Isolated from the West, Seki Takakazu churned out some of the finest mathematical work of his time. Centuries later, scholars are finally giving him his due.

Isolated from the West, Seki Takakazu churned out some of the finest mathematical work of his time. Centuries later, scholars are finally giving him his due

**TOKYO—**Seki Takakazu, arguably Japan's greatest mathematician ever, labored in isolation at a time when Japan had cut itself off from the rest of the world. Yet his discoveries rivaled and often anticipated those of European mathematicians he never heard of. With the 300th anniversary of Seki's death approaching on 24 October, historians met here^{*} to assess his work and discuss why the tradition he embodied—an indigenous Japanese mathematics known as *wasan*—languished after he was gone.

“There are many new discoveries” about Seki's life, says Shigeru Jochi, a specialist in early East Asian mathematics at National Kaohsiung First University of Science and Technology in Kaohsiung City, Taiwan. For example, Jochi says, Seki worked as a professional mathematician—contrary to previous thinking that mathematics in feudal Japan was a mere pastime. Hikosaburo Komatsu, a science historian at Tokyo University of Science, says Seki's most important discovery—a general theory of solving systems of equations—has been overlooked because it led to calculations “almost beyond human capabilities.”

Seki was born into a samurai family around 1640 in Fujioka, a town about 90 kilometers northwest of Tokyo (then known as Edo). After centuries of internal warfare, the Tokugawa clan had unified and pacified Japan at the start of the 17th century and ruled through military commanders known as shoguns, while Japan's emperor became a figurehead. Japan's legendary samurai warriors remained a privileged elite atop a rigid, feudal class hierarchy.

No one knows who introduced Seki to mathematics. It appears he was largely self-taught, partly using Chinese texts; the shoguns' policy of isolation kept *wasan* almost free of influences from other parts of the world. Seki's mathematical proficiency landed him a job keeping accounts and mapping for a Tokugawa lord whose fief lay just west of Edo. When that lord's son became the shogun, Seki joined the shogunate in the capital.

Science historians have long believed *wasan* was more of a refined accomplishment—like mastering the tea ceremony—than a serious scientific pursuit. Jochi says that was probably true during the late 18th and early 19th centuries. But for Seki and other contemporary samurai, who had to develop skills useful for a peaceful bureaucracy, knowledge of *wasan* “was a tool for success in life.”

The accounting jobs apparently gave Seki the leisure to work on mathematics. Among other advances, he devised new notation for handling equations with several variables and developed solutions for equations with an unknown raised to the fifth power. His most significant work focused on determinants, numbers that capture characteristics of matrices, a field he pioneered a year or two ahead of his European contemporary Gottfried Leibniz.

Seki published only one book during his lifetime. After his death in 1708, two of his students, the brothers Katahiro and Takaakira Takebe, gathered Seki's more erudite work into 20 volumes; another follower, Nushizumi Yamaji, published a book on Seki's methods of measuring circles and arcs and founded the Seki School of Mathematics.

Mining those posthumous texts, Komatsu says he turned up a significant overlooked achievement: Seki's discovery around the year 1680 of a general theory of elimination, a method of solving simultaneous equations by whittling down the number of unknown quantities one by one. Komatsu says René Descartes had hinted at an elimination theory in 1627. Another French mathematician, étienne Bézout, advanced the theory 150 years later.

Unlike Descartes, however, Seki had no real successors; his work was the high-water mark of *wasan*. After the Takebe brothers, the Japanese mathematical tradition hit a dead end. Later in the Tokugawa era, says Tatsuhiko Kobayashi, a science historian at the Maebashi Institute of Technology in Japan, texts entering the country through the tightly controlled trade with China and Holland began bringing news of advances made elsewhere in the world. The isolation policy ended in 1853. In 1868, dissident lords overthrew the Tokugawa shogunate and restored imperial rule, ushering in a modernization drive that included an 1872 governmental order to replace *wasan* with Western mathematics in school curricula.

“The demise of *wasan* stemmed largely from its divorce from the natural sciences,” Mark Ravina, a historian at Emory University in Atlanta, Georgia, wrote in the Summer 1993 issue of the journal *Monumenta Nipponica*. Scientists of the Tokugawa era, Ravina wrote, considered *wasan* intriguing but useless for astronomy, calendar-making, and surveying. Rivalries among the schools that formed around *wasan* masters also hurt Japanese mathematics. Many schools had their own schemes of mathematical notation and treated problem-solving techniques as trade secrets.

More simply, Komatsu says, Seki's more erudite work “was too difficult for people to pick up and carry forward.” For example, Komatsu believes Seki's elimination theory fell into obscurity because it required computations that were impossible without number-crunching computers.

Could things have turned out differently? Historians can only speculate. Seki worked on determinants simultaneously with Leibniz, another mathematician whose work went unrecognized for decades because he never published it. “There were striking similarities in mathematical thinking” between the two men, says Eberhard Knobloch, a Leibniz scholar at the Berlin University of Technology. If the Eastern and Western mathematical sages had been in contact, Knobloch says, it probably would have advanced mathematics worldwide.

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^{*}International Conference on History of Mathematics in Memory of Seki Takakazu, 25-31 August 2008, Tokyo University of Science.