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The Influence of Chinese Mathematical Arts on Seki Kowa THE INFLUENCE OF CHINESE MATHEMATICAL ARTS ON SEKI KOWA b y SHIGERU JOCHI, M.A. (Tokai) Thesis submitted for the degree of Ph.D. School of Oriental and African Studies, University of London. 1 9 9 3 ProQuest Number: 10673061 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10673061 Published by ProQuest LLC(2017). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C ode Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106- 1346 ABSTRACT I will consider the influence of Chinese mathematics on Seki Kowa. For this purpose, my thesis is constructed in four parts, introduction, I the studies of editions; Shu Shn Jin Zhang and Yang Uni S u m Fa, II the conception and extension of method for making magic squares, and 1 the analysis for solving indeterminate equations. In the introduction, I will explain some similarities between Chinese mathematics in the Song dynasty and Seki Kowa's works. It will become clear that the latter was influenced by Chinese mathematics. Then I introduce some former opinions concerning which Chinese mathematical book influenced him. I shall show that two Chinese mathematical books, Shn Shn Jin Zhang and Yang Hni S u m Fa, are particularly important. Some Chinese mathematical books were republished and studied by Japanese mathematicians, but these two books were not accessible to Japanese mathematicians. Thus we must study them for considering questions of influence. I will consider two subjects, the treatment of magic squares in Yang Hni Snan Fa in chapter H and the method of solving indeterminate equations in Shn Shn Jin Zhang in chapter M. Before considering the contents of these subjects, we must know more about the available versions of these two books in chapter I , otherwise we cannot know whether Seki Kowa could have obtained them. It seems certain that Seki Kowa studied the Yang Hni Snan Fa, but I camot know whether he studied the Shn Shn Jin Zhang. However, Seki Kowa’s method of solving indeterminate equations is very similar to that of Qin Jiushao, especially when their methods of changing negative constants into positive are similar. Thus I would like to propose that Seki Kowa studied the Chinese method of solving indeterminate equations. - 2 - ACKNOWLEDGEMENTS My first debt of gratitude is to my supervisor, Dr. Christopher Cullen, whose unfailing helpfulness and encouragement made this thesis possible. Members of the Needham Research Institute, Dr. Joseph Needham, Prof. Ho Peng- Yoke, Dr. Huang Xingzong and other menbers, were kind to help my research. When I studied history of mathematics at Beijing Normal University, Prof. Bai Shangshu very kindly guided me. At least in spirit he can also be regarded as a supervisor of this thesis. I would like to thank two researchers who corrected my English carefully, Mr. Craig Rodine who is studying history of mathematics and Mr. Paul Charity, mathematician. Of course, all errors and omissions are my responsibility alone. Finally, I would like to thank the Anglo-Daiwa Foundation for financial support. - 3 ~ CONTENTS ABSIRACT.................................................................. 2 ACKN0WITIX2MMS............................................................................................................ 3 INTRODUCTION (1) Aims of this thesis ...................................... 9 (2) Biographical study of Seki Kowa............................ 10 (3) Similarity between Chinese mathematics and Japanese mathematics * • 16 (4) Opinions of how Chinese mathematics influenced Seki K o w a ...... 19 (a) Opinions that Seki Kowa studied Chinese Mathematics * * * • 19 (b) Opinion concerning the S u m Xus Qi Meng {Introduction to Mathematical Studies)......... 21 (c) Opinion concerning the Ce Yuan Hai jing (Sea Mirror of Circle Measurement)............ 22 (d) Opinion concerning the Zhui Shu (Bound Methods)......... * .................. 23 (e) Opinion concerning the Yang Hni Sum-Fa (Yang Hui's Method of Computation)............ 24 (f) New Opinion; opinion concerning the Shu Shu Jiu Zhang (Mathematical Treatise in Nine Sections)....... 25 N o t e s .................... * ................................26 - 4 - [ : THE STUDY OF EDITIONS (1) The Shn. Shu J in Zhang..................................... 28 (a) Before completion of the Si Kn Qu m Sh u ................ 29 (b) From completion of the Si Ku Q m m Shu to the publication of Yi-JiarTang CongrShu................ 30 (c) After publication of the Yi-JiarTang Ccmg-Shu........... 33 (d) Conclusion to section I -1 ........................... 34 (2) The Yang Hni Suan F a ..................................... 35 (a) Versions of Qindetang press and its related editions • * * * 37 (b) Versions of the Yong Ls Da Diau Ed i t i o n............... 42 (c) Conclusion to section 1 - 2 ........................... 45 N o t e s ...................................................... 47 Diagram of manuscript tradition................................ 59 Biography of Ruan Yun and Li R u i................................ 62 H : TIE CONCEPTION AND EXTENSION OF tfflHQD FDR MAKING MAGIC SQUARE (1) Introduction............................................ 71 (a) Magic squares as “mathematics” ........................ 71 (b) History of Chinese magic squares...................... 73 (2) Before the Yang Hni Snan F a ................................ 75 - 5 - (a) Characteristics of “Luo Shu"; three degree magic square * * 75 (b) Origin of “Luo S h u " ................................ 77 (c) “He Tu" and "Luo S h u " .............................. 78 (d) Making magic squares and the concept of "Changes"........ 84 (3) Concerning Yang Hui S u m Fa (Yang Hui’s Method of Computation) • * 87 (a) Term used by Yang Hui for magic squares................ 87 (b) Composing the three degree magic square ("Luo Shu”) • * * • 89 (c) Composing four degree magic squares........ * ........ 90 (d) Composing five degree magic squares................... 93 (e) Conposing six degree magic squares..................... 96 (f) Conposing seven degree magic squares.................. 98 (g) Conposing eight degree magic squares......... 101 (h) Conposing a nine degree magic square................. 105 (i) Conposing a ten degree magic s q u a r e ..................107 (j) Conclusion....................................... 109 (4) Before Seki Kowa............................... 113 (a) Before the Edo p e r i o d ............................. 113 (b) The early Edo period............................... 116 (5) Works of Seki Kowa..................................... 121 (a) Terms used in magic square.......................... 121 (b) Odd number degree magic squares..................... 125 (c) Doubly-even degree magic squares..................... 128 (d) Oddly-even degree magic squares..................... 131 - 6 - (6) Conclusion 134 (a) Studies of magic squares in the Edo period, Japan * * * * 134 (b) Studies of magic squares in the Qing dynasty, China * • * 139 Notes..................................................... 142 m : IDE ANALYSIS FOR SOLVING INDETERMINATE EQUATIONS (1) Study history........................................... 148 (2) “The Sunzi Theorem” (Chinese Remainder Theorem)................. 150 (a) In China...................................... 150 (b) In Japan...................................... 156 (3) The General Solution of Da-Yan Problems...................... 158 (a) The name of "Da-Yan Shu"......................... 158 (b) The content of "Da-Yan Zong Shu Shu" (the General Solution of Dayan Problem)........... 159 (c) ‘Da-Yan Jiu Yi Shu" (The Technique of Acquiring "One" in Dayan)....... 160 (d) Changing the divisors of the question into mutually prime numbers.......... 168 (e) "Fu Cheng" (multiply a g a i n ) ..................... 180 (f) “Jian Guan Shu" (the Technique of Cutting Lengths of Tube) in the Yang Hui S u m F a ............. 184 (g) Before Seki Kowa in J a p a n ....................... 187 - 7 (h) Works of Seki Kowa................................. 192 (i) Advanced works in China and Japan in 19th and 20th Centuries................. 198 (4) Conclusion • - ....................................... 203 N o t e s ..................................................... 205 CONCLUSION..................................................... 213 CHRONOLOGICAL TABLES OF EASTERN ASIA............................... 216 ABBREVIATIONS.................................................. 217 BIBLIOGRAPHIES (1) Classics, by title ............................ 218 (2) Other works, by a u t h o r ................................... 237 - 8 - INTRODUCTION (1) Aims of this thesis The influence of Chinese mathematics was felt in most East Asian countries. In the case of Japan, it was introduced into the country two times. The first time was in the eigjhth century, when many mathematical arts were introduced and taught in the University, but Japanese mathematicians only imitated the work of Chinese authorities, and its level was limited. The second time was at the aid of the sixteenth century— at which point, Japanese mathematicians applied Chinese mathematics, and were to produce brilliant achievements. It is
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