Bharatiya Trikonamiti Sastra (Indian Trigonometry)

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Bharatiya Trikonamiti Sastra - Indian Trigonometry was developed as a powerful mathematical tool for Siddhantic Astronomy.

The book, Bharatiya Trikonamiti Sastra deals with all the relevant topics of indian trigonometry, including trignometrical identities and other formulas, trigonmetrical tables, methods of interpolation and trigonometrical series, etc. The subject matter is discussed in 11 chapters divided into 80 sections and 50 sub-sections, involving translations of 250 verses spread over in 38 classical Sanskrit works and based on 42 research articles published in 16 research journals.

By including material of the late Aryabhata School or Kerala Aryabhata School, the author has made the book comprehensive and up-to-date. indeed, the book is fascinating abd signicificant. It is a definite contribution in the study of the history of indian mathematics.

The book is the English version of the Kannada book of the Same title.

Venugopal D Heroor, an Engineer by profession, is a keen and enthusiatic scholar of Ancient Indian Mathematics. He has brought pout 12 books related to indian mathematics, including Development of Combinatorics from the Pratyayas in Sanskrrti Prosody and translattion of the Sanskrit works: Trisatika pr Patiganitasara by sridharacarya, jyotpatti by Bhaskaracharya, and Ganitadhyaya of Brahmagupta's Brahma-sphuta-siddhanta into both Kannda and English.

He has translated Narayana Pandits's Ganita Kaumudi, Sridharacharya's patiganita, bhaskaracharya's Lilavati, Sripati's Ganitatilakam, Citrabhanu's Ekavimsati Prasnoattara Bhakhshali manuscript, into Kannada, and Bhaskaracarya's Bijaganitam into both English and Kannada, which are yet to be published.

He has translated many articles and research papers, and has also contributed original articles. He has conducted classes for teachers and research scholars, presented papers at various universities, and in various national and international seminars.

FOREWORD A grand contribution of India to world mathematics is the introduction of the predecessor of the modern trigonometric function called "sine" and the development of a system of the trigonometrical science based on it. In due course, the new Indian trigonometry (based on sine) drove away the Greek trigonometry (based on chords) just as the Indian decimal place-value system of numerals tiumphed over the other system of numerals.

The world history of the Victorious spread of the Indian sine (/vd or jiva) is quite interesting, although it is full of misadventures of translation due to lack of proper understanding of the terms used. In fact, the English technical word "sine" itself was evolved from the original Sanskrit term jiva (denoting Indian-sine) via the deformed Arabic transliteration (in 8th century CE) as j-y-b for jib which was further read wrongly as jaib or jeb (4, pocket or cavity) by the Latins in the 12th century Spain. Latins, therefore, translated jeb as sinus (cavity) whence came the English term sine! Indian trigonometry was developed as a powerful mathematical tool for siddhantic astronomy and the very first pauruseya (i.e., historically-dated-type) work Aryabhatiya (499 CE) has a sine table.

Currently we are celebrating the 900th birth anniversary of the great Bhaskaracarya (or Bhaskara M1). He deserves full credit for giving us a standard set of works on ancient Indian mathematical sciences namely Lilavati (on arithmetic and geometry), Bijaganita (on algebra), Siddhanta-Siromani (on astronomy) including the Jyotpatti which is a tract on trigonometry, etc.

Engineer V D Heroor is a very keen, enthusiastic and devoted scholar of Indian mathematics. He has already benefitted students, teachers and other scholars by his edition and translation of the Jyotpatti.

His book Bharatiya (Hindu) Ti rikonamiti-Sastra (in Kannada) is bound to prove very useful to the readers of various categories. It is a definite contribution in the study of history of Indian mathematics. It deals with all the relevant topics of Indian trigonometry, including trigonometrical identities and other formulas, trigonometrical tables, methods of interpolation and trigonometrical series, etc. By including material of the late Aryabhata School, he has made the book comprehensive and up-to-date. Indeed, the book is fascinating and significant.

Obviously, there is a need to bring out the English version of the book for wider spread and appreciation of the achievements of the Indian (Hindu) Science of trigonometry. The English version of the book would also make-up for the deficiency or gap caused by the non-publication of R C Gupta’s doctoral thesis on Trigonometry in Ancient and Medieval India. .

Despite India’s rich heritage of mathematics, the public in general and student community in particular are not fully aware of Indian contribution to the development of the subject. Though studies and researches have been carried out in this field, and a lot of material is accumulated in the past century, it is contained in rare books and journals which are hardly available. There is thus, a great need of appealing popular books which provide authentic material on "History and development of Indian Mathematics" in a chronological order.

I took to writing first in the local language Kannada, and then in English, with the purpose of explaining the development of mathematics as a science, focusing on great mathematicians in India in the past four millenia. Earlier in 2004, I had written a book in Kannada Bharatiya (Hindu) Trikonamiti Sastra, adopting the scheme followed in the Hindu Trigonometry by Bibhutibhusan Datta and Avadhesh Narayan Singh, revised by Kripa Shankar Shukla, and further elaborating it by incorporating the matter culled from the articles of R C Gupta, and collating the Sanskrit verses with the readings found in the critical editions of the orginal works. Prof R C Gupta has written foreword for that book in July 2014, which is in press now. In his foreword, he has highlighted the fact that "there is a need to bring out the English version of the book for wide spread and appreciation of the achievements of the Indian (Hindu) science of trigonometry. The English version of the book would also make-up tor the deficiencey or gap caused by non-publication of R C Gupta’s doctoral thesis on "Trigonometry in Ancient and Medieval India." Accordingly. the English version is prepared.

In the present work Bharatiya Trikonamiti Sastra (Hindu Trigonometry), the subject matter is discussed in eleven chapters.

At the outset, after explaining the concept of the jyd, on which the Hindu-Tngonometry is based, its significance, scope and the source works are given in the first chapter. In the second, after defining the three fundamental functions jv (R sin 8), kojya (R cos 8), utkramajya (versed Rsin 8), and the relations between them, variation in the value of the functions, and rules indicating change in the sign of them depending on the quadrant in which the argument lies is discussed in detail. Next, in the third chapter, Tngonometric Identities are stated in the form of rules, and rationales of some of them are also given.

Different forms of the Addition and Subtraction Theorems for jya (Rsin), kojya ‘Rcos functions are detailed in the fourth chapter along with their proofs. As corollaries to these theorems, functions of arcs involving multiple angles are derived in the fifth. Next, converse to the functions of arc involving multiple angles, rules for the sanctions of arcs of sub-multiple angles are given in the sixth chapter. Rules stated to obtain the values of functions of the five arcs containing the particular angles: 30°, 60°, 45°, 36° and 18° (pajica jydka), and also the rules involving formula for the calculation of the approximate values of jya (Rsin 8), kojya (R cos ) functions of an arc without the help of Rsine-table are dealt with in detail in the seventh.

In the eighth chapter, various methods for the construction of Rsine-table is deliberated in detail. Next, in the ninth, rules involving the formulas for finding the tabular Rsine- differences in their different forms based on the property that "the second order Rsine- differences are proportional to the Rsines themselves" given in the Kerala Aryabhatiya School is discussed in detail.

Methods of interpolation involving first-difference and second differences for finding the trigonometrical functions of an arc, other than whose values have been tabulated at equal or unequal intervals are detailed in the tenth chapter. While in the last, Series Expansion of Rsine, Rcosine and tan~* x and also several infinite series to represent 7m are discussed in detail.

As the Spherical Trigonometry found in the Siddhantic works is concerned with Hindu- astronomy, and requires its elementary Knowledge, is not discussed here. In compiling this work, I have been indebted to and relied on the expository source works by great savants like B B Datta and A N Singh, K S Shukla, K V Sarma, R C Gupta, T A Sarasvati Amma, D A Somayaji, A K Bag and C N Srinivas Iengar. Articles of R C Gupta have been the main source of inspiration for me. I express my sincere gratitude to all those stalwarts in the field.

Detailed Index and Bibliography are included in the book to enhance the usefulness of the book.

I am greatful to R C Gupta for gracing this book with his valuable foreword.

I express my profound gratitude to Manipal Universal Press for taking up the task of publishing the work.

The author hopes that this book satisfies a long-felt need at least partially.

Book's Contents and Sample Pages