Seki Kōwa

Japanese mathematician and government official

• Born: March 1, 1642
• Birthplace: Fujioka, Kozuke province, Japan
• Died: October 24, 1708
• Place of death: Edo (now Tokyo), Japan

Seki Kōwa was a key figure in the development of premodern Japanese mathematics. The expansion of commerce under Tokugawa rule created a new need for practical mathematical accounting techniques. Seki and his student Takebe Katahiro were leaders in developing such techniques, and they independently duplicated some major discoveries of leading Western mathematicians, such as Gottfried Wilhelm Leibniz and Sir Isaac Newton.

Early Life

Seki Kōwa (sehk-ee koh-wo) was born in 1642, the second son of Uchiyama Shichibei. Uchiyama had been an adviser to a once-powerful domain lord who fell out of favor with the shogun ten years before Seki’s birth, and that lord was forced to commit suicide. As a result, Uchiyama’s own future prospects were in doubt. Consequently, it may have significantly benefited Seki when he was adopted at an early age by a well-placed samurai named Seki Gorozaemon.

There were a number of retainers in attendance at Gorozaemon’s mansion, and they took care of young Seki and kept him company. One day, one of these retainers was reading a book on abacus calculation that happened to attract Seki’s attention. The retainer taught Seki the basics of abacus use, and Seki then obtained an abacus and worked through all the exercises in the book on his own. From that point on, he spent most of his waking hours learning as much about mathematics as he could, from every book he could obtain. Seki is known to have studied Yoshida Mitsuyoshi’s Jinkōki (1627; Book of Large and Small Numbers , 2000), working through it page by page on his own.

It was the custom at that time for Japanese mathematicians to write books containing some mathematical problems without solutions, as a challenge for readers. One book Seki read, containing one hundred challenging problems by another brilliant young mathematics prodigy, Isomura Yoshinori, stimulated Seki to work out solutions to all one hundred problems. Throughout his career, Seki worked out original solutions to mathematics problems by Isomura and others, though he seems not to have created many such problems himself.

Seki obtained a modest official position as an accountant at the Edo mansion of Tokugawa Tsunashige, the domain lord of Kōfu, in modern Yamanashi prefecture. When Tsunashige was made the heir to the reigning shogun, Tokugawa Ietsuna, Seki rose in status along with the rest of Tsunashige’s household. In addition to his accounting work, Seki served as the mathematics tutor of the offspring of Tsunashige, his retainers, and other Edo aristocrats.

Life’s Work

The chances of Seki Kōwa’s patron Tokugawa Tsunashige becoming shogun were reduced over time as the result of court intrigues by his rivals, and this in turn lowered Seki’s own prospects for advancement in Edo. Tsunashige died in 1678, two years before Ietsuna, but this in fact turned out to enhance Seki’s prospects. Tokugawa Ienobu, Tsunashige’s son, to whom Seki had given lessons and for whom he had subsequently worked as accountant, was in a much better position at the shogun’s court than Tsunashige had been. Ienobu succeeded to the position of domain lord of Kōfu in 1678, and he was later adopted as the heir of Shogun Ietsuna’s successor, Tokugawa Tsunayoshi , in 1704. Ienobu did not actually become shogun until 1709, a year after Seki’s death, but Seki’s close relationship with Ienobu assured him a good position as an accountant and tutor at the shogun’s court. He enjoyed this position in Edo from 1678 to 1708, providing Seki with both the leisure and the resources he needed to conduct his research.

The early Tokugawa era was an important developmental period for mathematics in Japan. The Tokugawa shogunate had restored order to the country after more than 150 years of civil war. Commerce had once again begun to develop and flourish, and this created a pressing national need for accounting, inventory, and other commercial and financial forms of record-keeping. Practical mathematics had fallen into relative disuse in Japan, so the latest techniques were imported from China. A number of new abacus schools (soroban-juku), using Chinese methods, soon began to appear in Edo, the Kyōto- Ōsaka area, and elsewhere.

One of these abacus schools was run by Mōri Shigeyoshi, a former samurai (rōnin) said to have fought on the losing side against the Tokugawas. Mōri was the author of the earliest Japanese mathematics book still extant, the Warizansho (1622; writings on division), though it was actually a general abacus calculation manual. Yoshida Mitsuyoshi, who had learned mathematics at Mōri’s academy, became the mathematics tutor of the feudal domain lord of Kumamoto and wrote a derivative work of his own, the Jinkōki. Because Chinese methods still largely focused on basic abacus calculation methods, however, aspiring Japanese mathematicians began to see a need to move beyond this stage. Building on the familiar Chinese base, these mathematicians created a new set of uniquely Japanese mathematical techniques, known as wasan.

Yoshida Mitsuyoshi had established his reputation as a mathematical scholar and tutor to the aristocracy, so it was relatively easier for Seki to achieve the same sort of status a generation later. Most of Seki’s posthumous fame, in fact, resulted from accounts of his life and work by his aristocratic students. Unlike most of his contemporaries, Seki did not actively seek to publicize his own mathematical accomplishments. In fact, though he wrote many manuscripts on mathematics, Seki published only one book during his lifetime, Hatsubi sanpō (1674), a book of solutions to mathematical problems.

Among Seki’s aristocratic students were Takebe Katahiro and Katahiro’s older brother Katakira, the sons of an important official at the shogun court in Edo. Together with Seki, they worked on an encyclopedic study of all current Japanese mathematical knowledge, the Taisei sankei (1710; collection of classic mathematical texts). It took twenty-eight years to complete this twenty-volume work, which was finally published two years after Seki’s death.

Among Seki’s other pioneering accomplishments in Japanese mathematics, he developed a workable system of algebraic notation, a theory of determinants, and a formula for calculating the circumference of a circle. Seki also approximated roots of higher-order equations and is believed to have discovered the concept eventually named “Bernoulli numbers” even before Jakob I Bernoulli did. Seki also developed a form of calculus and anticipated important discoveries by Gottfried Wilhelm Leibniz and Sir Isaac Newton.

Takebe Katahiro is considered by historians to be Seki’s true intellectual heir, who refined and further developed his teacher’s mathematical ideas and methods. Katahiro carefully went through Seki’s many manuscripts, systematizing his ideas and preserving his writings for future generations. Katahiro also wrote his own mathematics books and made additional independent discoveries, succeeding Seki as the court mathematician for several more shoguns until his own death in 1739. The period of more than six decades during which Seki and his students dominated mathematics in Edo assured the continued predominance of Seki’s methods and concepts, until Western mathematics replaced traditional Japanese mathematics completely.

Significance

From the seventeenth century to the second half of the nineteenth century, the field of Japanese mathematics was dominated by wasan , as developed by Seki Kōwa, Takebe Katahiro, and others. Wasan was also the form of mathematics initially chosen by Japanese modernizers for use in the national school system, until it was officially replaced by European-style mathematics in 1873. A preexisting knowledge of wasan methods actually functioned as a bridge to the future, serving as the foundation upon which Meiji-era students were able to learn the new Western style of mathematics. Like Seki and his followers, many nineteenth century wasan experts were also from the samurai class, well-placed and well-educated people who often made the transition to become teachers of the new Western-style mathematics.

Kikuchi Dairoku (1855-1917), a pioneer in developing the study of modern mathematics in Japan, began learning Western mathematics at an early age, but later, as a leading mathematics teacher in Japan, he took advantage of his students’ basic wasan knowledge, using it as a basis on which to teach them equivalent new Western methods. Fujisawa Rikitarō (1861-1933), one of Kikuchi’s earliest students who had learned wasan as a child, subsequently became a leading figure in modern Japanese mathematics. Far from dismissing the importance of the wasan methods developed by Seki Kōwa, Fujisawa gave lectures on those methods in Japan and Europe. When the Tokyo Sugaku Kaisha, the first modern mathematical society in Japan, was established in 1877, more than half of its members had originally been wasan mathematicians.

Bibliography

Horiuchi, Annick. Les mathematiques japonaises a l’epoque d’Edo, 1600-1868: Une étude des travaux de Seki Takakazu, ?-1708, et de Takebe Katahiro, 1664-1739. Paris: Mathesis, 1994. A four-hundred-page study of the work of Seki and Takebe, the only book on this subject so far in a Western language.

Morris-Suzuki, Tessa. The Technological Transformation of Japan: From the Seventeenth to the Twenty-first Century. New York: Cambridge University Press, 1994. Contains an essay on Tokugawa technological development.

Nakayama, Shigeru. Academic and Scientific Traditions in China, Japan, and the West. Tokyo: University of Tokyo Press, 1984. A standard comparative history of scientific thought and research by a leading Japanese authority.

Smith, David, and Mikami Yoshio. A History of Japanese Mathematics. Mansfield Center, Conn.: Martino, 2002. The definitive history of Japanese mathematics up to the early twentieth century in English.