ब्रह्मगुप्तगणितम्: Brahmagupta's Ganita (Ganitadhyaya of Brahmasphuta Siddhanta)

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Sri Venugopal D. Heroor, an Engineer by profession, is a keen and enthusiastic scholar of Ancient Indian Mathematics. He has brought out eleven books related to Indian mathematics, which includes translation of the Sanskrit works Sridhara’s Trisatika or Patiganita Sara, Bhaskaracarya Jyotpatti, and Ganitadhyaya of Brahmagupta’s brahmasphuta-siddhanta into both Kannada and English. He has translated Narayana Pandita’s Ganita Kaumudi, Sridhara’s Patiganita, Baksali Manuscript, Bhaskaracarya’s Lilavati and Citrabhanu’s Eka Vimsati Prasnottara into Kannada which are yet to be published. He has conducted classes for teachers and research scholars; presented papers at various university’s, National and International Seminars.

The Present work; Brahmagupta’s Ganita contains the Sanskrit text of Ganitadhyaya of Brahmasphutasiddhanta with introduction, English translation and notes along with illustrative examples of prthudaka Svami and others. Rational for the geometric theorems and rules are provided with relevant figures, Parallel rules and examples found in other available work on Hindu Mathematics have been indicated and complete solution of the examples are given in modern notation and symbols for the benefit of the students.

Brahmagupta (598-670 CE) was one of the most popular mathematicians of ancient India. Prof. Sachau says, 'Brahmagupta occupies an important place in the history of oriental culture. Brahmagupta taught astronomy to the Arabs before they came to know of Ptolemy's works, Sindhind and Al-Arkand frequently occur in Arabic literature; these are the translations of Brahmagupta' s works. Brahmasphutasiddhanta and Khandakhadyaka, the two of his well- known works.

Brahmagupta's understanding of the number systems was indeed remarkable. He introduced the concept of zero as the result of subtracting a number from itself. He gave arithmetical rules in terms of fortunes and debts for positive and negative numbers respectively. But it is misfortune that he failed in the case of explaining division of number by Zero. The algorithm he used for multiplication, exploiting the place value system. is almost the same as we use it now. His algorithms to compute square root, to solve quadratic equations, indeterminate equations, etc . are highly commendable. He gave remarkable formulas for the area of cyclic quadrilaterals and for the lengths of their diagonals in terms of their sides.

He gave the formula for ∑n2 and ∑n3 of course without proofs. Historians of mathematics hail his solution to the second order indeterminate equation of the form of Nx:+ 1 =y2. This equation is attributed to one John Pell (1611-1685 CE) and is famous as PeIl' s equation in the present day mathematical community. But the truth is that it is Brahmagupta' s equation. Later in the 12th century Bhaskaracarya (1150 CE) refined and perfected the algorithm by his Cakravala method. Thus the equation should rightly be called Brahmagupta-Bhaskara equation. Brahmagupta gave the second- difference interpolation formula, 'a thousand years before the rediscovery of the popular Newton -Stirling formula' .

Brahmagupta's ingenious method of using continued fractions to find integral solution of indeterminate equation like ax +c=by commands our admiration. Prthudakasvami (830-890 CE) is best known for his work on solving equations. He wrote an important commentary on Brahmasphutesiddhanta, which contains many examples to illustrate the theoretical results of Brahmagupta. 'A considerable. difference of opinion prevails whether these examples are those of Brahmagupta or Prthudakasvami' .

Prthudakasvami's commentary also throws light to know how 'Algebra', that is the method of calculating with the unknown was developed in India. Prthudakasvami called his method of calculating with unknown quantities as the kutteka method.

Shri Venugopal D. Heroor, Charted Engineer, Gulbarga, is a serious and passionate researcher in the field of Ancient Indian Mathematics. I had known him since a long time and I read a few of his books in this field. They are the History of Methemetics and Mathematicians of India (2006) and Bhaskaracarya's Jyotpatti (2007), translation of the original Sanskrit text into English and Hindi, with necessary explanation and notes Ganitsbharati (2008) - A Quiz book on Mathematics and Mathematicians of India. I also read the book, Development of Combinstorics form the Pratyayas in Sanskrit prosody (2011).

Shri Heroor goes to various Universities and institutions of Research, on invitation to deliver lectures on Ancient Indian Mathematics. He received number awards in recognition of his unique contribution to popularize Ancient Indian Mathematics. I have read with abiding interests his recent work (in the press) on Sridharacaryas Trisatika.

With missionary zeal Shri Venugopal has given us now Brahmagupta's Ganits. It is an authentic translation of Ganitadhyaya part of Brahmasphutasiddhants, with notes and illustrative examples of Prthudakasvami, The Sanskrit text adopted is the one edited by Ram Swarup Sharma (1966 CE) and collated with the text, edited earlier (1902 CE) by Sudhakar Dwivedi.

Shri Venugopal has done yeoman service to the mathematical community in presenting a part of Brahmagupta' s Ganits in the most faithful and commendable manner. To narrate briefly its salient features, we should start with its valuable introduction which contains an account of various commentaries on Brahmagupta. Then we go through some definitions, (Rasivyavahara) and Shadow of a Gnomon (Chayavyavahara)

These are followed by sections titled special things, which deal with various methods of multiplication, division, squaring of numbers in sexagesimal parts and place value system, based on an identity. Finally the author concludes with an Appendix and Bibliography. The Author's efforts in making an effective presentation of the theme without injuring the original are marvelous and highly appreciable.

Some interesting and useful features which attracted my attention are, the modern notation, anecdotes, footnotes, clarity and brevity in presenting the mathematical arguments. One can perceive mastery of the subject matter and craftsmanship on the part of Shri Heroor. I am delighted in going through the pages on the book and I feel privileged to write these few lines as foreword to this phenomenal work of Shri Venugopal D. Heroor.l recommend that every teacher/ student, of mathematics should study such works as this, which inspire us and make us feel delighted and proud of our national scientific heritage treasures.

The Present work contains the Sanskrit text of Ganitadhyaya of Brahmasphutasiddhanta with introduction', English translation and notes along with Illustrative examples of Prthudakasvami and others. The Sanskrit text adopted is as published in 1966 AD edited by Ram Swarup Sharma with its Hindi translation and also collated with the text edited by Sudhakar Dvivedi published in 1902 AD.

The whole text of Ganitadhyaya is rendered in English. Technical terms which have their English equivalents have been translated into English; others have been kept as they are and have been explained. The portions of the English translation enclosed within brackets do not occur in the text and have been given in the translation to make it understandable, and are at places explanatory.

The translation to each rule is preceded by a sentence or two giving in brief the contents of the rule and is followed. where necessary, by relevant notes and comments. Parallel rules and example found in other available works on Hindu mathematics have been indicated in the foot notes. Headings and sub-headings have been provided to facilitate consultation. Complete solution of the examples are given here for the benefit of the students.

In compiling this work, I have been indebted to and relied on the expository source works by great savants like Sudhakar Dvivedi. B. Datta and A.N. Smgh, K. S. Shukla. Bina Chaterji. T. A. Saraswati mma, R. C. Gupta and Takao Hayashi. Works of Datta Singh have been the main source of inspiration for me in this work. Broadly speaking. mathematical matter round in the source works, and major highlight of historic matter found in the articles and research papers. have been collated and presented in this book. I express my sincere gratitude to all those stalwarts in the field.

I am grateful to Prof. A. V. Arunachalam, Former Vice-chancellor for gracing this book with his valuable foreword.

I wish to record my deep appreciation to Smt. Karuna Suresh Bhat for excellent DTP work of the text.

I express my profound gratitude to authorities of Chinmaya International Foundation Shodha Sansthan for taking up the task of publishing the work.

The author hopes that this book will be quite useful and interesting.

Brahmagupta

Brahmagupta the most celebrated mathematician belonging to the school of Ujjain was born in 598 A.D. According to his own statement: 'In the reign of Vyaghramukha, a great king of capa dynasty, when 550 years of Saka era had elapsed, Brahmagupta, son of Jisnu at the age of thirty (30), composed Brahmasphutasiddhanta for the pleasure of good mathematicians and astronomers.

Date and works

Thus Brahmagupta was 30 years old in Saka 550 or AD 628 when he wrote Brahmasphutasiddhanta (Br. Sp. Si) his masterpiece. He wrote the next work Khandakhadyaka (K. K)3 in Saka 587 or AD 665, i.e. 37 years after the Brahmasphutasiddhanta. According to Brahmasphutasiddhanta XXIV -9, Brahmagupta also composed a small tract called the Dhyanagraha (625 AD) in 72 verses; but the author did not include this in his Brahmasphutasiddhantas of 24 chapters. Alberuni (C. 1030 AD), however, regarded it as the 25th chapter of Br. Sp. Si No other work of Brahmagupta is known.

Place

Brahmagupta's commentators Prthudaka (864 A. D.) and Amasarma or Amaraja (1180 A. D.) in their commentaries. call him Bhillamalavakacarya and Billamalakacarya respectively, which shows that he came from Bilamala. This village Bhilmala was also known as Srimala. According to Alberuni, Bhilmala was between Multan and Anhilwara, sixteen yojanas from the latter. Buhler identifies it with pi-lo-mi-lo, mentioned by Hiuen-T-Sang (600-664 A.D.) as the capital of Kiu-Che-lo. That is the northern Gurjar.

This place has been identified with the modern village Bhinamal, situated between Mount Abu and the River Luni (Lat. 25O N long 72° 19' E) close to the Rajasthan-Gujarat border. Dvivedi calls Brahmagupta a Vaisya (Vide Ganakatarangini 18), as his name ends in Gupta but Alberuni calls him a Brahmana, Brahmagupta was probably a worshipper of Siva, whom he propitiates in the beginning of his works. We have no knowledge of Brahmagupta' s teachers, or of his education, but we know he studied the five traditional siddhantas of Indian astronomy. His sources also included the works of Aryabhata I. Latadeva , Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Visnucandra, He was however quite critical of most of these authors.

Brahmasphutasiddhanta

The Brahmasphutasiddhanta is a standard treatise on ancient Indian astronomy, containing twenty four chapters and a total of 1008 verses in Arya meter. The Brahmasphutasiddhanta claimed to be an improvement over the ancient work of the Brahmapaksa, which did great deal of originality in his revision. Brahmagupta not only describes many astronomical instruments, but also teaches methods of computing various astronomical elements from the readings taken with these instruments." He examined and criticized the views of his predecessors, especially Aryabhata I (499 A.D.). He devoted two chapters to mathematics.

Brahmagupta has called the twelfth chapter as Ganits and the eighteenth chapter as Kuttaka. According to tradition of those times, Ganits includes matter pertaining to arithmetic problems on mixtures, plane figures, shadows, series piles and excavations.

The eighteenth chapter of the Brahmasphutasiddhanta, the Kuttakadhyaya, contains solutions of the indeterminate equations of both first and second degree. The Spastadhikara chapter contains trigonometrical notations including standard tables of sines and versed sines.

Khandakhadyaka

The Khandakhadyaka is a practical manual of Indian astronomy of the Karana category.

I bow to Mahadeva, the cause of creation, existence and destruction of the universe. I now write the Khandakhadyaka, which gives the same results as those obtained from Aryabhatas formulas. His rules are lengthy and hence impracticable for daily purposes, such as marriage, birth and the like. Mine, on the other hand, are brief, yet yield similar results.

The work Khandakhadyaka consists of two distinct parts, viz., the Khandakhadyaka proper and the Uttara Khandakhadyaka. In the first part the astronomical constants are the same as those of Aryabhatas ardharatrika system, but the methods of spherical astronomy, calculations of eclipses and other topics are almost the same as in the Brahmasphutasiddhanta in the Utters Khandakhadyaka Brahmagupta gives corrections to the Khandakhadyaka proper. We find in this work, the neat and original methods of interpolation and correction to the longitudes of the aphelia, as also to the dimensions to the epicycles of apsis of the sun and the moon, while a few additional chapters supply what else is necessary to the seven chapters of the first part, to make the whole a complete treatise on Hindu scientific astronomy.