The fact that Seki Kowa was known during his own lifetime as the "Arithmetical Sage," a tribute that later graced his tombstone, indicates the magnitude of his genius. He is generally considered to be the founder of the Japanese mathematical tradition known as Wasan. Among his many accomplishments he is best known for being the first person to study determinants and for discovering what came to be known as Bernoulli numbers even before Jacob Bernoulli did. Because there was a tradition of strict secrecy in the Japanese schools of the period (due to competition among them and the constraints of modesty imposed by his noble upbringing), some of Seki's contributions may still be unknown. However, there is no doubt that he was a major factor in the popularization and development of modern mathematics. He is also credited with transforming the discipline from an esoteric art form practiced by aristocrats into a science worthy of respect.
Seki's natural parents were hereditary samurai warriors who lived in Fujioka, Kozuke, Japan, although some sources say Seki was born in Edo, now Tokyo. He was born the second son of Nagaakira Utiyama. When he was a young child, a nobleman and accountant named Seki Gorozayemon adopted him, giving him the name by which he is known today. In fact, some sources give Seki's first name as Takakazu. The circumstances surrounding the adoption are not clear today. Seki's prodigious talent for mathematics was apparent from the time he was an infant. An educated household servant began the boy's formal introduction to the discipline when he realized the young nobleman's interest and astounding grasp of mathematical concepts.
As Seki's understanding of and passion for mathematics increased with the years, he began studying under his first academic mentor, a man named Yositane Takahara, who himself was a disciple of the great Sigeyosi Mori (author of A Book on Division in 1622). One of Seki's most famous mathematical feats came in his thirties, when he took up a challenge posed by Chu Shih-chieh. Chu had written An Introduction to Mathematical Studies in 1299, which contained numerous problems that he had solved using the "method of the celestial element." This technique was based on changing the problem into an algebraic equation with one variable. At the end of his book, Chu presented 150 problems that he believed could not be solved with this technique. Another mathematician of the period, Kazuyuki Sawaguchi, solved 135 of those, claiming in 1670 that the last 15 were truly unsolvable. However, Seki stunned the academic world in 1674 when he published his solutions to these stubborn problems using entirely new means.
The solutions appeared in his book Hatubi sanpo, which revealed Seki's design of a completely new system of notation. This system was so innovative that most mathematicians did not even understand it until a Seki disciple published a guide to the notation in 1685. Seki's system, which he called endan, was based on Chinese ideographs and allowed him to eliminate the cumbersome traditional use of Chinese calculating rods. The rods would not allow calculation of algebraic expressions with more than one variable, whereas ideographs permitted the representation of known and unknown quantities and the use of several variables.
Seki's method was limited only by his algebraic alphabet, which allowed him to work with equations only from the second through fifth degrees. Seki discovered a procedure, very similar to that used much later by William George Horner, that let him solve second-degree algebraic equations with numerical coefficients; he also introduced the concept of a discriminant. In addition, in about 1683 Seki discovered determinants, the mathematical notations that are still used today in virtually every branch of mathematics and in many of the natural sciences. Also in 1683, Seki wrote about magic squares, becoming the first Japanese to treat the topic. Among the mathematician's other important contributions were his discovery of Bernoulli numbers and his exact definition ("rectification") of a circle's circumference, for which he obtained a value for pi () that was correct to eighteen decimal places.
Late in life, Seki made significant contributions to the area of calculus. Mainly, he is credited with developing a way to find the approximate value of a numerical equation's root. He passed on many of his methods to disciples at his school of mathematics, which had long been renowned as the best in Japan. Like the other Japanese schools, however, Seki's had a code of strict secrecy and permitted only his very best students to learn about his discoveries and ideas.
For the last four years of his life, Seki served as master of ceremonies in the household of a shogun, having become a shogunate samurai himself some time earlier. As a descendant of the samurai class, he had also spent years serving the public as examiner of accounts to the Lord of Koshu. Seki died in Edo on October 24, 1708.
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