ŚRĪPATI’S ARITHMETIC IN THE SIDDHĀNTAŚEKHARA AND GAṆITATILAKA: EDITION, TRANSLATION, AND MATHEMATICAL AND HISTORICAL ANALYSIS Rev. Jambugahapitiye Dhammaloka Thero A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics Department of Mathematics and Statistics University of Canterbury New Zealand 2019 Abstract The literature on the exact sciences in Sanskrit in the second millennium is significant from a his- torical point of view because of the emergence of innovative ideas, systematization of the traditional knowledge, advanced technicality, creativity of the scientific writing style, and so on. Śrīpati, a mathematician-astronomer, is the first known writer of exact sciences in this era. He lived inthe 11th century most probably in Maharashtra and he wrote in a vast range of subjects from astrology to astronomy. In spite of his prolific writings and influence in the early stages of the second millennium, Śrīpati has been understudied. Śrīpati is the first author who wrote a separate mathematical text while still retaining all the main mathematical rules in theoretical astronomical texts. It is Śrīpati who used different elegant metrical forms in versification of mathematical rules. Most importantly, he invented several rules in arithmetic and provided inspiration for successive mathematicians and astronomers especially Bhāskara II. In this thesis we provide a critical edition of the 13th chapter of Śrīpati’s Siddhāntaśekhara, an astronomical work, consulting the published edition and three manuscripts and his Gaṇitatilaka, his arithmetic text, based on the published edition. They are followed by the critical translation and the commentary where mathematical analysis of all the rules is given. These mathematical rules, procedures, and executions are compared with that of other preceding and succeeding mathematical authors mainly in identifying Śrīpati’s contribution and innovation. We attempt to understand Śrīpati’s role in the history of Indian mathematics, how he was influenced by his predecessors, his influence on succeeding mathematicians, and to contextualize the mathematical rules given inboth texts. This research will also examine, wherever possible, the use of mathematical rules given in the texts in daily practices, similarity or differences of the same rule in different texts, the characteristics of the executions of rules, and the commentators’ approaches to the base texts. i Acknowledgement One day (this is 20 May, 2014) I received an email from Dr. Rohana Seneviratne with a link leading to a project that is known as History of Indian Mathematical Sciences Project at the University of Canterbury, New Zealand. By the time I came to know about that, it had called for applications for a scholarship. I applied for that. On receiving the scholarship I started my research in December, 2015. It has been four years by now. I spent most of my time in the office reading Sanskrit texts, com- mentaries, working through mathematical formulas and examples. I traveled to India and Europe, visited manuscripts libraries, and participated in conferences, seminars, and workshops. Further, I have been privileged to meet many leading scholars in the field and to learn from them. I have met wonderful people in New Zealand. Finally, I have been able to reach to the end of this journey. Any of these would not have been possible without the kind and generous support of many people. Therefore, this is the time I must thank everyone who was the strength behind this. Though the words cannot do the justice, I have to mention that I am indebted to Prof. Clemency Montelle, my chief supervisor, for her brilliant guidance for last four years in my research. I met her for the first time on the 11th of December, 2015 at the Christchurch international airport. She was waiting for me at the airport together with Prof. Kim Plofker. From that moment onward she has been taking a great care of me. Without her right directions, I would not be writing this acknowledgment now. Her enthusiasm and fondness for the subject always inspire me. Therefore, my first thank goes to Prof. Clemency Montelle. I am also very grateful to Prof. Kim Plofker,my co-supervisor, for her comprehensive supervision. During her yearly-stay in New Zealand, we read a few chapters on mathematics and astronomy of the Siddhāntaśekhara which was a crucial training in my research apart from all the other regular correspondences and meetings. Her suggestions, critical thoughts, comments and directions led the thesis up to this level. Further, she sent me a copy of the manuscript of the Siddhāntaśekhara in Brown University, USA. Hence, she deserves a lot of thanks. Then, I would like to thank Dr. Agathe Keller, my associate supervisor. I met her at the Université Paris 7 in July, 2019 and got one-on-one feedback in addition to regular correspondence. Her erudition on the mathematical traditions in India was an indispensable source of this research. My appreciation also goes to Prof. K. Ramasubramanian, Prof. Takao Hayashi, Prof. Glen Van Brummelen, Prof. Kenneth Zysk, Dr. John Hannah, Ass. Prof. Toke Knudsen, Dr. Anuj Misra, Dr. K. Mahesh, Dr. Aditya Kolachana, Dr. Keshav Melnad, Liz Cornwell, Sahana Cidambi, and Aditya Jha for their guidance, suggestions, and all the supports in the past four years for the research and other academic activities. All their support have been immensely helpful and I firmly believe that it gave my thesis the standard shape. I am also very grateful to Dr. Rohana Seneviratne thanks to whom I came to know about the History of Indian Mathematical Sciences Project, Dr. Sanath Wijesundara, and Ven. Prof. Nepalaye Sankicca Thero, my university Sanskrit teachers, who always encouraged and strengthened me. This PhD research was financially supported by the University of Canterbury and the Rutherford Discovery Fellows, administered by the Royal Society of New Zealand through the HAMSI project. I am very grateful for these supports. Further, all the facilities to carry out my research were given by the School of Mathematics and Statistics so I am appreciative for their supports as well. I would ii Acknowledgement iii like to thank the Head of the School Prof. Jennifer Brown, former postgraduate coordinator Dr. Miguel A. Moyers Gonzalez, and the current postgraduate coordinator Dr. Deniel Gerhard, Helen Rowley, Leigh Davidson, Matt McPherson, Paul Brouwers, Penelope Goode and in general, all the academic and administrative staff who made this easy for me. My gratefulness is also towards the library staff who assisted with supplying sources. I am also very appreciative for granting methe study leave for four years by the University of Peradeniya, Sri Lanka for undertaking this research here. In addition, my acknowledgments are due to Prof. Siniruddha Dash, Prof. M.D. Siriniwas, Dr. Venkateshvara Pai, the staff and the members of the Baroda Oriental Institute (Baroda), British library (London) for their support in providing some manuscripts and texts. I thank with all my heart to my great teacher under whose patronage I became a monk. He accepted me as his student at my thirteen. He raised me, took care of me, taught me and gave me love of both father and mother. I would also like to thank all my fellow monks in the monastery who took care of all the duties on behalf of me in my absence. The source of my perseverance is my family who also raised me and nourished me with love. Even though I left home long time ago, their affection never left me, and will never leave me. Therefore, my wholehearted thanks go to them. I met many friends in New Zealand in and outside the university. I thank them all especially, Montelle family for making my life easy in New Zealand. In addition, my gratefulness is to all my other friends who listened to me in my hard times, who encouraged me when I was bored, who made me laugh when I was sad, in short, who were always there for me especially to Charitha Bandara Weerasinhe, and Dr. Sarath Samaraweera. In my last few words I would like to thank Sri Lankan community in Christchurch mainly Most Venerable Makuludeniye Somaratane Thero, the chief incumbent of the Samadhi Buddhist Vihara, Christchurch for the great support. I am also thankful to Dr. Duy Ho, Duttatreya Srivastava, Phillipa Graham, Brad and Raquel, Patricia Trujillo, Nataly Lopez, Ilayaraja, and Maurine who filled me with kindness, friendliness, and affection. You allmademy leaving for home difficult. Jambugahapitiye Dhammaloka 2019.11.30 Contents Abstract i Acknowledgement ii List of Figures vii List of Tables viii Abbreviations x 1 Introduction 1 1.1 Arithmetic and algebra in Sanskrit texts . 2 1.2 Literature review on Śrīpati’s mathematics . 5 1.3 Objectives . 7 1.4 Methodology . 9 1.4.1 Conventions . 10 1.5 Śrīpati and his works . 11 1.5.1 Biographical data . 11 1.5.2 Literary contribution . 13 1.5.2.1 Astronomical works . 13 1.5.2.2 Astrological works . 14 1.5.2.3 Mathematical work . 15 2 Vyaktagaṇitādhyāya of the Siddhāntaśekhara 16 2.1 Introduction to arithmetic . 16 2.2 Operations (parikarmans)................................. 17 2.2.1 Operations of integers . 17 2.2.1.1 Multiplication . 17 2.2.1.2 Division . 22 2.2.1.3 Square and cube . 23 2.2.1.4 Square root . 24 2.2.1.5 Cube root . 28 2.2.2 Operations of fractions . 31 2.2.2.1 Addition and subtraction . 31 2.2.2.2 Multiplication and square . 32 2.2.2.3 Division . 34 2.2.3 Classes of fractions . 34 2.2.3.1 Fraction-sub-class of the sum of integer and fraction and fraction- sub-class of the difference of integer and fraction . 35 2.2.3.2 Class of sub-fractions, fraction-sub-class of the sum of fraction and sub-fraction, and fraction-sub-class of the difference of fraction and sub-fraction .