Takakazu Seki (関 孝和)

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History of Mathematics Biographical Paper

Takakazu Seki

(関 孝和)

By: Reiji Kukihara


Takakazu Seki is the most famous Japanese mathematician. His mathematical accomplishments were great and Mikami (1913) called him the “Japanese Newton.” However, his life is not well known for his major works. Therefore I examined the historical background of his period and examined why the Japanese Kanji system mathematics could be improved so much by Seki while the Chinese Kanji System mathematics was declining.

General historical background of Seki’s period

The period that Takakazu Seki lived in is called the Edo period which was ruled by the Edo shogunate. Before the Edo period, Japan had been in the Warring State Period for over 100 years. There were so many daimyo (Japanese feudal lord) in Japan, and they contended with each other for supremacy of Japan. Finally, Ieyasu Tokugawa who was a daimyo of Edo, that was a part of eastern Japan (the present Tokyo area) succeeded in national unification and established the Edo shogunate in 1603. The shogunate is a government system and the head is a shogun. Essentially, “shogun” means a military leader of samurai in Japan and it later was switched to mean the leader of Japan because Japan was ruled by samurai at that time. The Edo shogunate lasted for over 250 years until 1867 and it was a peaceful time except for the first couple of decades (Meyer, 1993). This peaceful age urged developments of science and learning, and mathematics was no exception.
Mathematic historical background of Seki’s period

Traditional Japanese mathematics is included in the Kanji system of mathematics (Wang, 1999). The Kanji system of mathematics is defined by Wang (1999) so as to avoid confusion with the eastern Asia mathematic culture and other Asian mathematic cultures such as the Indian mathematic culture. China, Japan, Korea and Vietnam are included in the Kanji system of mathematics. The first Kanji system of mathematic that was imported into Japan is the Chiu-chang through Korea in the 6th~7th centuries when Buddhism and the calendar were imported (Kotera, 1996; Meyer 1993). Basically the Japanese Kanji system of mathematics had not developed by itself in Japan at that time. It had depended on imports of the Chinese Kanji system of mathematics until the 17th century. Previously Japanese Kanji system of mathematics had been practical mathematics until the 16th century. Theoretical mathematics was stagnant from the 7th Century to 12th century in China (Makino, 1998). The t’ien yuan shu algebra (天元術) that provided the notion of algebra to the Kanji system mathematics was invented in the 12th century (Libbrecht, 1973). However, t’ien yuan shu algebra was not imported to Japan soon after its invention. The most influential book about t’ien yuan shu algebra for Japan is Chu Shi-chieh’s the Suan-hsueh ch’I-meng (算学啓蒙, Introduction to mathematical studies) (Libbrecht, 1973). Although it was published in 1299, it was imported to Japan through Korea at the end of 16th century while t’ien yuan shu algebra perished in China in the 13th ~17th centuries because of the intellectual effervescence policy of the Ming Dynasty (1368-1644) that ruled China at that time (Kotera, 1996; Makino, 1998; Martzloff 1997). On the other hand, the practical Kanji system of mathematics had survived. Especially abacus calculation developed well and many books about abacus calculation were published in the 14th ~16th centuries (Makino. 1998). The Suan-fa t’ung-tsng (算学統宗, Systematic treatise on arithmetic), that was written by Ch’eng Ta-wei in 1593, was the most influential book of all. This book was imported to Japan soon after publication, and abacus calculation was popularized very quickly throughout Japan (Mikami, 1947).

There were also some imports of western mathematics from the middle of 16th to the early 17th century. In 1543, Portuguese sailors voyaging along the south China coast were blown far off course to the shores of Tanegashima, a small island off the southern coast of southern Japan and brought firearms to Japan (Meyer, 1993). This was the first direct contact with Europeans. Then the first missionary St. Francisco de Xavier reached Japan in 1549 and Japan started trading with European countries. At first, missionaries asked each daimyo for protection and permission to conduct missionary work in exchange for priority trading rights. Many daimyo accepted this, and missionaries made many churches and taught not only Christianity but also European derived science including mathematics. Missionaries worked pretty well and sometimes they succeeded in converting some daimyo into Christians. However, after the establishment of the Edo shogunate in 1603, the shogunate had been afraid that Christianity which places Jesus Christ on top would shake the feudalism that has shogun on top. They also saw that the cultivation of Christianity was the colonization strategy of European countries. Therefore the Edo shogunate issued a prohibition against Christianity in 1612 and started the persecution of Christians and gradual national isolation. Then the outbreak of the biggest revolt by Japanese Christians (the revolt of Shimabara) in 1637 became a strong decisive factor and the shogunate completed its national isolation policy (Meyer, 1993). The shogunate drove out all Europeans except for the Dutch who promised that they would never bring missionaries. They killed the remaining missionaries and Christians. Studying of all western books including non-Christian books was prohibited, and books were burned. The Dutch were gathered in Nagasaki that is in western part of Japan, and allowed to trade only with the Edo shogunate. Therefore the way to get western knowledge for civilians was completely lost. Of course the western knowledge that was already taught was also lost because people who had it could not write or teach their knowledge, since if it emerged they would be punished or killed. Finally, imported western mathematic knowledge and books were almost completely lost.

Thus, the mathematical background of Seki’s period is characterized by peace enough to study and mathematic textbooks limited to only Kanji system of mathematics. This background enabled the Japanese Kanji system of mathematics to develop originally while the original Chinese Kanji system of mathematics was perishing in the face of the advance of western mathematics.

There were some notable Japanese mathematicians before Seki in the early Edo period. The growing tendency of the Japanese Kanji system mathematics occurred as a result of imports of Chinese textbooks in the 16th century, especially the Suan-fa t’ung-tsng (算学統宗, Systematic treatise on arithmetic) that is about the Chinese abacus and the Suan-hsueh ch’I-meng (算学啓蒙, Introduction to mathematical studies) that is about the t’ien yuan shu algebra as I mentioned above (Mikami, 1947). The first Japanese mathematician noted in history is Shigetaka (Kambei) Mouri who is considered to have brought the Chinese abacus suan-pan, which the Japanese call soroban from China for the first time (Mikami, 1913). Although his books were lost, Mitsuyoshi (Koyu) Yoshida who was one of his students wrote the Jinkoki based on the Suan-fa t’ung-tsng in 1627. This was a quite successful publication. After this publication, the soroban arithmetic experienced rapid progress. Thanks to this book, Japanese people’s ability in calculation and their level of mathematical education achieved a much higher standard than before (Wang, 1999). Because the book was so popular, if someone spoke of a math textbook, it meant the Jinkoki at that time (Mikami, 1947). However, popularization of soroban was not the only accomplishment of the Jinkoki. The other accomplishment of the Jinkoki is that it became a trigger of the Idai-keishou (遺題継承, succession of remaining questions) (Wang, 1999). After the publish of Jinkoki, some more good mathematical books were published such as the Kengairoku (1639) and the Inkisanka (1640) both by Chisho Imamura who was also one of Mouri’s student. Yoshida, who was shocked by these works, tried to make a new Jinkoki and published it in 1641 (Wang, 1999). In this book, he added 12 problems without questions for his readers. One mathematician Kittoku Yoshimura solved these questions in his book Ketugisho and added some other questions in 1660. These were in turn solved in some succeeding publications that were printed a few years later. For mathematicians who wanted to show their mathematical skills, the Idai-keishou was repeated again and again, and questions became more difficult. This helped the development of the Japanese Kanji system of mathematics to a great degree. For example, Yoshida asked about π in his 10th remaining question and the answer in Shigekiyo Muramatsu’s the Sanso (1663) was 3.141592648777698869248 (Wang, 1999). In the course of time, another mathematical trend emerged from the Idai-keishou. It was the Sangaku-hounou (算額奉納, dedication of mathematics tablets) (Kotera, 1996; Wang, 1999). The problem of the Idai-keishou was that publishing was essential. It was very expensive and difficult to publish books for even famous mathematicians. People who wanted to publish mathematical books had to solve so many questions. Therefore nameless and poor mathematicians who could not publish books settled on the method of dedicating tablets in shrines. There has been a custom that people write their wishes on tablets and put them on boards and trees to dedicate at shrines even now in Japan. They decided to write their answers on tablets instead of their wishes. Those tablets with mathematical answers are called Sangaku (算額). It was cheap to make Sangaku and they could very easily be shown to many people who came to shrines. Actually, the Sangaku-hounou worked as journals for announcements. Shrines were very useful places for exchanges of mathematicians, especially for those who lived locally. Sometimes Sangaku were carried by tourists and dedicated in local shrines which helped mathematical knowledge to spread all over Japan. Later not only answers but also new questions (Idai; 遺題) could be seen on Sangaku as Idai-keishou, those Sangaku were called Sangaku-Idai (算額遺題; Wang, 1999). Not only special mathematicians but also normal people enjoyed these two mathematical trends. They enjoyed mathematics as their hobby just like flower arrangement or Japanese chess. Wang (1999) pointed out that this joy enabled the Japanese Kanji system of mathematics to be independent from the Chinese Kanji system mathematics that valued practicality. In 1658, the Suan-hsueh ch’I-meng that was imported earlier was reprinted and this time the t’ien yuan shu algebra was popularized in Japan. The Japanese Kanji system mathematics was reaching a transition time from the practical arithmetic period to the theoretical algebraic period. Takakazu Seki appeared in this dawn of the theoretical algebraic period.

Life of Takakazu Seki

Despite his great works, Seki’s life is not known so well. He did not leave any diaries and there are no notable anecdotes that tell us of his personality. He is considered to have been born around 1637~44. Although sometimes his birth year is set in 1642, this assumption is only a kind of desire by Japanese Mathematical historians who want to set his birth the same year as Newton’s (Wang, 1999). His correct birth place is not known either. There are two possibilities, Edo (Tokyo) and Fujioka (Gunma). He was a second son of Nagaakira Uchiyama, a samurai, but as he was adopted by the patriarch of the Seki family, he was named Takakazu Seki. Sometimes he is called Kowa Seki because of different ways of reading the Japanese characters in his name. Although his name can be called both ways, the original reading of his name is Takakazu. It is said that he had begun the study of mathematics under Yoshitane (Kisshu) Takahara who was one of the students of Mouri like Yoshida and Imamura (Kobori, 1975). However, little is known of Takahara, and there is another belief that he studied mathematics by himself (Wang, 1999). Probably, as mathematics was getting more and more popular and many mathematical books and Sangaku flooded into Japan at that time, he could study mathematics by himself. He started serving Tsunashige Tokugawa who was a daimyo of Koufu in 1676, then he served Ienobu Tokugawa who became the 6th shogun in 1709 as a chief of the Bureau of Audit after Tsunashige’s death in 1678. He was transferred to a sinecure because of old age in 1706, and died in 1708 (Kobori, 1975). Although Seki did much mathematical work in his life, he published only one book entitled the Hatubi sanpo in 1674. But this book brought him much fame and maybe his Bureau job also.

Process and contents of the Hatubi sanpo

When the Chinese Kanji system of mathematicians of the 13th ~14th centuries were studying t’ien yuan shu algebra, they thought there should be only one solution for an equation whatever its degree is. Therefore they did not come to discover the existence of other roots in an equation (Mikami, 1913). For example, when they solved x^2=4, they took 2 as the only solution and –2 was ignored. The reason for this ignorance is not demonstrated. Makino (1998) said that the Chinese traditional trend that prefers practical mathematics to theoretical mathematics can not be the reason for this because they knew negative numbers and they accepted only one solution even when equations had two positive solutions. At this time, Japanese mathematicians started realizing that the roots of an equation are not necessary a single one (Mikami, 1913). Masaoki (Seiko) Sato published the Sanpo Kongenki with 150 questions (idai) that included polynomial equations with one unknown in 1666. Then Kazuyuki Sawaguchi who is considered to be the first Japanese mathematician who understood the t’ien yuan shu algebra correctly answered Sato’s questions by the t’ien yuan shu algebra in his book the Kokon sanpoki published in 1671. Sawaguchi also added his 15 questions in his book. However, his questions were different from former ones. They were hypercomplex polynomial equations. T’ien yuan shu algebra is nor done by figures but by using sangi (counting rods). Sangi is a kind of calculator that was invented in China. Sangi looks like match sticks and are used on a special board called Sanban. Old Chinese people could do addition, subtraction, multiplication, division, square root, and polynomial equations with one unknown by t’ien yuan shu algebra by using sangi (Kotera, 1996; Fig.1). Although sangi is a good calculator, it can not be used for hypercomplex polynomial equations because of its functional restriction. Seki, who was trying to solve Sawaguchi’s questions realized that those questions could not be solved by traditional t’ien yuan shu algebra. Then he started reforming t’ien yuan shu algebra. Seki realized that the limitation of t’ien yuan shu algebra is due to using sangi, so he invented the bousho-ho (傍書法) and the hojomichisu-ho (補助未知数法). The bosho-ho means “writing besides method” and is the way to show equations by writing letters on the right of a vertical line. For example, a+b is written as |a|b and a*b is written as |ab (Wang, 1999). Seki succeeded in solving hypercomplex polynomial equations by this calculation method using figures. Although it looks natural to calculate by figure or symbols from our current perspective, the bousho-ho was a quite revolutionary invention in the Kanji system of mathematics that had not taken heed of calculation processes. Answers traditionally had depended on calculators such as sangi and soroban because original Kanji numerals are not as good as Arabic numerals for calculation with figures (Wang, 1999). The hojomichisu-ho is the way to give new letters to raised letters. For example, polynomial equations can be simplified by setting X=x^2. He answered Sawaguchi’s questions by these methods in the Hatubi sanpo. These Seki’s inventions that were extended from t’ien yuan shu algebra served as the first corner-stone of the gigantic mathematic building he accomplished in the course of his famous life (Mikami, 1913). A detailed analysis of this work was attempted a dozen years afterwards by his student Katahiro (Kenko) Takebe in 1685 and then his method became generally known to mathematicians (Kobori, 1975).
After the Hatubi sanpo

Although he would never publish another book after the Hatubi sanpo, he left many works which were not published. The most famous results in those works are the discovery of determinants and Bernoulli numbers in 1683 for the first time in the world (Mikami, 1913; O’Connor and Robertson, 1997). His works were succeeded, improved and new theories were added by his students. His school was called “Sekiryu” and it was the largest school in the Japanese Kanji system of mathematics of the Edo period. After the fall of the Edo shogunate in 1867, Japan opened trading with other countries. Then the new government decided to adopt western mathematics to import and study western science, techniques and especially tactics. Therefore the Japanese Kanji system of mathematic went out of use and declined. Although Seki’s mathematics are not used now any more, it is interesting that Seki could improve the Kanji system of mathematics and find some new mathematical notions at almost same time as western mathematicians but under a special peaceful situation when there was no western mathematics influence.


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Kobori, A., 1975, Seki, Takakazu, in Dictionary of Science Biography Vol.XII, Gillipie, C.C. eds, Charles Scribner’s, New York, pp. 290-292

Kotera, H., 1996, Sangaku, http://www.wasan.jp/english/index.html

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Mikami, Y., 1947, Japanese Mathematics in History of Culture (in Japanese), Iwanami shoten, Tokyo, 341pp.

O'Connor, J.J. and Robertson, E.F., (1997), Takakazu Seki Kowa,


Wang, C., 1999, The man who exceeded Sangi (in Japanese), Toyo shoten, Tokyo, 208pp.
The cover figure was cited from Wang (1999)

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