange to hear claims that Indians received the idea of the sine function from the Greeks, especially when they had discovered it by their own methods, and which Europe took note of, only a good 1100 years later. Datta and Singh quote a French historian of astronomy Jean Baptiste Joseph Delambre, who in turn states that in Europe, the second order difference formula for the sine function became known only by the time of Henry Briggs, sometime in the late 16th century.
MSS: Regarding the story behind the sine function getting its name, its original name was `jyā’. And it had another name too, called `jīvā’. When the Arabs started using it, they pronounced it as `jaib’, which also means a pocket, or a fold. When this word `jaib’ reached Europe, the Europeans introduced the Latin equivalent of a pocket, which is a `sinus’. This `sinus’ is now the `sine’ function. This is how the `jyā’ or `jīvā’ from India became the `sine’ function known to all of us. The story does not end there. The `jyā’ had a close cousin, its complement, which the Indians used to refer to as the `koṭijyā’. This `ko’ from `koṭijyā’ is now prefixed to the `sine’, to create the well known `cosine’ function.
KR: The way the sine function is dealt with, varies significantly from the time of Āryabhaṭa to the works of Nityānanda. For instance, Nityānanda has considered the computation of sines to be a very important topic, and has devoted more than 60 verses to explaining the rationale behind its computation in his Sarvasiddhāntarāja.29 Its importance is echoed in the works of Bhāskara II too. In fact, Bhāskara has about 30 verses on jyotpatti. To elucidate this point, Indian astronomers have come up with a few beautiful similes which highlight how they saw this function playing a central role in this subject. For instance, Munīśvara says – just as the human body has various nerves and veins passing through its own self, so too does one find `sines’ in astronomical computations, cutting across and passing through its entire being. Metaphorically, he equates the pervasiveness of the sine function in astronomy to the ubiquitous nerves and veins of the physical body. This is one of the beautiful similes to effectively drive the point in. Earlier, Bhāskara had in his own style invoked the idea of an Indian `dhoti’. Just as the threads of a dhoti run back and forth while it is being woven, thus making up the fabric of the dhoti, the rich tapestry of astronomical calculations is also formed by these sine functions, which similarly run back and forth throughout the calculations. In fact, Nityānanda even says that if one does not know how to compute sines starting from first principles, then such a person cannot be considered a mathematician – ācāryavaryā gaṇakāsta eva. This is one aspect. Another aspect that I want to speak about is with regard to the varieties of sine tables that have been used since those days. Āryabhaṭa gives values of the sine function for angles differing by every 3 degrees and 45 minutes. This means there are 24 values of sines in the table presented by Āryabhaṭa, whereas later astronomers thought that similar tabular values may be needed for every change in degree, in which case there will be 90 entries. Thus, based on the interval chosen, there is one kind of variation seen. Another kind of variation that we find in sine tables available in Indian astronomical texts has to do with the choice made by the astronomer in representing numerals. Certain tables are available, as given by Mādhava, in the Kaṭapayādi notation, whereas others are given in the Bhūtasaṅkhyā system.
Munīśvara equates the pervasiveness of the sine function in astronomy to the ubiquitous nerves and veins of the physical body.
Some other beautiful literary inventions associated with the sine function and its values may also be mentioned here. Mādhava uses every single quarter of a verse to express one sine value; and therefore we have 24 sine values, each correct to the third sexagesimal place, condensed in just six verses. Āryabhaṭa by listing out only the first order sine differences, cleverly manages to present the entire sine table in one single verse. There is yet another striking difference that I wanted to point out, and this concerns the choices exercised by various authors in choosing the maximum value sinus totus or the radius. In Pañca-Siddhāntikā, the maximum is taken as 120. Āryabhaṭa takes the maximum value of the radius to be 3438. Others use 90, as well.
MSS: It is important to note that the ideas of spherical trigonometry were also known and extensively used for calculations. In Āryabhaṭīya, which has only 121 verses, and without much fuss, Āryabhaṭa uses ideas from spherical trigonometry to calculate the duration of the day-time, which depends upon the latitude of a location, and Sun’s declination (its distance from the celestial equator).
MDS: There is a baseless claim in some circles of modern scholarship, especially from the past 40–50 years, that Indians borrowed their trigonometry from the Greeks. There was some such talk earlier too, at the time of Colebrooke. But seeing the sophisticated works on trigonometry, especially the detailed sine tables from the time of Āryabhaṭa onwards, such claims had been subdued for a while. But it seems to have resurfaced.
To understand this, let’s first see what the Greeks themselves had done. It is well known that till the time of Hipparchus, the Greeks had no trigonometry. It is said that Hipparchus prepared a table of chords, but the only table of chords that has come down from the Greeks is due to Ptolemy. Ptolemy assumes the radius to have a value of 60 “parts”, and then poses the question that if one is given an arc of such a circle, how much will be the corresponding chord? Ptolemy’s calculation is quite good; he has the formula for the chord of sum of two arcs, and also the equivalent of the half angle formula for the chords. In this context, Ptolemy also comes up with what is well known as Ptolemy’s theorem, which is that the sum of the product of opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. Further, he calculates the value of the chord for every half degree to fairly good accuracy.
The key point to note is that all of this has nothing to do with the way the sine function is dealt with in the Aryabhatan tradition. In the Aryabhatan tradition, the circumference is given the value of 360 degrees or 21,600 minutes. Indians did not use angles at all, and instead only used arcs of a circle, or essentially the radian measure. So from the famous approximate relation between circumference and the radius given by Āryabhaṭa, the radius turns out to be nearly 3437 minutes and 44 seconds. This is approximated to 3438 minutes. With a better value of \pi, Mādhava comes up with the value of the radius to be 3437 minutes, 44 seconds and 48 thirds which is stated in kaṭapayādi form as Devo viśvasthalī bhṛguḥ. Incidentally, the value 3438 was used by al-Khwarizmi in the ninth century when he was translating Indian astronomical methods to Arabic.
For those who did not want to credit Indians with any originality for their knowledge of trigonometry, the following question became important: “How does one cleverly make it appear that the Indians acquired the knowledge of the sine function from the Greeks?” So, what was done was to launch an effort to find the number 3438 mentioned in some ancient Greek text, which if found, would give the Greeks primacy in the discovery of trigonometry.
Theoretically speaking, a logical basis for the appearance of the number 3438 as a value for the radius is not available in Greek thought at all. Calculations involving the idea of the radian measure became possible in Europe only from medieval times onwards. For the Greeks, there was always a separate measure for the radius, and a separate measure for angles. As we noted, Ptolemy divided the radius into 60 “parts” and calculated all the chords in the units of these “parts”. In these units, we no longer have the relation that the sine of a small angle (in the radian measure) is approximated by the angle itself. Further, it must be noted that philosophically the arc-length was not a quantity considered amenable to precise measurement in Greek thought.
In Ptolemy’s “Almagest”, there is a calculation in the chapter on lunar motion where the number 3423 occurs. This number 3423 appears when Ptolemy is referring to a related work of Hipparchus. Based on this, a way was found to show that Hipparchus must have used the value 3438 for the radius. Further, with some back calculations, a chord table was constructed based on the radius value of 3438, and that was attributed to Hipparchus; and then, lo and behold, we have a proof that Āryabhaṭa had used this mythical Hipparchan chord table, and converted it into a sine table. All this is in accordance with one of David Pingree’s most profound dictums on Indian astronomy: “The preservation of this Greek material is one of India’s greatest contributions to the history of science”.30
MSS: Indian epicycle models also involve the sine function. Further, there is a formula for longitude, a formula for the duration of the day, a formula for the relation between the length of the shadow and the time of its measurement, and so on. On the other hand, all that one has in Greek works are tables after tables. There are also no such explicitly mentioned formulae, as far as I know.
It’s heartening to note that historically, Indian kings and royalty have been supportive of scholars. In your collective view, who have been the most significant and benevolent patrons of mathematical studies down the ages?
MDS: It is said that Āryabhaṭa was a kulapati, who in today’s terminology was possibly a university vice-chancellor, maybe of Nalanda. He lived in Kusumapura, which is said to be the same as Pāṭalīputra, or present-day Patna. He lived and worked in the Gupta period, a Golden Age.
The most well-known grant awarded to promote the study of astronomy is the grant given to the grandson of Bhāskarācārya. Ramasubramanian has even gone to the village, by the name of Chaalisgaon which is in modern-day Maharashtra, where a related stone inscription can be seen. The name of the generous king who made the grant in the 13th century is Caṅgadeva. He even urged people residing in that area to encourage and support such scholars. Records of these events are available in a detailed inscription, which has several verses. One of the interesting things in that inscription is that Bhāskara’s scholarship is praised in superlative terms. Such acts supporting and nurturing scholarship must have been dependent also on the prosperity of the kingdom.
There were two kinds of education in ancient India. One was the basic elementary education which children received, and that was supported at the village level by contributions made to the teacher from the produce of the village, by the village community itself. We have detailed records of these happening in the 18th century. Arithmetic, commercial mathematics and elementary geometry were some of the main components of elementary education the children were receiving. The other component was reading, writing and recitation.
For higher education in specialised disciplines, students from one part of the country even travelled to famous centres such as Varanasi or Navadvipa to study with famous gurus, as a part of their Gurukula. But everywhere in the country, the local community and the local nobility nurtured and supported scholars who were pursuing their studies in that area. If the times were good, grants and support would be generous and the expenses borne by the teacher would be taken care of by the community. If the times were bad, and the teacher didn’t have this kind of a grant, the students would still be fed by the households of the village where the teacher lived. In essence, education was supported by the village community, in both good and bad times.
KR: Extraordinary scholars would even be honoured by kings with impressive titles such as the Āsthāna Vidvān. In addition, places of religious worship, the Maṭhas, themselves had a strong and unbroken tradition dating back to many centuries, of promoting study and scholarship at various levels.
MDS: From the 18th century records, it is seen that many villages had their own `pañcāṅgam brāhmin’ who computed the `pañcāṅgam’. This scholar would typically employ a standard `karaṇa text’ relevant to his period. He would sometimes even improvise upon it, while simultaneously being engaged in perhaps some astronomical observations too. All of this disappeared after 1820 or so, during British rule. In later times, most scholars would only copy earlier calculations, and the tradition of observational astronomy got thoroughly disrupted in the course of a few decades.
Today, India stands at the cusp of many potentially exciting future outcomes. But a peek into an event that unfolded nearly two centuries ago, reveals Thomas Babington Macaulay advocating a shift away from native and traditional Indian education, and imposing English medium education. How important is it to find a balance now, as we simultaneously seek to find inspiration from within our ancient heritage, even as we imbibe the best practices and scholarly traditions of the current era?
MSS: Of course, best practices from other civilisations have to be incorporated here, while ensuring that our traditions are not neglected. The key here is balance, and I am sure this balance can be achieved in India.
KR: I will say just a couple of things based on my own experience teaching bright engineering students who come to IIT Bombay. When I teach them the intricacies of the sine function for instance, they get excited because they now have a greater feel for what is actually happening at a physical and a mathematical level. When they realise that all this was done right here in India, nearly a millennium and a half ago, one can feel a sense of awe and excitement in them. I believe that we have to find more and more examples that we can present to our youngsters that convince them of the power and the depth of knowledge that ancient Indians had explored. One does not usually have a feel for what exactly is the connection between \pi, sine function, and the circle. Laying it all bare on the board and bringing it alive with examples from astronomy makes their understanding real and concrete.
Another example that I like very much is the discrete version of the second-order harmonic equation, which we may call Āryabhaṭa’s equation. Scholars have shown that merely by a careful study of similar triangles that get formed, when we geometrically represent the arcs and chords, one can arrive at a profound result such as Āryabhaṭa’s equation. Students find this quite exciting when they learn such completely novel and elegant approaches found in our scientific literature composed several centuries ago, and that too presented in beautiful metrical composition. Finally, based on my own experience, I can say with reasonable confidence that modern day gadgetry has completely stripped us of our ability to perform quick mental computations, which were fairly commonplace before these gadgets arrived. I think rediscovering this lost skill may be really useful.
MDS: Since this is one of the core issues that we have sought to address all these years, let us put it in its proper historical context.
Today, we study everything that has been done in the last three centuries in Europe, without learning anything about what happened anywhere else in the world, either before or during that time. So, right now there is no balance, and things are fully skewed; and sadly, our curriculum is totally silent on even the Indian contributions. How did this happen? How did a society which had a rich history in science, mathematics, arts, philosophy etc., arrive at this smṛti-vibhrama?
Our tragic decline began during the early decades of the 19th century. The older or the indigenous education system, along with its social support base was starved of all resources. Whatever help was extended by powerful and benevolent kings was going away, and whatever our village communities were contributing by way of monetary or grain allocations and donations, was simply taken away under British rule. The British also implanted here a new system of education which was designed to introduce modern European knowledge and denigrate all Indian knowledge. And, further, this was all to be learnt in English. Over time, the public space available to people who were steeped in our own traditional sciences and knowledge systems, became smaller and smaller. When the rulers also mandated that knowledge of English and other subjects taught in the imposed scheme was a prerequisite for all jobs, it was so to say, the final assault as far as our own indigenous knowledge was concerned.
What typically happens after a great defeat and foreign subjugation of a civilisation, is not just disorientation and impoverishment at the level of individuals; it is that an entire culture, along with its people, gets enslaved. We too got brainwashed that all our ancient knowledge is of little value, and that only new knowledge from outside can lead us out of the desperate situation that we had found ourselves in. Incidentally, this is exactly how Rajaram Mohan Roy seems to have felt when he wrote to the British Governor-General Lord Amherst in 1823, beseeching that they should stop patronising Sanskrit and Indian knowledge, and teach us modern European knowledge, and that too in English.
Around the time when India became independent, it was found that 18 percent of the population was literate in Indian languages (as per census of 1951), while only one percent was literate in English (as per census of 1931). This was roughly the situation in 1947. Later, we expanded our elementary and higher education. Today, most of our children attend schools and a fair percentage of them study in colleges.
Historically, when we see the Greeks and their adoption of mathematical and astronomical knowledge from the Babylonians, we also notice that after borrowing ideas, they developed those very ideas further, but gave them an appropriately Greek interpretation. So, adoption was followed by transformation and adaptation. This is incidentally a pattern seen in most other ancient cultures too. In a similar vein, the Arabs did the same. When a flourishing civilisation emerged with Baghdad at its centre, the Arabs liberally borrowed knowledge of astronomy and mathematics from India, even translating various Indian books too. They took ideas also from the Greeks, and found that Greek philosophy as modified by later Christian thought was closer to their own philosophy and religion. So, Arab thought evolved more in the Greek mould, but with suitable modification and native adaptation. Again, when modern Europe embarked upon the creative era of new discoveries, theoretical formulations and technological innovations, they seem to have borrowed a great deal from the Islamic, Chinese and perhaps Indian precedents, but transformed them appropriately to suit their own quests, preferences and worldview so that these now became an integral part of the newly emerging modern science and technology of the West. This civilisational behavioural pattern is seen repeated in far too many instances for a keen observer to miss it.
The famous Japanese scholar of history of mathematics, Takao Hayashi, who did his PhD with David Pingree at Brown University, summarises this sentiment beautifully, but from an essentially Japanese context. And though the context is essentially Japanese, the wisdom emanating from the Japanese viewpoint is universally relevant. According to Hayashi:
Japanese have been looking to the West ever since the middle of the Edo period [1603–1868]. This not only holds true with the Western culture in general, but in particular in the fields of science and technology. Certainly the discipline of modern science originated in the seventeenth century in Western countries. Before that, however, perspectives of nature, as well as approaches to it, differed considerably according to place, nationality and time. This fact suggests that the modern-scientific view of, and approach to, nature is neither unique nor absolutely correct, and that there are alternatives as to the direction modern science should take. We hope that the study of the history of sciences in India, China, and Korea, which have all had a great influence upon the Japanese culture including the indigenous science, will make us consider the past, present, and future of our own culture [and] science and enhance our understanding…
Returning to our own Indian context, and learning from other people’s experiences, it is imperative that we have to seriously study and comprehend our own civilisational knowledge in various domains such as sciences and technologies, arts and philosophies etc., and update all of them after carefully examining and analysing what is known in the present day, and through this process, try and adapt whatever is necessary and appropriate from the modern, to our own civilisational knowledge. We shall then be in a position to embark on our own quest for new knowledge as a resurgent Indian civilisation which can, as an equal, freely borrow from, and contribute to all other contemporary civilisations and cultures.
On a concluding note, are there any caveats that you would like to sound regarding any of the ongoing works in this direction?
MSS: To associate every modern discovery to an idea that was supposedly already discovered in the ancient past, is an obsession seen in some quarters. I think this should go. Not everything found today has an echo from the past. We have to recognise this. Anyone indulging in this unfair and unwanted comparison is committing a disservice to both the modern and the ancient.
There’s some serious work being done in Indian astronomy today, but the quantum is far too less, considering the size of this country. Another vastly overlooked aspect of Indian Knowledge Systems is the area of popular literature. There is a need for well-researched, fact-based, aesthetically designed books meant for the interested, lay public. This is a hugely underserved area, and its absence is both glaring and conspicuous. Think of a potential book dealing with the works of Āryabhaṭa, Bhāskara, Brahmagupta and Mādhava, and which compares with the wider and larger outreach effects of Robert Kanigel’s The Man Who Knew Infinity.
As we draw to a close, we note the according of a distinguished honour to Ramasubramanian, being chosen as an invited speaker at the upcoming International Congress of Mathematicians (ICM). This is the first time a scholar from India has been invited to present their work in an ICM in the section on “History of Mathematics”. It is a rare distinction, and something that makes all of us very happy and proud. Please accept our hearty congratulations on this honour Ramasubramanian, even as you prepare to address the global mathematics community at the ICM, in July 2022.
KR: Thank you for the kind words. It was indeed a pleasant surprise for me to receive the invitation from the organisers of ICM to deliver an invited talk at ICM 2022. This event which was supposed to be held at St. Petersburg has now been moved to Europe due to various developments in the political scenario. I may also mention here that it is for the first time in the history of the International Mathematical Union (IMU) that the Fields Medal Award Ceremony is going to be held in the General Assembly of IMU instead of the inaugural ceremony of ICM. As a nominee for one of the positions in the Executive Committee of the IMU, I have been invited to attend the general assembly that is going to be held in Helsinki on 3–4 July 2022.
Regarding the recognition, I must say that it is an honour for the work done by us together as a team. I have benefitted immensely from my association with Srinivas and Sriram. I personally perceive this invitation as a recognition of the work done by the three of us over the last three decades. In a way, I owe it to all my gurus in different domains of knowledge, all of whom remain a constant source of high inspiration for me.
On that celebratory note, we thank you for your time, effort and patience. We also gratefully acknowledge the eagerness and the warmth with which all three of you have participated in this interaction, sharing your insights, freely and enthusiastically.\blacksquare
Footnotes
- Lilavati of Sri Bhaskaracharya, translated into Kannada by K.S. Nagarajan, S.S.V.M. Publishers, Bengaluru, 2nd ed., 1986. ↩
- The Institute of Mathematical Sciences. ↩
- A Statement on Scientific Temper, Mainstream, 25 July 1981, pp 06–10. ↩
- Dharampal, Indian Science and Technology in the Eighteenth century, Some Contemporary European Accounts, Dharampal Classics Series, Rashtrotthana Sahitya, Bengaluru 2021 (originally published by Impex India, New Delhi 1971). ↩
- M.D. Srinivas, Measurements and Quantum Probabilities, Universities Press, Hyderabad, 2001. ↩
- M.D. Srinivas, `The Untapped Wealth of Manuscripts on Indian Astronomy and Mathematics’, Ind. Jour. Hist. Sc., 54, 2019, pp.243–268. ↩
- M.D. Srinivas, `The Methodology of Indian Mathematics and its Contemporary Relevance’, PPST Bulletin, 12, 1987, pp.1–35. ↩
- The dictionary meaning for upapatti is: ascertained or demonstrated conclusion, proof, evidence, argument. ↩
- K. Elfering, The Area of a Triangle and the Volume of a Pyramid as well as the Area of a Circle and the Volume of a Sphere in the Mathematics of Aryabhata I, Ind. Jour. Hist. Sc.,12, 1976, pp.232–236. ↩
- M.D. Srinivas, Proofs in Indian mathematics, in G. Emch et al (Eds.), Contributions to the History of Indian Mathematics, Hindustan Book Agency, Delhi, 2005, pp. 209–248. ↩
- K. Ramasubramanian, M.D. Srinivas and M.S. Sriram, Modification of the Early Planetary Theory by Kerala Astronomers (c.1500 AD) and the Implied Heliocentric Picture of Planetary Motion, Current Science, 66, 1994, pp.784–790. ↩
- Karaṇa is a Sanskrit technical term used in ancient Indian sciences such as astronomy, mathematics and geometry, to mean a handbook for astronomical computation or astronomical manual. ↩
- K.V. Sarma and S. Hariharan, “Yuktibhāṣā of Jyeṣṭhadeva, A Book of Rationales in Indian Mathematics and Astronomy: An Appraisal”, Ind. Jour. Hist. Sc., 26, 1991, pp.185–207. ↩
- K. Ramasubramanian and M.S. Sriram, Corrections to the Terrestrial Latitudes in the Tantrasaṅgraha, Ind. Jour. Hist. Sc., 38, 2003, 129–144. ↩
- A monastic tradition, referring to a place for an initiate to stay in for an extended period for the purposes of study, with a guru offering teaching. ↩
- K. Ramasubramanian, Takao Hayashi and Clemency Montelle (Editors), Bhāskara-Prabhā, Hindustan Book Agency, Delhi, 2019. ↩
- Suśrutasaṃhitā, edited and translated by P.V. Sarma, Vol. I, Chaukhambha Viswabharati, Varanasi 1999, p.380. ↩
- Vākyapadīyam of Bhartṛhari with Puṇyarāja Commentary, edited by Raghunatha Sharma, Part II, Sampurnanand Sanskrit University, Varanasi 1980, pp.78–81. ↩
- For an overview of the epistemology of Indian sciences, see M.D. Srinivas, The methodology of Indian sciences as expounded in the disciplines of Nyāya, Vyākaraṇa, Gaṇita and Jyotiṣa, in K. Gopinath and Shailaja D. Sharma (eds.), The Computation Meme: Explorations in Indic Computational Thinking, Indian Institute of Science, Bengaluru, 2022 (in press). ↩
- D. Mumford, “Review of Mathematics in India”, Notices of AMS, 57 (2010), pp.385–390. ↩
- See for instance, Āryabhaṭīya-bhāṣya of Nīlakaṇṭha, Part I Gaṇitapāda, edited by K. Sambasiva Sastri, Government Press, Trivandrum 1930, p. 3, and Siddhānta-darpaṇa of Nīlakaṇṭha with auto-commentary, edited and translated by K.V. Sarma, VVRI, Hoshiarpur 1976, pp. 20–24. ↩
- See for instance R.N. Iyengar, Concept of Śruti, Svara and Rāga of Classical Music in Sanskrit Texts, Ind. Jour. Hist. Sc., 53, 2018, pp.131–147. ↩
- For details see Raja Sridharan, R. Sridharan and M.D. Srinivas, “Combinatorial methods in Indian music: Pratyayas in Saṅgīta-Ratnākara of Śārṅgadeva”, in C.S. Seshadri (ed.), Studies in the History of Indian Mathematics, Hindustan Publishing Agency, Delhi, 2010, pp.55–112. ↩
- Jolly John, “Derivation of the Saṃskāras applied to the Mādhava Series in Yuktibhāṣā”, in M.S. Sriram, K. Ramasubramanian and M.D. Srinivas, eds., 500 Years of Tantrasaṅgraha. ↩
- G. Rajarajeswari and M.S. Sriram, Mahājyānayanaprakāraḥ: Infinite series for Sine and Cosine Functions in the Kerala Works, In History and Development of Mathematics in India, Eds. Sita Sundar Ram and Ramakalyani, V., National Mission for Manuscripts and DK Printworlds, New Delhi, 2022, pp. 293–306. ↩
- Kapil Kapoor and Michel Danino, Eds., Knowledge Traditions and Practices of India, 2 Volumes, CBSE (2012). ↩
- A recent effort in this direction is the book, V. Ramakalyani, M.V. Mohan, R.S. Venkatakrishnan, N. Karthik and M.D. Srinivas, Bhāratīya-Gaṇita-Praveśa I: Introduction to Indian Mathematics for Students of Classes VII – VIII, Samskrit Promotion Foundation, New Delhi, 2021. ↩
- See for instance the discussion in, K. Ramasubramanian and M.D. Srinivas, Development of Calculus in India, in C.S. Seshadri (ed.), Studies in History of Indian Mathematics, Hindustan Book Agency, Delhi, 2010, p.266. ↩
- See for instance the discussion in, Clemency Montelle and K. Ramasubramanian, “Determining the Sine of One Degree in the Sarvasiddhāntarāja of Nityānanda”, SCIAMVS 19 (2018), 1–52. ↩
- D. Pingree, “Indian Astronomy”, Proc. Amer. Phil. Soc. 122 (1978), pp.361–4. ↩