Sources and Studies in the History of and Physical Sciences

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Jed Z. Buchwald, Division of the Humanities and Social Sciences, Caltech, Pasadena, CA, USA

Associate Editors

A. Jones, Department of Classics, Institute for the Study of the Ancient World, New York, NY, USA J. Lützen, Koebenhavn OE, Denmark J. Renn, Max Planck Institute for the History of Science, Berlin, Germany

Advisory Board

C. Fraser T. Sauer A. Shapiro Sources and Studies in the and Physical Sciences was inaugurated as two series in 1975 with the publication in Studies of Otto Neugebauer’s seminal three-volume History of Ancient Mathematical Astronomy, which remains the central history of the subject. This publication was followed the next year in Sources by Gerald Toomer’s transcription, translation (from the Arabic), and commentary of Diocles on Burning Mirrors. The two series were eventually amalgamated under a single editorial board led originally by Martin Klein (d. 2009) and Gerald Toomer, respectively two of the foremost historians of modern and ancient physical science. The goal of the joint series, as of its two predecessors, is to publish probing histories and thorough editions of technical developments in mathematics and physics, broadly construed. Its scope covers all relevant work from pre-classical antiquity through the last century, ranging from Babylonian mathematics to the scientific correspondence of H. A. Lorentz. Books in this series will interest scholars in the history of mathematics and physics, mathematicians, physicists, engineers, and anyone who seeks to understand the historical underpinnings of the modern physical sciences.

More information about this series at http://www.springer.com/series/4142 Aditya Kolachana • K. Mahesh • K. Ramasubramanian Editors

Studies in and Astronomy Selected Articles of Kripa Shankar Shukla

123 Editors Aditya Kolachana K. Mahesh Indian Institute of Technology Bombay Department of Humanities Mumbai, Maharashtra, and Social Sciences Indian Institute of Technology Bombay K. Ramasubramanian Mumbai, Maharashtra, India Department of Humanities and Social Sciences Indian Institute of Technology Bombay Mumbai, Maharashtra, India

ISSN 2196-8810 ISSN 2196-8829 (electronic) Sources and Studies in the History of Mathematics and Physical Sciences ISBN 978-981-13-7325-1 ISBN 978-981-13-7326-8 (eBook) https://doi.org/10.1007/978-981-13-7326-8

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K. S. Shukla (1918–2007)

दैदवभूतमणतभ ाावधेशाुदा नैरयगवेषणेन मुनव काशे रतः । ोतवकृतीः भूषयदयं ीयानुसानतः वलकेसर महमा-ः कृपाशरः ॥

Kripa Shankara Shukla, having received the treasure of the lumi- nescence of mathematics bequeathed by [Bibhutibhusana] Datta through Avadhesha [Narayana Singh], happily dedicating himself to research, relentlessly worked towards the publication of [mul- tiple] treatises like a sage. Through his investigations, he embel- lished the writings of astronomers, and was truly a lion in the assembly of scholars, shining (śukla) through his fame. ॥ ��समप�ण� ॥

���ाचाय�पदे �ह ल�णपुरे �व�ा�भरामः कृपा- ��ो �ो�तषशा�गूढ�वषया� लोकाय चोपाहर� । लेख��� सुधी��या� सुम�नभा� स�ृ� माला�मम� सौल�ाय बुभु�ुभृ�ततये द�ो वयं तु�ये ॥

While remaining in city of Lucknow (lakṣmaṇapura) as an Ācārya, Kripa [Shankar] Shukla was indeed a source of delight for the whole world (viśvābhirāma), [as] he also presented the subtler aspects of [Indian] astronomy [in a manner comprehensible] to the entire mankind (loka). Having collected his articles resembling a bunch of fowers, which are a source of delight to the men of wisdom (sudhīpriya), we present them, for the sake of easy accessibility, as a garland for the gratifcation of the swarm of bees constituted by the [community of] knowledge-seekers (bubhutsu). Contents

Preface ...... xiii

Acknowledgements ...... xix

I The Oeuvre of Kripa Shankar Shukla 1

Reminiscences of Prof. K. S. Shukla Yukio Ohashi ...... 3

Dr. Kripa Shankar Shukla, veteran historian of Hindu astronomy and mathematics R. C. Gupta ...... 5

Obituary: Kripa Shankar Shukla (1918–2007) Yukio Ohashi ...... 13

Prof. Shukla’s contribution to the study of the history of Hindu astronomy Yukio Ohashi ...... 23

The seminal contribution of K. S. Shukla to our under- standing of Indian astronomy and mathematics M. D. Srinivas ...... 39

II Studies in Indian Mathematics: Bhāskara I to Nārāyaṇa Paṇḍita 71

Hindu mathematics in the 7th century as found in Bhāskara I’s commentary on the Āryabhaṭīya (I) ..... 73

Hindu mathematics in the 7th century as found in Bhāskara I’s commentary on the Āryabhaṭīya (II) .... 88

Hindu mathematics in the 7th century as found in Bhāskara I’s commentary on the Āryabhaṭīya (III) .... 107 x Contents

Hindu mathematics in the 7th century as found in Bhāskara I’s commentary on the Āryabhaṭīya (IV) .... 110

On Śrīdhara’s rational solution of Nx2 + 1 = y2 ...... 120

Ācārya Jayadeva, the mathematician ...... 133

Series with fractional number of terms ...... 153

Hindu methods for fnding factors or divisors of a number 160

Magic squares in Indian mathematics ...... 169

III Revised version of the Manuscript of the Third Volume of Datta and Singh 187

Hindu geometry ...... 189

Hindu trigonometry ...... 268

Use of calculus in Hindu mathematics ...... 345

Use of permutations and combinations in India ...... 356

Magic squares in India ...... 377

Use of series in India ...... 438

Surds in Hindu mathematics ...... 462

Approximate values of surds in Hindu mathematics ... 475

IV Studies in Indian Astronomy: From Vedic Period to the Emergence of Siddhāntas 485

Astronomy in ancient and medieval India ...... 487

Main characteristics and achievements of ancient Indian astronomy in historical perspective ...... 495

On three stanzas from the Pañcasiddhāntikā ...... 510

The Pañcasiddhāntikā of (1) ...... 517 Contents xi

The Pañcasiddhāntikā of Varāhamihira (2) ...... 533

Āryabhaṭa I’s astronomy with midnight day-reckoning . 548

Glimpses from the Āryabhaṭasiddhānta ...... 569

V Development of Siddhāntic Astronomy: Some Highlights 577

Early Hindu methods in spherical astronomy ...... 579

Use of hypotenuse in the computation of the equation of the centre under the epicyclic theory in the school of Āryabhaṭa I ??? ...... 600

Hindu astronomer Vaṭeśvara and his works ...... 616

The evection and the defcit of the equation of the centre of the Moon in Hindu astronomy ...... 625

Phases of the Moon, rising and setting of planets and stars and their conjunctions ...... 646

VI Reviews and Responses 695

Vedic Mathematics: The deceptive title of Swamiji’s book 697

A note on the Rājamṛgāṅka of Bhoja published by the Adyar Library ...... 705

Review of Rājamṛgāṅka of Bhojarāja ...... 707

Review of Karaṇaratna of Devācārya Raymond P. Mercier ...... 709

A note on Raymond P. Mercier’s review of “Karaṇaratna of Devācārya” ...... 715

The yuga of the Yavanajātaka: ’s text and translation reviewed ...... 719

Review of Vaṭeśvarasiddhānta and Gola of Vaṭeśvara David Pingree ...... 732

Appendix ...... 734 Preface

The rich history of the origin and development of science in India and the voluminous literature produced in this pursuit stand in stark contrast to the prevalent ignorance and meagre attention paid to this heritage in current times. Despite the neglect of the discipline of history of science, our under- standing of India’s scientifc heritage—especially that of mathematics and astronomy—has progressed due to the tireless eforts of several stalwarts such as Bapudeva Sastri (1821–1900), Shankar Balakrishna Dikshit (1853–1898), Sudhakara Dvivedi (1855–1910), M. Rangacarya (1861–1916), P. C. Sengupta (1876–1962), B. B. Datta (1888–1958), A. A. Krishnaswamy Ayyangar (1892– 1953), A. N. Singh (1901–1954), C. T. Rajagopal (1903–1978), T. A. Saraswati Amma (1918–2000), S. N. Sen (1918–1992), K. S. Shukla (1918–2007) and K. V. Sarma (1919–2005). Prof. K. S. Shukla was one of India’s leading historians of science. Com- bining a fair for mathematics with a strong grasp of , Prof. Shukla made immense contributions to advancing our understanding of the history and development of mathematics and astronomy in India. On the occasion of his birth centenary, we have taken the opportunity to collate and compile some of Prof. Shukla’s most important papers in the form of the volume in front of you. Such a volume naturally demands a brief introduction of the individual who has authored its contents, and in the following sections we set out to do the same. Subsequently, in the fnal section of the preface we briefy discuss the structure of the volume as well as the editorial practices adopted in compiling it.

Early life

The biographical details pertaining to the early phase of Prof. Shukla’s life has been succinctly brought out by Prof. R. C. Gupta, an eminent historian of mathematics himself, and a student of Prof. Shukla on the occasion of the latter’s 80th birth anniversary:1

1See the paper entitled “Dr. Kripa Shankar Shukla, veteran historian of Hindu astronomy and mathematics” by Prof. Gupta in Part I of this volume. Subsequent quotes and many xiv Preface

Kripa Shankar Shukla’s birth took place at Lucknow on July 10, 1918. From the very early years, he was a brilliant student of Mathematics and Sanskrit. He passed the High School Examina- tion of U.P. Board in 1934 in First Division with Distinction in Mathematics and Sanskrit and the Intermediate Examination of that Board again in First Division with Distinction in Mathemat- ics. He had his higher education at Allahabad, passing the B.A. examination in the second division from Allahabad University in 1938. From the same University, he obtained his Master of Arts degree in Mathematics in the First Division in 1941.

Career and contributions

Just as Prof. Shukla was completing his undergraduate studies, Prof. Bib- hutibhusan Datta and Prof. Avadhesh Narayan Singh published the second part of their monumental work “History of Hindu Mathematics” in 1938. Subsequently, in 1939, Prof. Singh launched a Scheme of Research in Hindu Mathematics in the Department of Mathematics and Astronomy at Lucknow University, where he was a Lecturer. After his post-graduation, Prof. Shukla joined this scheme. Prof. Gupta writes: Dr. K. S. Shukla joined the Department and the Scheme in 1941 ... Dr. Shukla investigated thoroughly the works of Bhāskara I and studied other relevant primary and secondary material. Under the supervision of Dr. A. N. Singh, Shukla prepared a thesis on “Astronomy in the Seventh Century India: Bhāskara I and His Works” ... The signifcance of the thesis lies not only in providing a genuine additional source for the history of early Indian exact sciences but also in bringing to light many new historical and methodological facts. Over the course of his career at Lucknow University, Prof. Shukla carried for- ward his investigations into the history of Indian mathematics and astronomy, publishing a number of critically edited texts (often with English translation and commentary), as well as research papers, in addition to supervising the re- search of fve doctoral scholars. Prof. Gupta as well as Prof. Ohashi2 (another student of Prof. Shukla) have given detailed bibliographies of Prof. Shukla’s publications. The major source works brought out by Prof. Shukla are listed in Table 1. He also wrote over 40 important articles and reviews which not only brought

details in this preface regarding Prof. Shukla’s life and work are also borrowed from here. 2See Prof. Ohashi’s “Obituary” in Part I of this volume. Subsequent quotes from Prof. Ohashi in this preface are also borrowed from this paper. xv

Table 1: Source works brought out by Prof. Shukla.

No. Title of the works and their authors Year 1 Sūrya-siddhānta with the commentary of Parameśvara 1957 2 Pāṭīgaṇita of Śrīdharācārya 1959 3 Mahābhāskarīya of Bhāskara I 1960 4 Laghubhāskarīya of Bhāskara I 1963 5 Dhīkoṭida-karaṇa of 1969 6 Bījagaṇitāvataṃsa of Nārāyaṇa Paṇḍita 1970 7 Āryabhaṭīya of Āryabhaṭa 1976 8 Āryabhaṭīya of Āryabhaṭa with the commentary of 1976 Bhāskara I and Someśvara 9 Karaṇaratna of Devācārya 1979 10 Vaṭeśvarasiddhānta and Gola of Vaṭeśvara (2 Vols) 1985–86 11 Laghumānasa of Mañjula 1990 12 Gaṇitapañcaviṃśī (published posthumously) 2017 forth the numerous contributions of Indian mathematicians and astronomers, but also served to demolish certain wrong conceptions regarding the origin, technical soundness, and depth of the Indian works. Some of Prof. Shukla’s most important contributions include (i) his study of Varāhamihira’s Pañca- siddhāntikā, (ii) bringing to light Jayadeva’s verses on the brilliant cakravāla method of solving second order indeterminate equations, (iii) clearing miscon- ceptions among modern scholars regarding the use of the iterated hypotenuse by Indian astronomers in determining the equation of centre, and (iv) revising and publishing the third and fnal part of the “History of Hindu mathematics” by B. B. Datta and A. N. Singh.3

Scholarship and commitment

In praise of Prof. Shukla’s scholarship, Prof. Gupta notes:

Working wholeheartedly with single minded devotion for more than half a century, Dr. Shukla’s contribution in the feld of history of ancient and medieval Indian mathematics forms a pioneer work which will continue to motivate future research and investigations. He gave new interpretations of many obscure Sanskrit passages and corrected misinterpretations and other errors committed by others. 3For a detailed discussion on some of Prof. Shukla’s seminal contributions, see the articles of Yukio Ohashi (1995) and M. D. Srinivas (2018) in Part I of this volume. xvi Preface

Extolling Prof. Shukla’s meticulous way of maintaining notes, and his com- mitment towards his doctoral students, Prof. Ohashi writes:

When we read his notes, we feel as if we are being taught by him directly. It should also be mentioned that he noted several parallel statements in other Sanskrit texts in the footnotes of his English translations. So, his English translations can also be used as a kind of annotated index of Sanskrit astronomical and mathemati- cal texts. Only Prof. Shukla could do this ... I studied the history of Indian astronomy and mathematics un- der the guidance of Prof. Shukla from 1983 to 1987 as a research scholar (Ph. D. student) of Lucknow University ... Prof. Shukla already had retired but kindly taught me how to read Sanskrit astronomical texts, both printed texts and manuscripts. I saw several people were visiting Lucknow to meet Prof. Shukla.

Tireless eforts to attain perfection

Unlike today, in the India of the 1980s, printing was done using the letterpress, wherein a worker used to compose the text in a metallic frame of a given dimension, employing a variety of metallic fonts stored in a huge type case. Also printing of a volume could not happen all in one go as we do it today. At most 16 or 32 sheets could be printed at one time, and if there were to be any slip in proof reading, it could not be corrected again since the frame would have been dismantled once the pages were printed. Hence the author had to be all the more careful in proof-reading the text. Reminiscing how punctiliously and tirelessly Professor Shukla worked to ensure that the books he edited were error-free, Prof. Ohashi observes:

When I was in Lucknow, the Vaṭeśvara Siddhānta and Gola of Vaṭeśvara was being printed at a press in Lucknow. Prof. Shukla visited the press almost every day, supervised its printing work by himself, and read its proofs very carefully. From this fact, we can understand why his edition is so reliable. These original sources are the most important foundation for future research.

Awards and accolades

In recognition of his scholarship and lifelong contributions, Prof. Shukla re- ceived many awards and was associated with several prestigious institutions. Some of these have been detailed by Prof. Gupta: xvii

Dr. Shukla was awarded the Banerji Research Prize of the Luc- know University. He was associated with the editorial work of the Journal Gaṇita of the Bhārata Gaṇita Pariṣad (formerly the Benaras Mathematical Society) for many years. He was elected Fellow of the National Academy of Sciences, India in 1984, and the Corresponding Member of the International Academy of His- tory of Science, Paris, in 1988. He served as a member of several national and international committees.

The current volume

Prof. Shukla passed away on September 22, 2007. In his memory, on the occasion of the centenary of his birth, the current volume presents a collection of his papers highlighting the wide range of his scholarship.

Structure of the volume

This volume consists of six parts. Part I consists of fve introductory articles which give an overview of the life and work of Prof. Shukla. They include detailed bibliographies of his publications, and reminiscences from his former students Prof. Yukio Ohashi and Prof. R. C. Gupta. The last two articles by Prof. Yukio Ohashi and Prof. M. D. Srinivas highlight the important contributions made by Prof. Shukla to improve our understanding of Indian mathematics and astronomy. Part II consists of a collection of articles penned by Prof. Shukla related to various aspects of Indian mathematics. Part III consists of revised version of articles on Indian mathematics by Bibhutibhu- san Datta and Avadhesh Narayan Singh, which together constitute the third unpublished part of their “History of Hindu Mathematics”. As noted earlier, these articles were revised and updated by Prof. Shukla and published in the Indian Journal for History of Science between 1980 and 1993. Parts IV and V consist of a number of articles penned by Prof. Shukla on diferent aspects of Indian astronomy. Part VI includes some of Prof. Shukla’s reviews of works related to Indian mathematics and astronomy authored by various scholars. This part also includes a few reviews of Prof. Shukla’s publications by other scholars, and in one instance, his response to a review.

A note on the editorial practices adopted

While preparing this volume, we have emended the original text in a number of places. These emendations are generally accompanied with an editorial note prefaced with the abbreviation “ed.”. Occasionally, we have also emended the text silently for a better reading experience. For instance, typographical xviii Preface errors in the English text as well as the Sanskrit verses have been silently emended. In a few instances, tables and fgures which originally occur in be- tween running text have been given numbers and placed elsewhere in the text, with the appropriate reference, for better typesetting. Footnotes to mathe- matical equations have been moved to the adjacent text. In the interests of standardising the style of the volume, we have (i) redrawn all the fgures, (ii) presented Sanskrit verses in the Devanāgarī script, (iii) largely made uni- form the diferent styles of transliterating Sanskrit words and the names of Sanskrit texts into the roman script, (iv) modifed the section numbers in a few instances, (v) standardised table styles across papers, and (vi) converted all end-notes to footnotes. We hope that this volume serves to familiarize the reader with the wide range of research carried out by Prof. Shukla, and also inspires young scholars to seriously pursue research in Indian mathematics and astronomy.

Aditya Kolachana �वल��-माग� शर��ष�ी K Mahesh गतक��ाः ५११९ K Ramasubramanian क�हग�णः १८७०००० (अ�ा�ननः सदीपाः) December 13, 2018 IIT Bombay, India Acknowledgements

At the outset, we would like to express our sincere gratitude to the various publishers of Prof. Shukla’s research articles for giving their kind permission to compile this volume. For the sheer volume of articles sourced from these journals, we would like to especially thank the publishers of Gaṇita, Gaṇita- Bhāratī, and the Indian Journal of History of Science. Given the wide range of scholarship and prolifc output of Prof. Shukla, the process of sorting through and selecting the appropriate assortment of his papers for this volume, which are representative of the depth and range of his scholarship, was not an easy task. We are quite grateful to Prof. M. D. Srinivas for his invaluable guidance and helpful insights in this process. Some of Prof. Shukla’s important papers that were published in Lucknow University’s Journal Gaṇita were not accessible to us as they were not dig- itally archived. In this connection, we approached Prof. Poonam Sharma, Department of Mathematics and Astronomy, Lucknow University. She read- ily agreed and immediately arranged to procure copies of these articles. We deeply appreciate her enthusiastic help and express our sincere gratitude to her for this kind gesture. We would also like to thank Prof. Yukio Ohashi for contributing a note on his Reminiscences, heeding to our request in a very short notice. Currently, the personal library of Prof. Shukla is safeguarded by his son Sri Ratan Shukla. As we wanted to consult this library in connection with the preparation of this volume, we contacted Sri Shukla. He warmly welcomed us into his home, and provided us open access to the entire library as well as Prof. Shukla’s personal communications. This gave us greater insights into the working style as well as the personality of Prof. Shukla, and helped us make informed choices while editing this volume. We express our heartfelt thanks to Sri Ratan Shukla for his generous hospitality. Preparing this volume involved the diligent efort of several people who assisted with typing, drawing fgures, typesetting, proof-reading, technical support, and overall coordination. For helping with all these variety of tasks— without which this volume would not have been possible—we are greatly xx Acknowledgements thankful to Dr. Dinesh Mohan Joshi, Smt. Sushma Sonak, Smt. Sreelekshmy Ranjit, Sri G. Periasamy, Sri Vikas Uttekar, and Smt. T. Mehtaj. We would also like to express our sincere gratitude to the Ministry of Hu- man Resource Development, Government of India, for the generous support extended to carry out research activities on Indian science and technology by way of initiating the Science and Heritage Initiative (SandHI) at IIT Bombay. Finally, we convey our gratitude to the Hindustan Book Agency, New Delhi, for enthusiastically coming forward to publish this volume as a part of their series on Culture and History of Mathematics. About the Editors

Aditya Kolachana is a post-doctoral researcher at the Indian Institute of Technology Bombay, India, studying various aspects of the history of mathematics and astronomy in India. He obtained his PhD in the history of science from Indian Institute of Technology Bombay, India. He studied at the Indian Institute of Technology Kharagpur and Indian Institute of Management Ahmedabad, India. Earlier, he worked for over four years with Bain & Company, Clinton Foundation, and Procter & Gamble and also briefly taught finance at the Birla Institute of Technology and Science, Pilani, Hyderabad Campus, India.

K. Mahesh is a research scientist at the Cell for the Indian Science and Technology in Sanskrit, Department of Humanities and Social Sciences, Indian Institute of Technology Bombay, India. He acquired his PhD degree from Indian Institute of Technology Bombay, India, in 2010, by working on the Indian astronomy (Siddhānta Jyotiṣa). He went to the Centre Nationale de la Recherche Scientifique (CNRS), Paris, for doing his post-doctoral research in the history of numerical tables. On returning from Paris, he joined the Samskrit Promotion Foundation, New Delhi, and served there for three years. In 2015, the Indian National Science Academy, New Delhi, bestowed upon him the Young Historian of Science Award.

K. Ramasubramanian is a professor at the Cell for the Indian Science and Technology in Sanskrit, Department of Humanities and Social Sciences, Indian Institute of Technology Bombay, India. He holds a doctorate degree in theoretical physics, a master’s degree in Sanskrit, and a bachelor’s degree in engineering—a weird but formidable combination of subjects to do multidisciplinary research. He was honoured with the coveted title “Vidvat Pravara” by the Shankaracharya of Sri Sringeri Sharada Peetham, Karnataka, India, for completing a rigorous course in Advaita Vedānta (a 14-semester program), in 2003. He is one of the authors who prepared detailed explanatory notes of the celebrated works Gaṇita-yuktibhāṣā (rationales in mathematical astronomy), Tantrasaṅgraha and Karaṇa-paddhati, which bring out the seminal contributions of the Kerala School of Astronomers and Mathematicians. He was conferred with the prestigious award of the Mahaṛṣi xxii About the Editors

Bādarāyaṇa Vyas Samman by the then President of India, in 2008, and the R.C. Gupta Endowment Lecture Award by the National Academy of Sciences India, in 2010. He is a recipient of several other awards and coveted titles as well. From 2013, he has been serving as an elected council member of the International Union of History and Philosophy of Science and Technology. He is also a member of various other national and international bodies.