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EditorsEberhard Knobloch • Hikosaburo Komatsu • Dun Liu

Seki, Founder of ModernMathematics in Japan

A Commemoration o His Tercentenaryn

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Technische Universität BerlinBerlin 10623, Germany

44

Eberhard Knobloch

2013932551

Hikosaburo Komatsu

The University of Tokyo

Beijing 100190, P. R. ChinaInstitute for the History of Natural Science

ISSN 2194-1009 ISSN 2194-1017 (electronic)

Typesetting by volume editors using a Springer TeX macro package

Mathematics Subject Classification (2010): , 11C2

Professor Emeritus

Dun Liu

01-06, 01A05, 01A27 0, 11D72, 13P15, 01A25, 11D41, 01-08

Professor

Professor

Dedicated to the spirit of Takakazu Seki

Postal Stamp (Japan)T. Seki (mathematician, 1642–1708)The background shows his expansion of a determinant of order four.

To the best knowledge of Hikosaburo Komatsu, one of the editors, there are no authentic portraitsof T. Seki, even though there are two existing portrayals of him. The image on the postal stampshown on the preceding page is based on a painting of T. Seki that is now in the Ichinoseki CityMuseum. The image seen above is a scroll painting (kakejiku) of T. Seki that Ishikuro Nobuyoshi(1760–1836) had an artist draw for ceremonies in his school. (Owned by the Koju Foundation;now in the Imizu City Shinminato Museum, Toyama, Japan)

The Use of Sans-serif Type

Emphasized texts in Japanese are usually set in sans-serif type (see [1], p. 226 or [2],p. 160.) in place of boldface type, which is too thick for Japanese texts.In this book sans-serif type is used for the same purpose, but with two additionaluses:One is as an identifier in English of the title of a book written in a language otherthan English, or of the name of a person, place, etc. The reader may find the original(title, etc.) in the Index at the end of the book.In the Supplements, five of Seki ’s works are published for the first time as a col-lated edition. Here a character of classical Chinese in sans-serif type means that thecharacter in question is not taken from any of the documents listed in the notes oncollation following the text, but is chosen at the discretion of the collators.

Editor

References[1] L. Lamport: LaTeX: A Document Preparation System, User’s Guide and Refer-ence Manual, Addison-Wesley Publ. Co., 1994.[2] M. Goossens, F. Mittelbach and A. Samarin: The LaTeX Companion, Addison-Wesley Publ. Co., 1994.

Foreword

These proceedings are a collection of papers written by lecturers at the InternationalConference on the History of Mathematics in Memory of Seki Takakazu (1642?–1708), held August 25–31, 2008. The conference was one of the main events ofSeki’s 300th memorial. The papers are faithfully based on the lectures, except thefirst paper, entitled “Seki Takakazu, His Life and Bibliography”, which containsnew and significant facts discovered later by studying some documents that wereunknown at the time of the original lecture. We are deeply grateful to the document-holder.

Seki Takakazu was a Japanese mathematician in the early part of the Edo eraand was known as the first to discover the so-called determinants in the world. Hepassed away on October 24 in the fifth year of the Hoei period, which correspondsto December 5, 1708, in the current solar calendar. The year 2008 thus correspondsto his 300th posthumous anniversary.

In 1907, according to Buddhist custom, the Tokyo Mathematico-Physical Societyobserved his 200th posthumous anniversary with a commemorative ceremony onthe occasion of an annual meeting, publishing a book of seven main articles of hisschool, and with a conference on Japanese mathematics for a general audience andpublishing the conference proceedings.

In 1958, the Mathematical Society of Japan observed his posthumous 250th an-niversary as a memorial by organizing a commemorative conference. After a com-memorative service in front of his tombstone at Jorinji temple in Benten-cho, Shin-juku Ward, Tokyo, a conference with lectures on the old Japanese mathematics washeld by the society jointly with the Tokyo Metropolitan Board of Education, theShinjuku Ward Board of Education, and others. In the same year, the small area sur-rounding his tombstone including the monument was designated as a historic siteby the Tokyo Metropolitan Government.

A year later, in 1959, an association of Japanese traditional mathematics wasformed and, in three years, was reorganized as the History of Mathematics Societyof Japan.

In order to commemorate his 300th posthumous anniversary, the executive com-mittee of Seki’s 300th memorial was organized on May 6, 2007, and decided to

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xii Foreword

organize the following commemorative events: a commemorative Buddhist serviceat Jorinji temple; exhibitions in several museums; the International Conference onthe History of Mathematics on the Kagurazaka Campus of the Tokyo University ofScience; lectures and events in connection with the exhibitions; and other events.The aim of these commemorative events was to honor Seki Takakazu and his math-ematical works, to inform people about the mathematical achievements in the Edoera and their later developments, and thus to encourage young mathematical talentin Japan.

The following is a list of our activities in addition to the International Conference:

• On December 2, 2007, a commemorative Buddhist service at Jorinji temple, fol-lowed by a conference on mathematical power in Japan, where a report on thepreparation of commemorative activities was given by representatives of the ex-ecutive committee, and lectures on education and research in mathematics weregiven;

• From January 19 to March 2, 2008, a special exhibition entitled “Seki Takakazuand the old Japanese mathematics” in the Shinjuku Historical Museum;

• On March 9, 2008, the 10th commendation ceremony of the Sangaku (mathemat-ical tablet) contest and lectures on old Japanese mathematics in the Edo-TokyoMuseum;

• From March 6 to April 10, 2008 (the first period), a special exhibition entitled“Seki Takakazu and the old Japanese mathematics” in the Museum of Science,Tokyo University of Science;

• From April 15 to May 18, 2008, a special exhibition entitled “Seki Takakazu andthe old Japanese mathematics” in the sea adventure pavilion, Osaka MaritimeMuseum;

• From April 26 to June 8, 2008, a special exhibition entitled “Seki Takakazu andthe old Japanese mathematics” in the Ichinoseki City Museum;

• From August 21 to November 3, 2008 (the second period), a special exhibitionentitled “Seki Takakazu and the old Japanese mathematics” in the Museum ofScience, Tokyo University of Science;

• Repair of the tomb and the monument of Seki Takakazu at Jorinji temple;• From November 22, 2008, to January 12, 2009, the 7th exhibition on Japanese

scientists and technicians entitled “Japanese Pioneers in Mathematics” in theJapan Gallery, National Museum of Nature and Science, Ueno, Tokyo;

• On December 6, 2008, a commemorative Buddhist service at Jorinji temple.

We are proud to report that all of these activities have been successfully carriedout and we are grateful to all people concerned for their heartfelt support and coop-eration.

Tokyo, October 2011 Hideyuki MajimaChairman of the Executive Committee of

Seki’s 300th Memorial

Preface

Seki Takakazu (1642?–1708) was a mathematician of the Edo period (1603–1868)of Japan who made outstanding contributions to the mathematics of the world —the first time for a Japanese to do so. The Mathematical Society of Japan and theHistory of Mathematics Society of Japan hosted the International Conference on theHistory of Mathematics in Commemoration of the 300th Posthumous Anniversaryof Seki Takakazu August 25–31, 2008, on the Kagurazaka Campus of the TokyoUniversity of Science in Tokyo. This book comprises the proceedings of the con-ference supplemented with collated texts of five of Seki’s writings and reprints ofpapers on these texts.

In ancient Japan, funerals lasted a very long time. It often took more than twoyears before the body was finally placed in the tomb. This period was called mog-ari [殯], during which relatives and others regularly met in front of the coffin andrecalled with tears [誄 shinobigoto wo su] the deeds and wishes of the departed.During these ceremonies the story of the deceased person was fixed and was laterrecorded as his history.

This custom was lost after the introduction of Buddhism and cremation in theeighth century. However, even today Buddhist families in Japan commonly con-tinue to have anniversaries 1, 3, 7, 13, 17, 23, and 33 years after a person’s death.The deceased cannot sleep quietly but are remembered regularly. For a great personsimilar ceremonies are held again every 50 or 100 years.

In the case of Seki Takakazu, the commissioner of finance [勘定奉行] in theShogunate presided over the 100th posthumous anniversary. Some years earlier,Honda Toshiaki [本多利明] and other mathematicians of Seki’s School erected amemorial epitaph beside the tomb. The 200th posthumous anniversary was held bythe Tokyo Mathematico-Physical Society on April 6, 1907. They held a memorialconference for citizens and published its proceedings and the reprint of the Septe-nary [七部書] of Seki’s advanced-level papers.

The purpose of our conference on this 300th posthumous anniversary is not dif-ferent from that of the preceding two.

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xiv Preface

Mencius [孟子] (ca. 372–289 B.C.) is regarded by many Confucians as the mostimportant successor of Confucius [孔子] (551–479 B.C.). He left the words: “Thelegacy of a great person is suddenly lost after five generations. The same is true of aminor person. I have never been able to be a student of Confucius. I secretly improvemyself by learning from the learned people.” [孟子曰君子之澤五世而斬小人之澤五世而斬予未得為孔子徒也予私淑諸人也].

The expression “secretly improve myself” [私淑] was understood by Japanese inthe old days to mean choosing a person, not necessarily a living person, as one’steacher and learn from the teacher’s writings.

The theories that Seki and his pupils established are far more advanced than waslong supposed. They applied those theories to various problems in geometry and soon, and left many manuscripts in classical Chinese, which was an international lan-guage at that time, but few of them had readers for these 300 years. The theoreticalparts, some of which were discovered 80 years before Europeans did, are readablebut their formula calculations in problem solving are almost beyond human abili-ties. The recent progress in computers and their usage makes it practical to continuetheir investigations. We hope that the reader will improve himself by reading theseproceedings.

The Conference was organized by the Organizing Committee consisting of Pro-fessor Henk Bos (Utrecht, the Netherlands), Professor Karine Chemla (CNRS,France), Professor Annick Horiuchi (Paris 7, France), Professor Hideki Kawahara(Tokyo, Japan), Professor Eberhard Knobloch (T. U. Berlin, Germany), ProfessorHikosaburo Komatsu (Tokyo, Japan), Professor Liu Dun (CAS, China) and Profes-sor Michio Yano (Kyoto Sangyo University, Japan), and by the Local Committeeconsisting of Mr. Ken’ichi Sato (President of the History of Mathematics Societyof Japan), Professor Katsuhiko Shimizu (Tokyo Universuty of Science, Japan), Pro-fessor Keitaro Sekine (Director of the Science Museum of TUS, Japan), and Mr.Kazuhiko Masuda (Science Museum TUS).

All speakers were invited by either of these Committees.

The Conference received the financial or moral support of the following organi-zations: The Tokyo Club; Inoue Foundation for Science; Japan–China Science andTechnology Exchange Association; Tokyo University of Science; The Ministry ofEducation, Culture, Sports, Science and Technology, Grant-in-aid for Scientific Re-search on Priority Areas 17083006; Japan Society for the Promotion of Science,Grant-in-Aid for Scientific Research (C) 20540107.

We would like to thank them for their generous assistance.

Tokyo,July, 2010

Hikosaburo Komatsu

Contents

Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Part I Contributed papers

Seki Takakazu, His Life and Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Hideyuki Majima

Some Reflections on Main Lines of Mathematical Development . . . . . . . . . 21Wenlin Li

Babylonian Number Theory and Trigonometric Functions:Trigonometric Table and Pythagorean Triples in the MathematicalTablet Plimpton 322 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Kazuo Muroi

Archimedes in China: Archimedes and His Works in Chinese Literatureof the Ming and Qing Dynasties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Liu Dun

The Nine Chapters on the Mathematical Procedures and Liu Hui’sMathematical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Guo Shuchun

On the Alternative Algorithm of the 7th Problem in the Sea IslandMathematical Canon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Hideki Kawahara

A Comparative Study on Traditional Mathematics of Korea and Japan . . 93Kim Yong-Woon

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xvi Contents

The Axes of Mathematical Methodology in the Song and YuanDynasties: The Construction of Mathematical Models . . . . . . . . . . . . . . . . . 105Guo Shirong

The Suanxue Qimeng and Its Influence on Japanese Mathematics . . . . . . 119Mitsuo Morimoto

Power Series Expansions in India Around A. D. 1400 . . . . . . . . . . . . . . . . . 133Setsuro Ikeyama

An Early Japanese Work on Chinese Mathematics in Vietnam: YoshioMikami’s Study of the Vietnamese Mathematical Treatise Chi Minh ToanPhap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Alexei Volkov

The Jinkoki of Yoshida Mitsuyoshi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Ken’ichi Sato

Resume of Works on Mathematics of Seki Takakazu . . . . . . . . . . . . . . . . . . 187Osamu Takenouchi

Seki Takakazu’s Measuring Process of the Volume of Solids Derivedfrom Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Toshio Sugimoto

Seki Takakazu’s Method on the Remainder Problem . . . . . . . . . . . . . . . . . . 207Sumie Tanabe

Seki Takakazu’s Method of Calculating the Volume of Solids ofRevolution and His Mathematical Object . . . . . . . . . . . . . . . . . . . . . . . . . . . 219Fumiaki Ozaki

Leibniz’s Theory of Elimination and Determinants . . . . . . . . . . . . . . . . . . . 229Eberhard Knobloch

Algebra, Elimination and the Complete Book of Mathematics . . . . . . . . . . 245Hikosaburo Komatsu

Some Questions and Observations Around the Mathematics of SekiTakakazu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275Silke Wimmer-Zagier and Don Zagier

Ming Antu and His Power Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . 299Luo Jianjin

Standing on the Shoulders of the Giant Influence of Seki Takakazu onTakebe Katahiro’s Mathematical Achievements . . . . . . . . . . . . . . . . . . . . . . 311Xu Zelin and Zhou Chang

Contents xvii

Takebe Katahiro’s Algorithms for Finding the Circular Arc Length . . . . . 331Mitsuo Morimoto

The Method of Successive Divisions by Takebe Katahiro and NakaneGenkei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343Tamotsu Tsuchikura

Manuscripts in the Edo Period: Preliminary Study on ManuscriptsWritten by Seki Takakazu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353Tomomi Nishida

Influence of European Mathematics on Pre-Meiji Japan . . . . . . . . . . . . . . . 357Tatsuhiko Kobayashi

On Contemporary Mathematics in Vietnam . . . . . . . . . . . . . . . . . . . . . . . . . 375Ha Huy Khoai

Part II Supplements

Notes on Complete Book of Mathematics Vol. 4: Three Essentials . . . . . . . 387Mitsuo Morimoto

Complete Book of Mathematics Vol. 4: Three Essentials, by SekiTakakazu, Takebe Kataakira and Takebe Katahiro, collated by FumiakiOzaki and Hikosaburo Komatsu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393Seki Takakazu, Takebe Kataakira and Takebe Katahiro

Seki’s Trilogy: Methods of Solving Explicit Problems, Methods ofSolving Implicit Problems and Methods of Solving Concealed Problems . . 437Hikosaburo Komatsu

Methods of Solving Explicit Problems, by Seki Takakazu, collated byHikosaburo Komatsu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441Seki Takakazu

Methods of Solving Implicit Problems, by Seki Takakazu, collated byHikosaburo Komatsu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457Seki Takakazu

Methods of Solving Concealed Problems, by Seki Takakazu, collated byHikosaburo Komatsu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469Seki Takakazu

Notes on Complete Book of Mathematics Vol. 10: Geometry . . . . . . . . . . . . 495Hikosaburo Komatsu

xviii Contents

Complete Book of Mathematics Vol. 10: Geometry, by Seki Takakazu,Takebe Kataakira and Takebe Katahiro, collated by HikosaburoKomatsu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499Seki Takakazu, Takebe Kataakira and Takebe Katahiro

Seki’s Theory of Elimination as Compared with the Others’ . . . . . . . . . . . . 553Takefumi Goto and Hikosaburo Komatsu

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

Program

August 25, 2008 (Monday)

9:00–12:00, 12:30–17:00 Registration

Introductory Lectures in Japanese

13:00–13:50佐藤健一 Sato Ken’ichi吉田光由の『塵劫記』の紹介 (On Yoshida Mitsuyoshi’s “Jinkoki”)

14:00–14:50川原秀城 Kawahara Hideki九章算術 (The Nine Chapters of Mathematical Procedures)

15:00–15:50森本光生Morimoto Mitsuo算学啓蒙について (On Zhu Shijie’s “Suanxue Qimeng”)

Lecture in Japanese

16:00–16:50金容雲 Kim Yong Woon韓日傳統数学の歴史と創造性の比較 (A comparative study on traditional math-ematics of Korea and Japan)

18:30Snack

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xx

August 26

9:00–12:00, 12:30–17:00 Registration

9:30–10:20Opening Ceremony

Address by Takeuchi Shin (竹内伸)President of Tokyo University of Science

Address by Komatsu Hikosaburo (小松彦三郎)Representative of the Organizing Committee of International Conference

Address by Majima Hideyuki (真島秀行)Director of the Executive Committee of Seki’s 300th Memorial

10:30–11:20Majima Hideyuki真島秀行Seki Takakazu, his life and bibliography

11:30–12:20Jochi Shigeru城地茂Seki Takakazu’s biography and “Kanjo-gata Wasan Era” (1674–1780)

14:00–14:50Eberhard KnoblochLeibniz’ theory of elimination and determinants

15:00–15:50Komatsu Hikosaburo小松彦三郎Algebra, elimination theory and “Complete Books of Mathematics大成算經”

16:00–16:50Matsumoto Takao松本尭生“Sanpo-Hakki 算法発揮,” the first published book on determinants, and the re-lated works by Seki and others

August 27

9:00–12:00, 12:30–17:00 Registration

9:30–10:20Karine ChemlaRemarks on the polynomial algebra of the Song-Yuan dynasties and its continu-ation in Edo Japan

Program

Program xxi

10:30–11:20Guo Shuchun郭書春A study on “The Nine Chapters of Mathematical Procedures” and Liu Hui’smathematical theory

11:30–12:20Muroi Kazuo室井和男Babylonian number theory and trigonometric functions: Trigonometric table andPythagorean triples in the mathematical tablet Plimpton 322

14:00–14:50Guo Shirong郭世栄Mathematics in the Song and Yuan Dynasties, “Axis of Mathematical Methods”

15:00–15:50Takenouchi Osamu竹之内脩“Katsuyo Sanpo括要算法”, resume of works on mathematics of Seki Takakazu

16:00–16:50Sugimoto Toshio杉本敏夫Seki Takakazu’s measuring process of the volume of solids derived from spheres

August 28

9:00–12:00, 12:30–17:00 Registration

9:30–10:20Liu Dun劉鈍Archimedes in China — Archimedes and his works in Chinese literature of theMing and Qing dynasties

10:30–11:20Kobayashi Tatsuhiko小林龍彦Influence of European mathematics on pre-Meiji Japan

11:30–12:20Feng Lisheng馮立昇Mathematics exchanges between China and Japan in modern times (1850’s–1920’s)

14:00–14:25Fumiaki Ozaki尾崎文秋Takakazu Seki’s method of calculating the volume of solids of revolution

15:00–17:00Excursion to Jorinji temple淨輪寺, where Seki is buried, and Eisei Bunko永青文庫

18:30–20:00Banquet at Bettei torijaya別亭鳥茶屋

xxii

August 29

9:00–12:00, 12:30–17:00 Registration

9:30–10:20Li Wenlin李文林Some reflections on “main lines” of mathematical development

10:30–11:20Luo Jianjin羅見今Ming Antu and his power series expansions

11:30–12:20Ikeyama Setsuro池山説郎Power series expansions in India around 1400

14:00–14:50Ogawa Tsukane小川束Theories of circles originated by Seki and Takebe Katahiro

15:00–15:50Morimoto Mitsuo森本光生Takebe Katahiro’s algorithms to find the circular arc length

16:00–16:25Yokotsuka Hiroyuki横塚啓之“Kohai-Setsuyaku-Shu弧背截約集” regarded as a work of Takebe Katahiro

16:30–16:55Don ZagierThe mathematics of and around Seki Takakazu from a modern mathematician’sviewpoint

August 30

9:00–12:00, 12:30–17:00 Registration

9:30–10:20Alexei K. VolkovVietnamese mathematics and mathematics education: adaptation or invention?

10:30–11:20Ha Huy KhoaiOn the history of mathematics in Vietnam

11:30–12:20Xu Zelin徐澤林Standing on the shoulders of a giant — Influence of Seki Takakazu on TakebeKatahiro’s mathematical achievement

14:00–14:25Free Discussion

Program

Program xxiii

14:30–14:55Tsuchikura Tamotsu土倉保“Ruiyaku-Jutsu累約術,” the method of successive divisions

15:00–15:50Qu Anjing曲安京How did Chinese deal with a scientific problem? — A case study of the solareclipse theory

16:00–16:25Ohashi Yukio大橋由紀夫Mathematical astronomy of Seki Takakazu關孝和 and Shibukawa Harumi澁川春海— Understanding and overcoming the Chinese traditional calendars

16:30–17:00Closing Ceremony

August 31

9:00–12:30 Registration

Lectures in Japanese

9:30–9:55西田知己 Nishida Tomomi江戸時代の写本・稿本—関孝和の稿本研究の予備的考察— (Manuscripts inthe Edo period)

10:00–10:25田辺寿美枝 Tanabe Sumie翦管術 (“Senkan-Jutsu”, Seki Takakazu’s method on the remainder problems)

10:30–10:55藤井康生 Fujii Yasuo奇零方垜と 1/sinxの展開について (The sum of the power of odd numbers andthe development of 1/sinx)

xxiv

List of Contributors

Karine ChemlaUniversite Paris 7 Paris Diderot& CNRS, REHSEIS UMR 7219,Laboratoire de Philosophieet d’Histoire des Sciences,Case 7064, 5 rue Thomas Mann,75205 PARIS CEDEX 13, Francee-mail: chemla@univ-paris-diderot.fr

Feng LishengInstitute for History of Scienceand Technology and Ancient Texts,Tsinghua University,113 Tsinghua University Library,Beijing 100084, P. R. Chinae-mail: fls@tsinghua.edu.cn

Yasuo FujiiTakarazuka Higashi High School,1-12-1, Nakayama Satsukidai,Takarazuka 665-0871, Japanhome address: 2-6-17, Funahara,Itami 664-0896, Japane-mail: yfujii@pearl.ocn.ne.jp

Guo ShirongCollege of the History of Science,Inner Mongolia Normal University,No.81, Road Zhaowuda, Huhhot,Inner Mongolia 010022, P. R. Chinae-mail: gsr@imnu.edu.cn

Guo ShuchunInstitute for the History of NaturalScience, 1-905# Building One,Xin Ke Xian Yuan, Ke Xue YuanNan Lu, Zhong Guan Cun,Beijing 100190, P. R. Chinae-mail: shuchunguo@gmail.com

Setsuro IkeyamaFaculty of Cultural Studies,Kyoto Sangyo University,Kamikamo-motoyama, Kita-ku,Kyoto 603-8555, Japane-mail: QWN11071@nifty.com

Shigeru JochiNational Kaohsiung First University ofScience and Technology,1, University Road, Yenchao,Kaohsiung, Taiwan, R. O. Chinae-mail: jochi@cc.osaka-kyoiku.ac.jp

Hideki KawaharaGraduate School of Humanities andSociology, University of Tokyo,7-3-1, Hongo, Bunkyo-ku,Tokyo 113-0033, Japane-mail: kawahara@1.u-tokyo.ac.jp

xxv

Ha Huy KhoaiInstitute of Mathematics, 18 HoangQuoc Viet., Hanoi 10307, Vietname-mail: hhkhoai@math.ac.vn

Kim Yong WoonHanyang University,17 Haengdang-dong, Seong dong-gu,Seoul 133-791, Koreae-mail: kyw06@hotmail.com

Eberhard KnoblochTechnische Universitat Berlin,Strasse des 17 Juni, 135,Berlin 10623, Germanye-mail: Eberhard.Knobloch@tu-Berlin.de

Tatsuhiko KobayashiMaebashi Institute of Technology,460-1, Kamisadori-machi, Maebashi,Gunma 371-0816, Japane-mail: koba@maebashi-it.ac.jp

Hikosaburo KomatsuDepartment of Mathematics,Faculty of Science,Tokyo University of Science,Kagurazaka 1-3, Shinjuku-ku,Tokyo 162-8601, Japane-mail: komatsu@ms.u-tokyo.ac.jp

Wenlin LiAcademy of Mathematics andSystem Science,55 Zhongguancun East Road,Beijing 100190, P. R. Chinae-mail: wli@math.ac.cn

Liu DunInstitute for the History of NaturalScience, 1-905# Building One,Xin Ke Xiang Yuan, Ke Xue YuanNan Lu, Zhong Guan Cun,Beijing 100190, P. R. Chinae-mail: liudun@ustc.edu.cn

Luo JianjinInstitute of the History ofNatural Science, Inner MongoliaNormal University,No 81, Road Zhaowuda, Huhhot,Inner Mongolia 010022, P.R. Chinae-mail: lluojj@163.com

Majima HideyukiOchanomizu University,2-1-1, Otsuka, Bunkyo-ku,Tokyo 112-8610, Japane-mail: majima.hideyuki@ocha.ac.jp

Takao MatumotoDepartment of Mathematics,Graduate School of Science,Hiroshima University,1-3-1, Kagamiyama,Higashi-Hiroshima 739-8526, Japane-mail: matumoto@math.sci.hiroshima-u.ac.jp

Morimoto MitsuoSophia University, 7-1, Kioicho,Chiyoda-ku, Tokyo 102-0094, Japane-mail: mormormor@jcom.home.ne.jp

Kazuo MuroiKawai-juku, Bunri,2-12-12, Honcho, Aoba-ku,Sendai 980-8405, Japane-mail: edubakazuo@ac.auone-net.jp

Tomomi Nishida3-7-10 Mama, Ichikawa,Chiba 272-0826, Japane-mail: sancan-cufu@goo.jp

Tsukane OgawaFaculty of Environmental InformationSciences, Yokkaichi University,Yokkaichi 512-8512, Japane-mail: ogawa@yokkaichi-u.ac.jp

List of Contributors

xxvi

Ohashi Yukio3-5-26, Hiroo, Shibuya-ku,Tokyo 150-0012, Japane-mail: yukio-ohashi@dk.pdx.ne.jp

Fumiaki OzakiFaculty of Mathematics,Kyushu University,6-10-1, Hakozaki, Higashi-ku,Fukuoka 819-8581, Japane-mail: f-ozaki@math.kyushu-u.ac.jp

Qu AnjingDepartment of Mathematics,Northwest University,Xian 710069, P. R. Chinae-mail: qaj@nwu.edu.cn

Sato Ken’ichiWasan Institute,48-23, Sakae-cho, Kita-ku,Tokyo 114-0005, Japane-mail: kensatoh@tt.rim.or.jp

Sugimoto ToshioJapan Women’s University,1-1-1, Nishi-ikuta, Tama-ku,Kawasaki 214-8565, Japan

Takenouchi OsamuOsaka University,1–3, Machikaneyama, Toyonaka,Osaka 560-8531, Japane-mail: osamu-ta@mxs.mesh.ne.jp

Tanabe SumieSacred Heart Senior High School,4-11-1, Shirokane, Minato-ku,Tokyo 108-0072, Japane-mail: tstanabe@da.airnet.ne.jp

Tamotsu TsuchikuraTohoku University,6-3 Aramaki, Aoba-ku,Sendai 980-8578, Japane-mail: o-tsuchikura@nifty.com

Alexei K. VolkovCenter for General Education andInstitute of History,National Tsing-Hua University,101, Sect. 2, Kuang-Fu Road,Hsinchu 30013, Taiwan, R. O. Chinae-mail: alexei.volkov@gmail.com

Xu ZelinCollege of Mathematics,Tianjin Normal University,Tianjin 300387, P. R. Chinae-mail: zelinxujp@yahoo.co.jp

Hiroyuki YokotsukaGraduate School of Mathematics andScience Education,Tokyo University of Science,1-3, Kagurazaka, Shinjuku-ku,Tokyo 162-8601, Japane-mail: BYS01005@nifty.ne.jp

Don Zagier and Silke Wimmer ZagierMax-Planck-Institut fur Mathematik,Vivatsgasse 7, Bonn 53111, Germanye-mail: don.zagier@mpim-bonn.mpg.de

List of Contributors