BOOK  REVIEW

500 years of Tantrasangraha and the outer planets. This was especially – A Landmark in the History significant for the calculation of latitudes of of Astronomy the planets. The struggle of the astronomers of the yesteryears and the ingenuity of people like Ptolemy, Copernicus and Kepler become B S Shylaja evident in this context. Nilakantha Somayaji of Kerala belongs to this category. Although 500 years of Tantrasangraha – A Landmark in the History of it was known that the Kerala group of Astronomy M S Sriram, K astronomers were quite active in the 16th and Ramasubramanian and M D 17th centuries, we were deprived of a detailed SrinivasSrinivas, 2002 study of their work till the last few decades. It Inter-University Center, Indian Institute of Advanced Study, Shimla. is a remarkable achievement that a model which appears as a transition from the All students of astronomy undergo a rigorous geocentric to the heliocentric version, was training in spherical trigonometry, which arrived at, in those years. It is unfortunate forms the basis of the fundamental compu- that such an important development was tational formulae and techniques of positional completely lost and had to be worked out all astronomy. However, one will still be left over again by Nicolas Copernicus. with many questions regarding the physical The monumental work of Nilakantha called universe leafing through the pages of the ‘Tantrasangraha’ was published in the year history of science. The turning point in our 1500. Its five hundredth anniversary was understanding of the heavens has been traced celebrated in Chennai during March 2000, by to the Copernican Revolution that took place the Indian Institute of Advanced Study, in the sixteenth century. From the knowledge Shimla. The occasion was very aptly used to procured in the undergraduate levels, one publicise the details of Nilakantha’s model, would understand the implications of as well as to enlighten the commoners about replacing the Earth with the Sun at the centre the life and achievements of the person. of the solar system. Only a deeper insight will reveal the difference between Ptolemy’s The book under review gives a comprehensive (geocentric) model and the Copernican (helio- outlook on the works of Nilakantha Somayaji. centric) model. It is a compilation of the papers presented at the conference. Prof. K V Sarma, well known It is well known that even prior to the for his contribution to the history of Indian Copernican era astronomers were able to astronomy and mathematics has described determine the positions of the planets and the life and works of Nilakantha Somayaji in predict the eclipses quite precisely. One had great detail. It has been possible to fix the to use different sets of formulae for the inner period, during which Nilakantha lived, quite

66 RESONANCE  March 2003 BOOK  REVIEW precisely as 1444-1545. His works include Although the first half reads like a chapter Golasara Siddhantha Darpana (two volumes), from a text book of spherical astronomy its Candrachaayaganitha and Vyakhya, Tantra- presence is justified. As one proceeds to sangraha, Aryabhatiyabhashya, Sundararaja Nilakantha’s methods, one needs to turn back Prasnoththara, Graha Parikshakrama and the pages constantly to decipher the various Jyotirmimamsa. K V Sarma has scrutinized technical terms. The words in italics for all the manuscripts and his painstaking efforts example, Seegrochcha, Manda Sanskara, etc. have shown that a Sanskrit work by name sound strange to us, exposed as we are to only Yukthibhashya is actually a translation from the modern (western) mathematical terms. Malayalam to Sanskrit. (He even points out Further, the formulae are written down as that the translation was done by a non-tech- verses. Here is an example (p. 76-77): “The nical person!) Prof. Sarma has described his sine of the difference in the longitudes of the difficulties he faced in tracing the originals nodes and the mandasphutagraha of Mars, because of the non-availibility of cross- etc. multiplied by the inclination of their references and the incompleteness of the orbits and divided by the iteratively obtained manuscripts. hypotenuse, gives the latitude of the planet at the desired instant”. This transforms into a The traditional method of teaching prevalent formula in India during the days of Nilakantha suffered \begin{equation} a setback like it happened in all other fields, \beta=frac{iRsin(\lambda_{m}-\lambda_ with the advent of colonisation and the even- {n})}/{karna} tual emergence of the British Empire. One of \end{equation} the many adverse effects was the loss of this heritage of astronomical calculation. Over This example demonstrates that precise the last couple of centuries, the tradition had transliteration is very essential for a good been forgotten. Therefore, to understand the comprehension. This paper, therefore, technique of Nilakantha, we constantly need provides the vital link to a forgotten past and to refer to the methods we learn today. This enables us to appreciate the mathematical happens to every student who is keen on skill of Nilakantha. understanding the traditional techniques. In this context the book comes as a handy tool to The next paper by K Ramasubramanian review the methods in the language of the continues the trend with a parallel explanation day. of the original text in modern language. The figures, which are perhaps absent totally in The paper by Prof. M S Sriram presents a the original, are drawn quite appropriately to comparative study of Nilakantha’s methods match the explanation. The need for the deciphered in modern text book language. model is very clearly explained. It appears

RESONANCE  March 2003 67 BOOK  REVIEW that this model fills the link needed between The heritage and tradition of these the Tychonic model and the Copernican mathematical skills were lost gradually and model. It may be recalled here that a complete the last member of this school was Samantha heliocentric model was created (in Europe) Chandrashekhara (1835-1904). Prof. Satpathy almost 100 years after the Copernican describes his works and achievements as an revolution. observational astronomer.

The use of the model to describe the motion The other papers in the volume include of the interior and the outer planets mathematical treatises of Nilakanta (Drs. V constitutes the next paper by Prof. M D Madhukar Mallaya and J K John) on the Srinivas. The difference in the procedure is summation of series. The astronomical tables explained. In fact it is the same technique of Indian origin have been traced in Iran by used today to predict the geocentric Farid Ghasemlon and Negar Nadei of Iran. coordinates of a newly discovered comet or Prof. S M R Ansari describes the influence of asteroid or any other solar system body. Islamic Astronomy. The book provides a very valuable guide to The other papers in the volume are of interest those who are not content with a superficial to astronomers who are serious about knowledge of our heritage in mathematical calculations. The determination of longitudes skills. One may pull out a pencil and paper of planets is discussed by Prof. S Balachandra and verify the formulae. Thus they do not Rao (with Padmaja Venugopal and S K Uma), remain as ‘black boxes’ any more. choosing the Moon as an example. The need for the determination of constants in the While going through the book, I had to equations of motion by observations has been compile my own cross-reference list for the emphasized as suggested by Nilakantha technical terms in Sanskrit. A glossary to this himself. effect and an index would have added to the value of the book. The attempt to introduce the elliptical orbits has been described by Dr. S Madhavan. This The occasion of the conference provided an is a new revelation since all along it was opportunity to honour Prof. K V Sarma, believed that the calculations were done with whose contributions to this field are very well epicycles. The associated difficulties with known. The citation presented to him is the use of elliptical orbits are explained and given at the end of the book. The language suggested as a possible reason for abandoning used is very simple and I hope there would be the technique. At this juncture, one naturally a wide readership to go through it and enjoy wonders, how things would have changed the flavour of a lost tradition. only if a communication existed with the Europeans! B S Shylaja, BASE, JN Planetarium, Bangalore

68 RESONANCE  March 2003