SEMINAR ON CURRENT RESEARCH IN INDIAN MATHEMATICS AND ASTRONOMY

Back Ground

In 1991, a major effort had been initiated to investigate the Foundations of Indian Theoretical Sciences'. At the initiative of the PPST Foundation, this has been taken up as a multi centered effort comprising of sixteen projects in which there are eleven participating institutions from all over India. As part of this effort, projects on various aspects of traditional Indian Mathematics and Astronomy are being carried out in different institutions in Madras. The titles of the projects: (with the names of the coordinators and institutions in parentheses) are:

1. Proofs in Indian Mathematics and Astronomy (K.V Sharma and M.D.5rinivas, PPST Foundation, Madras).
   
   
2. An analysis of the accuracy and optimality of the algorithms of Indian Astronomy (M.S.Sriram, Madras University).
   
   
3. Study of optimality of Indian Numerical Algorithms (CN.Krishnan, M.I.T., Anna University and Ashok Jhunjhuhwala, LIT. Madras) and
   
   
4. Design and testing of hardware for optimal algorithms with suggestions for applications (to be taken up later by Ashok Jhunjhunwala, LIT. Madras).
   

The projects have been in progress for about a year now and are of three years duration. From July 1991, there have been seminars/colloquia at fairly regular intervals on topics of current interest in mathematics and astronomy. The idea of holding a national conference involving persons actively engaged in research in traditional Indian mathematics and astronomy grew out of these discussion-group meetings.

Organization

The Seminar was organized by the PPST Foundation in collaboration with the School of Instrumentation and Electronics) Anna University, on March 6 and 7 and held at the MIT Campus of Anna University. It was funded by the Department of Science and Technology. Apart from lectures by the project personnel in Madras, there were lectures by other, researchers from institutions in Madras as well as outside. Here a special mention must be made of Sri A.B.Managoli, an 80 year old ex-school teacher from Karnataka who made it a point to attend the Seminar despite his advanced age and delicate health. He has worked on the 'Chakravak method' all on his own for many years. The audience consisted of teachers, scientists, research scholars and post graduate students. The total number of participants was over sixty.

Inaugural Function

In the inaugural function, Dr.CV.Seshadri, President, PPST Foundation and Director, Murugappa Chettiar Research Centre, Madras gave the background of the PPST Foundation and its activities and spoke about its relevance. Delivering the inaugural address, Dr.M.Anandakrishnan Vice Chancellor, Anna University emphasized the importance of comprehending our traditions in Science and Technology and taking them forward. (He said that we. would be rootless without them. There is need for good quality work in this area and he lauded the efforts of the PPST Foundation in this direction. He suggested that the Ramanujam endowment in Anna University can be utilized to invite scholars of high standing in traditional sciences.

Before the formal sessions, Sri Navjothi Singh, National Coordinator of the project gave an introductory lecture on the DST-funded research program on Foundations and Methodology of Theoretical Sciences in Indian Tradition'. Until now, most of the work in this area had been confined to cataloguing and glorifying (or denigrating) the achievements of India in various disciplines. Not enough attention has been paid to the methodology of the theoretical sciences in our tradition or to their foundations which are quite distinct from those of the Western tradition. Navya nyaya, Vyakarna and Ganita traditions have possible applications in natural language processing, numerical methods and algorithms. Their revitalization is necessary for the generation of new contemporary research problems and insights.

Mathematics

The first lecture in the Mathematics section was by Dr.M.D.Srinivas, who gave an overview of Indian mathematics. He traced the history of the development of Mathematics in India and pointed out how the problems that were dealt with arose in practical contexts. For instance, the Kuttaka algorithm for solving linear indeterminate equations is useful in solving problems in astronomy. There are different ways of expressing a number but the place value notation and the use of zero developed by Indians is the best method for most practical calculations. Various short cut methods in computations based on the place value system were devised by Indians and widely used by the common people like vegetable vendors, carpenters etc. 'Vedic Mathematics' by Jagadguru Sri Bharathikrishna Thirtha, a book which has become popular recently, makes use of the place value system in simplifying various kinds of manipulations like multiplication, division, squaring, extraction of square root etc. He felt that the possibilities of the place value system are still not exhausted and more research needs to be done in this direction. He introduced the problem of quadratic indeterminate equations (Varga Prakrithi) which were handled by Indian Mathematicians from the time of Brahmagupta (7th century A.D.) culminating in the 'Chakravaht method perfected by Bhaskaran. The Europeans could handle this sophisticated problem only in the eighteenth century.

Some of the papers presented in the mathematics session were pertaining to the 'ChakravaW ethod. Other topics covered were Aryabhata algorithm {Kuttaka), division algorithm based on Urdhya-Tiryak Sutra', algorithm for extraction of higher roots and analysis in Indian mathematics. The list of papers and a brief summary of each is given below:

1. 'A new variation in the Aryabhatta algorithm for finding modular inverses' was presented by B.R.Shankar, S.Jegannathan and Ashok Jhunjhunwala, of the Department of Electrical Engineering, LIT. Madras. A unique feature of the new algorithm is that it facilitates parallel computation.
   
   
2. 'Computer implementation of a simple division algorithm based on Urdhva-Tiiyak Sutra' was presented by Ashok Jhunjhunwala, Ranjani Parthasarathy and SJegannathan, of the Department of Electrical Engineering, LIT. Madras. In this algorithm, division involving a divisor of arbitrary size is performed in a simple manner using a small-number multiplier and divider. The implementation of this algorithm on a PC-AT shows an improvement in the time taken for computation by a factor of ten over the conventional method.
   
   
3. 'A generalization of the root extraction algorithm of Jagadguru Bharatikrishna Thirthaji' was presented by K.Ramasubramanian, M.D.Srinivas and M-S.Sriram of the Department of Theoretical Physics, University of Madras. It delineated a procedure for the calculation of cube roots, which is close in spirit to the Dwandwa Yoga of 'Vedic Mathematics'. The method can be generalized to find the nth root of a number.
   
   
4. 'A paper on the Optimality of the Chakravak algorithm for the solution of
   or- Dif = 1' was presented by M.D.Srinivas, Department of Theoretical Physics,
   University of Madras. Preliminary numerical estimates on the optimality properties of the Chakravak algorithm in relation to the Euler-Lagrange algorithm were presented. The estimates indicate that the Chakravak algorithm on the average needs only 69.4% of the number of steps in the Euler-La grange algorithm.
   
   
5. T.S.Bhanumurthy, of the University of Madras, presented a paper - 'On the Brah-magupta-Bhaskara equation Dxr-T= ± l.The Chakravak method may be described as continued fractions with reference to quadratic irrationalities. It ex¬ploits the direct use of the multipliers in preference to the coefficients of the expansion of root D and the proofs get considerably simplified. It does not need as prerequisite even the first lesson on continued fractions.
   
   
6. A.B.Managoli, Belgaum (Ex-School teacher, Kamataka) presented a paper - 'Chak- ravala method'. Consider the equation N ;r+ M~ y . The modern method for the solution of the equation requires quadratic expansion of root N. But the Chak- ravala method does not require expansion of root N. The old method has a vast range of applications and is not lacking in any way compared to the modern method.
   
   
7. M.S.Rangachari of Ramanujam Institute for Advanced Studies in Mathematics, University of Madras, presented a paper on Analysis in Indian Mathematics. The seeds of analysis were sown in India several centuries before Newton and Leib-nitz, in the works of Bhaskara II. Many texts of the medieval period have not been thoroughly studied e.g. Sadratnamala which contains traces of the bisection method, differentiation using power series and improvement of sine and cosine values.
   

Astronomy

On the second day of the Seminar, the proceedings began with an introductory overview of Indian Astronomy by Dr.M.S.Sriram. The history of Indian Astronomy from the Vedic times to the modern period was briefly outlined. There was a systematic lunisolar calendar in India even during the 'Vedanga Jyotisha' period (probably 1200 B.C.). Detailed mathematical treatment of astronomical problems begins with the Siddhanta texts, the earliest available of which is Aryabhatiya (499 A.D.) though according to traditional accounts there were Siddhanta texts even prior to that period. Some of the outstanding texts were Surya Siddhanta, Brahmasphuta Siddhanta, Khandakhadyaka, Laghumanasa, Siddhanta Shiromani etc. The essential features of the Siddhanta texts were outlined. In particular, the procedure for the calculation of the true longitudes of the sun, the moon and the planets was discussed. One calculates the mean longitude first using the sidereal period of the planet and the Ahargana and applies two corrections, namely the Manda correction (which takes into account the eccentricity of the orbit),and the Shighra correction (equivalent to conversion from the helio-centric system to the geo-centric system). The continuity in Indian astronomical tradition from the Vedic period to modern times was pointed out.

In the discussion that followed Dr.A.K Bag emphasized the necessity of making a comparative study of astronomy in India and other civilizations (Greek, Babylonian etc). Dr.C.V.Seshadri was struck by the concept of a Maha Yuga of 43,20,000 years (something peculiar to India) and felt that one should have a better understanding of the concept of time in .Indian tradition. The papers that followed dealt with a variety of topics such as - Mean and true longitudes in Indian astronomy, Spherical trigonometry, a novel approximation for the sine function due to Bhaskara, an outline of Ganita Yukthibhasha and algorithms in traditional Indian mathematics. The papers were:

1. 'Computations of mean and true positions of planets in Indian Astronomy' by S.Balachandra Rao, Department of Mathematics, National College, Bangalore. The Methodology of finding the positions of planets as discussed in the Madhyamaadhikara and Spashtadhikara sections of various Siddhanta texts was considered. The significance of the manda-phala and sigra phala in obtaining the true positions was discussed. A comparison of retrograde motion of planets as in the Siddhantic texts with modem results was made.
   
   
2. George Abraham, formerly of the Department of Mathematics, Madras Christian College, Madras, presented a paper titled 'Billiard's astronomies Indien'. Published in 1971, Roger Billiard's book sheds new light on Indian astronomy. In this paper an attempt has been made to assess the accuracy of mean motions calculated by some-leading Indian astronomers.
   
   
3. V.S.Narasimhan, of Vivekanda College, Tiruvedakam presented a paper titled 'Tantra Sangraha of Neelakanta Somayaji (AD 1444-1545)'. Tantra Sangraha gives simple trigonometric formulae and procedures for finding the site of an angle. It gives the equivalent of many modern results in astronomy. For any celestial body five coordinates are defined and the method of finding two of them given the other three is discussed. This is a novel result.
   
   
4. CN.Krishnan, A.Muthulakshmi and V.Srividya, of the School of Instrumentation and Electronics, M.I.T, Anna University, Madras presented a paper titled 'A-study of the rational approximation for the sine function due to Bhaskara I'. The rational approximation to sin theta due to Bhaskara I was examined from the point of view of accuracy and computational speed. Where small word lengths are used or high computational speeds are required, the Bhaskara approximation may be a useful alternative to the series summation technique.
   
   
5. K.V.Sharma, of PPST Foundation and Adyar Library and Research Centre, Madras, gave an outline of Ganita-Yukti Bhasha. There are several commentaries and full-fledged texts in Sanskrit which give the rationale of the various formulae and procedures used in Indian mathematics and astronomy. One such text is the Ganita Yuktibhaha ofjyeshta Deva of Kerala (A.D. 1500 - 1610). In this paper an outline of this highly important work was given.
   
   
6. A.K.Bag of Indian National Science Academy, New Delhi presented a paper on
   Algorithms in traditional Indian mathematics. He said that it was very important to investigate systematic 'steps' or 'phases' in the methods and their logical sequence in the development of specific results or discoveries. This will be useful for comparison of a method with methods in other cultures, for retrieval of results and estimation of the efficiency and limitation of a discovery.
   
   

Panel discussion

After the paper presentations, there was a panel discussion on the contemporary relevance of research in Indian theoretical sciences'. The panelists were: Dr.Rajiv Sangal, HT, Kanpur, Prof.M.Seetharaman, Madras University, Prof.Ramakrishnan, M.I.T. Madras, Prof.V.BJohri, IIT Madras and Dr.AXBag, INSA, New Delhi who chaired the session. Dr.Rajiv Sangal was positive in his opinion that an in-depth research in various theoretical disciplines is very essential and fruitful. He cited the example of Paninian grammar being used in machine translation and possible application in artificial intelligence. Dr.Seetharaman felt that a good result here and there in the traditional disciplines would be no match to the enormous modern knowledge system. There was also the problem of convincing youngsters of the desirability of studying Indian methods. Prof.Johri felt that the lectures in the two day seminar were revealing to him in many ways and made a strong plea for I highlighting Indian achievements in various areas to, our children. Prof.Ramakrishnan agreed with him and made out a case for including traditional Indian Science and Scientists in the text books. Dr.A.K.Bae stressed the need for vigorous research in the area of traditional science, done with a historical perspective. There are a lot of commentaries on important texts which are yet to be studied thoroughly.' He strongly felt that comparative studies of traditional sciences like mathematics and astronomy in different civilizations should be taken up. There was a general consensus that there should be institutions and scholars exclusively devoted to the study of traditional sciences and that our heritage in science and technology should be disseminated among the younger generation.

Author: M.S.Sriram

Note: The proceedings of the seminar are being published as a book. Details can be obtained from Dr.M.S.Sriram.