Indian Mathematics, An Introduction by Ashok Jhunjhunwala
(Wiley Eastern, New Delhi) 1993, pp.108 + vi, Price Rs.40/.

We are now in an era where Mathematics is supposed to be applied in unexpected fields of activity such as Economics, Biology and the like. If one looks at the history of Mathematics in the past, whether it is in the East or West, Mathematics has been developed on the basis of needs. A time has come when Mathematicians who consider their field 'sophisticated' justify their work as important because of later applications, be it theorizing, in other sciences. There is no doubt that India had a rich cultural background even several thousand years back as evidenced by the oldest extant literature, viz. the Vedas. With no literature of a comparable age in the rest of the world, one can safely say that there existed no civilization worth the name elsewhere. The Vedas do contain quantities and so Mathematics. Numbers as large as ten raised to sixteen power and as small as zero (in its emptiness as well as infinitesimal form) occur in the Vedas. This is no surprise since any civilization with so high an ethos and an incomparable wisdom cannot bypass basic Mathematical ideas. Even religions or philosophical literature in the ancient languages of India indicate traces of these ideas. There has been, in fact, a tradition in Science (Physical, Medical etc.) and Mathematics (in their modern connotation) in India, which has been embellished by dedicated scholars (perhaps to be called scientists in the current terminology) from time to time. Though some of their ideas have been documented in the form of works in Sanskrit, Malayalam and other Indian languages and have been studied by some, it is very likely that a good number of ideas have not been documented at all, or have been documented and has not reached posterity or is gathering dust in some private or public archives after documentation. After the invasion of the country by other races and its colonization by Europeans, there has been a setback in our tradition. Though one cannot be sure whether this was due to a policy of annihilation of the tradition by our rulers in the past three or four, centuries in a systematic way or because of a complex of inferiority over the ideas of these rulers which possessed our ancestors of the near past, the living generations of Indians have mostly not been aware of what existed in their country before the advent of these rulers. More specifically, they have believed that whatever they learn in their life as Science or Mathematics is completely the contribution of the Western hemisphere. Quite a few stilt believe that their ancestors did not have a scientific outlook. This is so in spite of exposition of hitherto unknown Indian sources time and again by an Indian scholar or? Foreign scholar. The dependence on western books of Science and History has become part and parcel of Indian education up to the present moment, so that even a mature Mathematician unhesitatingly believes a Vander Warden when he writes that the, word Kendra in Sanskrit is a deviation from some Greek term. It is quite heartening that a group of young scientists of the country have taken on themselves to tap the traditional sources, documented or otherwise, wherever they are preserved and, let the world at large know that Indians could have lived and developed their country in the best of Traditions without borrowed knowledge or Technology whether it is in Metallurgy or Medicine or Mathematics.

The present booklet is to be considered against the above backdrop. It concerns the basic operations of multiplication and division in the system of natural numbers 1,2,3….. otherwise known as Arithmetic in chapters 2 to 6 and square root extraction and divisibility in the last two chapters nunnbered-7,8 respectively. Chapter 1 is an introduction to the spirit of the booklet through a detailed discussion of a Carpenter's method of obtaining the area of a rectangular piece of wood and a method known as Navsesh of checking mistakes in addition and multiplication, in the accounts of a family in business. These instances are recorded to indicate a tradition which was in vogue before Westernization of Education in India and later in spite of it and, perhaps, now lost with the advent of the calculator and the computer. The author points out the likelihood of the techniques and algorithms described in Bharathi Krishna Thirtha's 'Vedic Mathematics' being widely used in India eighty or a hundred years ago. The rest of the book is devoted to a lucid exposition of the use of the "Nikhilam navatascarmam dasatah" and "Urdhva'tiryagbhyam" sutras propounded by the Thirtha. As regards squaring, the author illustrates in Chapter 4 how Dwandva yoga helps to simplify the Urdhva triyak method, in the context of the divisibility tests, it is the application of the sutra " Ekadhikena purvena" coupled by the upasutra vestanam" which is illustrated in Chapter 8.' The booklet has an appendage of notes and explanations, indicating the basis of some of the techniques exposed in the text, and an Index at the end. Each chapter goes with a set of exercises so that the booklet could be used in class.

Though all the ideas explained in the booklet are found in Krishna Thirtha's 'Vedic Mathematics', the motivation and presentation in the booklet are quite impressive. I have seen RanjaniSchari's book intended to popularize Vedic Mathematics and T.S.Bhanu Murthy's book- a part of which is devoted to justify some of the algorithms of 'Vedic Mathematics'. But I am convinced that Jhunjhunwala's booklet will popularize these algorithms much better. His provoking mathematicians, computer scientists and users of mathematics to get involved with them are timely. I hope he would supplement "the present booklet with another booklet relating to the machine implementation of the traditional techniques and their machine - time analysis to convince the 'modern' students as well.

The booklet has been very nicely brought out by Wiley Eastern but Jhunjhunwala's goal would be far-away if it is priced so high as Rs.40 per copy. The publishers would do well to get subsidy from National Book Trust or the like and publish the booklet with free translations in all languages of India but priced at not more than Rs.10 per copy.

I could notice only a few misprints, p. 13, k3 : 'arithemetic' for 'arithmetic'; p.64, 1.6 :
'repeatition' for 'repetition' ; p.72, 1.10 : omit 'a' at the end ; p.89, I.2 'Vetsana' for 'Vestana' also later; 1.1 : 'a' for 'the' ; p.54, 1.13 : 'SX'.for ',Sx' ; 1.8 : 'simplying' for 'simplifying' 'X' for 'x' : I have the following remarks to make: Navsesh means " Navatheetha sesham" and cannot be 9. It can be only one of 0,1,2, 8. If the term has been used it is a misnomer. 0 will not therefore be divisible by 9. In fact, the author remarks in one place that 9 is equivalent to 0 in the context of Navsesh. In view of the author's own remark earlier that Navsesh check cannot prove the answer to be correct the statement 'Navsesh is used for verification' ( p.28, I.4 ) is misleading. Instead of saying 'checking up' or 'verify' it is perhaps better to say that 'to see that the answer may be correct'. The comparison with 'commutation' on p.39 is not convincing and is not, any way, comprehensible to children to whom the booklet is addressed. In p.42, I.3 the author perhaps means 'carefully' when he writes 'competently'. In p.90, I.9 the author needs to write '4578 is divisible by.7 since 497 is divisible by 7' not the other way. The working at the top of p.92 is not easily comprehensible without relevant explanation. A list of references for further reading would help the mature reader.

I end up congratulating Professor Ashok Jhunjhunwala for his nice booklet and his service to uphold to originality of our ancestors.

Author:Dr.M.S.Rangachari, Professor,
The Ramanujam Institute, University of Madras.

" Pancavati - Indian Approach to Environment" by Banwari
(English Edition, translated by Asha Vohra)

Shri.Vinayaka Publications, B 17, Prashant Vihar, New Delhi - 110 085,185 pages. (1992)
ISBN: 81-85810-00-1. Price: Rs.200.00.

Our ancestors in ancient India led a lifestyle that merged inseparably with their natural environment. As a vast landmass, the natural environment of India varied from region to region and correspondingly the ethos and cultures of our people too varied. It unified culture and heritage. Because they identified themselves with Nature and never considered the human being as a more superior organism than other living beings, Nature ruled high providing adequately for everyone.1 Economic self-sufficiency was there offering scope for reasonable and modest growth. This economic "development" was, in principle, guided by the norms: selflessness and contentment. This unique status of India and her people attracted foreigners to establish trade and business contacts with India. With the passage of time, their influence systematically established the stable culture and ethos of our forefathers. Traders turned invaders and usurpers. They cleverly exploited the diversity among the people and implanted differences in opinions and actions. The rich cultural identity of unity in diversity was lost gradually. Transformation of mind, and consequently that of attitudes took place. We lost trust and faith in our hoary past and alien thoughts began to invade our minds. We began to change our attitudes changed our beliefs changed in short, every aspect of our life changed including our basic philosophy of selflessness and contentment.

Today, we are a bunch of confused people. Our confusion originates from misplaced priorities. We used to understand 'development' as a process of human existence that enabled the growth of the self and others, utilizing the available natural resources prudently through the practice of human values. But, currently, we measure 'development' of fiscal growth. Money has replaced values. This shift in attitudes has shattered peace, love, and coexistence Selfishness and greed. Maladies from mistrust between individuals to large-scale violence and terrorism have increased I am not referring to these as a matter of regret, but as a matter of utter shame.

In this undesirable 'development' context of modem India, Banwari's "Pancavati" appears relevant. Through several chapters he surveys the thoughts, beliefs, and practices that characterized India over the centuries. He analyzes the deep-seated philosophy and the theories behind each of such beliefs and practices. To a large extent, I am glad that he has succeeded in bringing out the finer aspects of the lifestyle of our ancestors who lived in harmony with Nature. He describes avidly the dimensions of lifestyle that were experimented and practiced by early Indians Banwari's statement in the preface creation is nothing but a manifestation of the Supreme Being. The Supreme Being and the world therefore are one and the same. There is an intrinsic bond between nature around us and the divinity. Since we have always visualized ourselves as integral part of the divine, and the divine has some thing which partakes in us, we have always 'maintained a harmonious relation with nature", more or less summarizes the entire book. Each chapter attempts to highlight this viewpoint variously, using diverse examples. Interestingly, I found every statement in the book to be apparently simple; in fact, each of it was pregnant with a deep philosophy and needed repeated reading to get at the roots of the concept. I am not unhappy about this nominal strain as a reader because this enabled me to see the relevance of the theory behind the practice of a glorious lifestyle of our ancestors, which is being labeled and rejected as superstitious by us.

Banwari drives home his interpretations through eighteen chapters. (Eighteen was auspicious to early Indians (Jaya). Eighteen book chapters, was it by design?) Each chapter is captivatingly titled with terms that echo the rich culture and traditions of ancient India: (1) Prakrti (Approach to Nature), (2) Mangaiaya (Universal Well Being), (3) Kalpavrksa (The Celestial Tree), (4) Vanasri (The Bounty of Nature), (5) Pancavati (The Grove), (6) Mathura (The City of Affluence), (7) Brajamandala (The Land of Sri Krsna), (8) Gramasri (Prosperous Village), (9) Soma (The God of Bliss), (10) Aranya (The Land of No War) (11) Asrama {Stages of Life), (12) Asvattha (The Cosmic Tree), (13) Khandavavana (The legend of Blazing Forest), (14) Upasana (The Divine Nature), (15) Utsava (Festivity Amidst Nature), (16) Uttarakuru (The Region of Affluence), (17) Upavanavinoda (An Old Text of Plant Science), (18) Vrksa Ayurveda (Therapeutic Care of Tree). The organization of chapters and captions are logical and impressive bringing out the substance of the book efficiently to anyone interested in the examination of our cultures and traditions, and relating them to our environment and its management.

But what bothered me was the tone of treatment of chapters: Some have been dealt with extreme care providing substantial number of illustrations and examples, while a few others are treated rather shallowly. An incisive and uniform treatment will certainly enable general readers to internalize the concepts better.

On the whole, I was happy to read this book. The translator has done a good job; I make this comment solely on the clarity that is made available in the book. A few language errors (Examples: page xvi, line 1 (our); page xvii, line 14 (decade-s?); page 5, line 17 (indicreetly of indiscriminately); page' 6, line 14 (colonizing or utilizing?); page 53, title line (Mathura - a city of effluence or affluence?); page 100, line 9 (questions put up to or questions put to…?) are glaring. Similarly, scientific concepts have been badly interpreted in some places. For example, in page 98, paragraph 2 is scientifically unacceptable. Moreover, the fourth paragraph contradicts the second. 1 wished Banwari had consulted a professional ecologist / environmentalist who shares his (Banwari's) philosophy of life and finalized the chapters. At certain points, miniatures of medieval India are reproduced in an effort to supplement the concepts. Unfortunately, the reproductions are bad and non-communicative. Banwari could have provided explanatory notes to these figures relating them with the points discussed in the text. Definitely that will help the reader to see the connections better.

Barring these few shortcomings, I am happy that this book represents a clear departure from our present thinking of environment and environmental management. I am personally convinced that our country has vast potential to play a lead role in teaching the World how to manage and conserve the fast deteriorating environment. People with the conviction that India's traditions are Nature-biased-will find this handy book very exciting and rewarding. A casual reader will find the explanations and examples a little difficult to digest, since Banwari has not effectively related the issues to perceivable and existing environmental problems probably he has taken it for granted that every modem Indian is aware of the underlying philosophy of India's heritage and culture and is proud of it.

I will recommend this book to libraries and to motivated readers with some commitment and background in Indian history, mythology, philosophy, and culture. I will consider even a thought to write such a book call for a lot of boldness and courage. Banwari has done it. It is a great job.

Author:Anantanarayanan Raman,
Loyola College, Madras.

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