The Ten Mathematical Classics: The Mathematics and Mathematical Education of Pre-Modern China by Ke-Xin Au Yong A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in History Department of History and Classics University of Alberta © Ke-Xin Au Yong, 2018 Abstract The Ten Mathematical Classics were the only imperially prescribed mathematics textbooks in pre-modern China. They were used during the Sui (581 – 618), Tang (618 – 906), and Northern Song (960 – 1127) dynasties at the imperial academy to structure the mathematical training of students. This dissertation explores the contents as well as information about the writers and commentators of these texts in order to arrive at a better understanding of Chinese mathematics and mathematical writers. It also analyzes how mathematical education actually took place, presenting a new perspective on why state-run mathematical education only existed at specific times. Lastly, this dissertation examines the circulation and transmission of the Ten Mathematical Classics. My thesis consists of three central points. Firstly, the history of Chinese mathematics should take into serious account the entire corpus of knowledge and endeavours, such as divination, that historical actors associated indivisibly with what we would consider pure mathematical knowledge. Secondly, the known writers, commentators, and readers of the Ten Mathematical Classics were all highly educated, many of whom were also government officials, so it is over-simplistic to attribute the lack of long-term state-run mathematical education to a general disdain among the Chinese literati for technical subjects. Thirdly, I argue that the state’s decisions to institute or drop mathematical education should be understood within the broader context of the state’s needs at the time, and were directly related to the availability of suitable mathematically skilled candidates who could be recruited into the bureaucracy. ii Acknowledgements I am deeply grateful to my co-supervisors, Professor Jennifer Jay and Professor Robert Smith, for their tireless support and encouragement as they oversaw this project from its conception to its completion. My heartfelt thanks also go to Professor Ryan Dunch, Professor Dominik Wujastyk, and Professor Yihong Pan for devoting so much time and care to reading my dissertation and for their invaluable comments and suggestions. I would also like to thank Mr. John Moffett of the Needham Research Institute for all his help during my research trip to Cambridge in 2015. In addition, this project could never have been completed without generous funding from the Social Sciences and Humanities Research Council of Canada and from the University of Alberta’s Department of History and Classics. Lastly, I would like to thank my family for providing me with every possible kind of support. Special thanks to my sister Ke Xian for helping me with the diagrams. iii Table of Contents List of Tables ...................................................................................... vii List of Figures ..................................................................................... viii List of the Ten Mathematical Classics of the Tang Dynasty .................. x Introduction ........................................................................................ 1 • Historical Background ............................................................... 5 • Purpose and Questions ............................................................. 8 • Working Assumptions and Thesis ........................................... 10 • State of the Field ..................................................................... 17 • Outline of Chapters ................................................................. 25 • Note on Sources and Translations .......................................... 27 Chapter One Putting Thought into Writing: The Formation of the Ten Mathematical Classics .......................................... 30 • Zhoubi suanjing ...................................................................... 35 • Jiuzhang suanshu ................................................................... 45 • Shushu jiyi .............................................................................. 68 • Haidao suanjing ...................................................................... 72 • Sunzi suanjing ........................................................................ 77 • Xiahou Yang suanjing ............................................................. 82 • Zhang Qiujian suanjing ........................................................... 84 • Wucao suanjing ...................................................................... 87 • Wujing suanshu ...................................................................... 89 • Jigu suanjing ........................................................................... 91 • Conclusion .............................................................................. 93 iv Chapter Two Writing About Mathematics in Pre-Modern China: The Writers and Commentators of the Ten Mathematical Classics .................................... 95 • Writers, Compilers, and Commentators Who are not Known to have Held Any Official Position ..................................................................... 100 • Writers, Compilers, and Commentators Who were Officials Outside the Bureau of Astronomy and Astrology ...................................... 110 • Writers and Commentators Who Served within the Bureau of Astronomy and Astrology ............................................................................... 115 • Conclusion ............................................................................ 119 Chapter Three State-Run Mathematical Education and the Ten Mathematical Classics as Textbooks ........................................ 120 • Mathematics Education Prior to the Sui Dynasty ................... 123 • The Sui Dynasty: The Creation of the School of Mathematics 125 • The Tang Dynasty: The Intertwined Relationship Between the School of Mathematics and the Civil Service Examination System ....... 133 • The Northern Song Dynasty: The Parting of Ways Between Mathematics Education and the Civil Service Examinations ...................... 147 • Conclusion ............................................................................ 152 Chapter Four Mathematical Education in China: The Complementary Roles of State-Run and Private Schools ...................... 154 • Mathematical Education after the Northern Song .................. 159 • Re-institution of State-Run Mathematical Education ............. 172 • Mathematics in Practice ........................................................ 173 • Conclusion ............................................................................ 175 v Chapter Five Mathematics in Motion: The Circulation and Transmission of the Ten Mathematical Classics .......................... 177 • Earliest Copies ...................................................................... 182 • First Imprints of the Ten Mathematical Classics .................... 185 • Printing of the 1213 Tingzhou Edition .................................... 199 • The Ten Mathematical Classics after the Song Dynasty ....... 203 • Transmission to Korea, Japan, and Vietnam ......................... 207 • Conclusion ............................................................................ 211 Concluding Remarks ...................................................................... 213 Bibliography ................................................................................... 219 Appendix: A Selection of Problems and Passages from the Ten Mathematical Classics ........................................................................................... 235 vi List of Tables 1.1 The Ten Mathematical Classics .............................................. 34 2.1 Known Authors and Commentators of the Ten Mathematical Classics ................................................................................. 99 3.1 Rank of Erudites and Teaching Assistants During the Sui Dynasty ............................................................................... 129 3.2 Number of Erudites, Teaching Assistants, and Students During the Sui Dynasty ........................................................................... 130 3.3 Number and Rank of Erudites and Teaching Assistants During the Tang Dynasty ........................................................................ 135 3.4 Student Composition of the Six Schools During the Tang Dynasty ............................................................................... 136 3.5 Official Capacities of the Six Schools During the Early Tang Dynasty ........................................................................ 138 3.6 Official Capacities Around the Year 765 ................................ 139 5.1 Names of the Carvers Who Worked on the Tingzhou Edition of the Ten Mathematical Classics ............................................. 202 vii List of Figures 1.1 A pictorial representation of Fuxi and Nüwa dating from the Tang dynasty .......................................................................... 37 1.2 Johannes Kepler holding a compass ....................................... 38 1.3 Replica of Han-dynasty counting rods made from animal bones ................................................................................. 48 1.4 Counting rods display digits by using these very specific