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Seki, Founder of Modern Mathematics in Japan

Eberhard Knobloch, Hikosaburo Komatsu, and Dun Liu, editors
Publication Date: 
Number of Pages: 
Springer Proceedings in Mathematics and Statistics 39
[Reviewed by
Toke Knudsen
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This book is the proceedings from the International Conference on the History of Mathematics in Memory of Seki Takakazu, which was held in Japan on August 25–31, 2008. The conference was one of many events to celebrate the 300th anniversary of Seki’s death. The foreword of the book by Hideyuki Majima provides a list of all the celebratory events, of which the conference was the main one.

Seki Takakazu was a Japanese mathematician, who lived during the Edo period. He was born between 1640 and 1645, and died in 1708 (for the date of his birth, see the paper “Seki Takakazu, His Life and Bibliography” by Hideyuki Majima, found in the book). During the Edo period, the study of mathematics and other sciences was encouraged.

Seki was a talented mathematician, who made significant contributions to the field of mathematics. He is perhaps best known for his early study of what we today call determinants to solve simultaneous linear equations, but other contributions of his include Bernoulli numbers, the Chinese remainder theorem, and regular polygons.

The book is divided into two parts: Part I contains 26 contributed papers from the conference and Part II contains seven supplements, primarily source material or notes on source material.

The papers of Part I are diverse. A good number of them are devoted to Seki and his works, but most deal with other topics, including Mesopotamian mathematics, Indian mathematics, and contemporary mathematics in Vietnam. Together the articles on Seki provide an excellent overview his life and mathematical contributions, as well as studies of technical details pertaining thereto. A reader interested in the life and mathematical works of Seki will find an abundance of information here.

In addition to papers on Seki, Part I of the book has interesting and worthwhile papers on other topics. The paper “Power Series Expansions in India Around A. D. 1400” by Setsuro Ikeyama studies how the Indian mathematician Mādhava (fl. 1400 CE) derived certain power series expansions; the paper “Leibniz’s Theory of Elimination and Determinants” by Eberhard Knobloch summarizes the main results of Leibniz on determinants; and the paper “On Contemporary Mathematics in Vietnam” by Ha Huy Khoai provides an interesting survey of the development of mathematics in Vietnam since 1947.

The supplements of Part II are all either the original texts of Seki’s works or notes on these. Having this primary source material available in the book is, of course, especially valuable to specialists who can read Japanese, but it is overall a strength of the book.

While the book contains valuable papers, it is disappointing that the papers were published without a proper introduction or commentary. A general introduction commenting on and synthesizing the contents of the various papers would have been helpful to both the specialist and the casual reader. This is especially so for the contributed papers on Seki and his mathematical contributions. An introduction summarizing all the contributions would have added significantly to the book. The book does, however, contain an index, which is useful to the reader.

Overall, this is a valuable book with good contributions to the field of history of mathematics. Specialists interested in the life and mathematical contributions of Seki will of course benefit from it, but interested readers will benefit from it also.

Toke Knudsen is Associate Professor of Mathematics at the State University of New York at Oneonta. He is the author of The Siddhāntasundara of Jñānarāja: An English Translation with Commentary.

Part I Contributed papers

Seki Takakazu, His Life and Bibliography
Some Reflections on Main Lines of Mathematical Development
Babylonian Number Theory and Trigonometric Functions: Trigonometric Table and Pythagorean Triples in the Mathematical Tablet Plimpton 322
Archimedes in China: Archimedes and His Works in Chinese Literature of the Ming and Qing Dynasties
The Nine Chapters on the Mathematical Procedures and Liu Hui’s Mathematical Theory
On the Alternative Algorithm of the 7th Problem in the Sea Island Mathematical Canon
A Comparative Study on Traditional Mathematics of Korea and Japan
The Axes of Mathematical Methodology in the Song and Yuan Dynasties: The Construction of Mathematical Models
The Suanxue Qimeng and Its Influence on Japanese Mathematics
Power Series Expansions in India Around A. D. 1400
An Early Japanese Work on Chinese Mathematics in Vietnam: Yoshio Mikami’s Study of the Vietnamese Mathematical Treatise Chi Minh Toan Phap
The Jinkōki of Yoshida Mitsuyoshi
Résumé of Works on Mathematics of Seki Takakazu
Seki Takakazu’s Measuring Process of the Volume of Solids Derived from Spheres
Seki Takakazu’s Method on the Remainder Problem
Seki Takakazu’s Method of Calculating the Volume of Solids of Revolution and His Mathematical Object
Leibniz’s Theory of Elimination and Determinants
Algebra, Elimination and the Complete Book of Mathematics
Some Questions and Observations Around the Mathematics of Seki Takakazu
Ming Antu and His Power Series Expansions
Standing on the Shoulders of the Giant Influence of Seki Takakazu on Takebe Katahiro’s Mathematical Achievements
Takebe Katahiro’s Algorithms for Finding the Circular Arc Length
The Method of Successive Divisions by Takebe Katahiro and Nakane Genkei
Manuscripts in the Edo Period: Preliminary Study on Manuscripts Written by Seki Takakazu
Influence of European Mathematics on Pre-Meiji Japan
On Contemporary Mathematics in Vietnam

Part II Supplements

Notes on Complete Book of Mathematics Vol. 4: Three Essentials
Complete Book of Mathematics Vol. 4: Three Essentials, by Seki Takakazu, Takebe Kataakira and Takebe Katahiro, collated by Fumiaki Ozaki and Hikosaburo Komatsu
Seki’s Trilogy: Methods of Solving Explicit Problems, Methods of Solving Implicit Problems and Methods of Solving Concealed Problems
Methods of Solving Explicit Problems, by Seki Takakazu, collated by Hikosaburo Komatsu
Methods of Solving Implicit Problems, by Seki Takakazu, collated by Hikosaburo Komatsu
Methods of Solving Concealed Problems, by Seki Takakazu, collated by Hikosaburo Komatsu
Notes on Complete Book of Mathematics Vol. 10: Geometry
Complete Book of Mathematics Vol. 10: Geometry, by Seki Takakazu, Takebe Kataakira and Takebe Katahiro, collated by Hikosaburo Komatsu
Seki’s Theory of Elimination as Compared with the Others’