Sources and Studies in the History of Mathematics and Physical Sciences Series Editors 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 History of Mathematics 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 Indian Mathematics and Astronomy Selected Articles of Kripa Shankar Shukla 123 Editors Aditya Kolachana K. Mahesh Indian Institute of Technology Bombay Department of Humanities Mumbai, Maharashtra, India 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 This work is a co-publication with Hindustan Book Agency, New Delhi, licensed for sale in all countries in electronic form, in print form only outside of India. Sold and distributed in print within India by Hindustan Book Agency, P-19 Green Park Extension, New Delhi 110016, India. ISBN: 978-93- 86279-78-1 © Hindustan Book Agency 2019. © Hindustan Book Agency 2019 and Springer Nature Singapore Pte Ltd. 2019 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore ॥ पसुमालः ॥ 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 Varāhamihira (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: David Pingree’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