Seki Takakazu

Not much is known about Kōwa's personal life. His birthplace has been indicated as either Fujioka in Gunma prefecture, or Edo. His birth date ranges from 1635 to 1643.

He was born to the Uchiyama clan, a subject of Ko-shu han, and adopted into the Seki family, a subject of the Shogun. While in Ko-shu han, he was involved in a surveying project to produce a reliable map of his employer's land. He spent many years in studying 13th-century Chinese calendars to replace the less accurate one used in Japan at that time.

Chinese mathematical roots

Seki Takakazu, from Tensai no Eikō to Zasetsu

His mathematics (and wasan as a whole) was based on mathematical knowledge from the 13th to 15th centuries. This consisted of algebra with numerical methods, polynomial interpolation and its applications, and indeterminate integer equations. Seki's work is more or less based on and related to these known methods.

Chinese algebra discovered numerical evaluation (Horner's method, re-established by William George Horner in the 19th century) of arbitrary degree algebraic equation with real coefficients. By using the Pythagorean theorem, they reduced geometric problems to algebra systematically. The number of unknowns in an equation was, however, quite limited. They used notations of an array of numbers to represent a formula; for example,

${\displaystyle (a\ b\ c)}$ for ${\displaystyle ax^{2}+bx+c.}$

Later, they developed a method that uses two-dimensional arrays, representing four variables at most, but the scope was limited. Hence, a target of Seki and his contemporary Japanese mathematicians was the development of general multi-variable algebraic equations and elimination theory.

In the Chinese approach to polynomial interpolation, the motivation was to predict the motion of celestial bodies from observed data. The method was also applied to find various mathematical formulas. Seki learned this technique, most likely, through his close examination of Chinese calendars.

Competing with contemporaries

Replica of Hatsubi-Sampo exhibited in the National Museum of Nature and Science, Tokyo, Japan.

In 1671, Sawaguchi Kazuyuki (沢口 一之), a pupil of Hashimoto Masakazu (橋本 正数) in Osaka, published Kokon-Sanpo-Ki　(古今算法記), in which he gave the first comprehensive account of Chinese algebra in Japan. He successfully applied it to problems suggested by his contemporaries. Before him, these problems were solved using arithmetical methods. In the end of the book, he challenged other mathematicians with 15 new problems, which require multi-variable algebraic equations.

In 1674, Seki published Hatsubi-Sampo (発微算法), giving "solutions" to all the 15 problems. The method he used is called bousho-hou. He introduced the use of kanji to represent unknowns and variables in equations. Although it was possible to represent equations of an arbitrary degree (he once treated the 1458th degree) with negative coefficients, there were no symbols corresponding to parentheses, equality, or division. For example, ${\displaystyle ax+b}$ could also mean ${\displaystyle ax+b=0}$. Later, the system was improved by other mathematicians, and in the end it became as expressive as the ones developed in Europe.

A page from Seki Kōwa's Katsuyo Sampo (1712), tabulating binomial coefficients and Bernoulli numbers

In his book of 1674, however, Seki only gave the single variable equations after the elimination, but no account of the process at all, nor his new system of algebraic symbols. Even worse, there were a few errors in the first edition. A mathematician in Hashimoto's school criticized him saying "only three out of 15 are correct." In 1678, Tanaka Yoshizane (田中 由真), who was from Hashimoto's school and was active in Kyoto, authored Sampo-meikai (算法明記), and gave new solutions to Sawaguchi's 15 problems, using his version of multi-variable algebra, similar to Seki's. To answer criticism, in 1685, Takebe Katahiro (建部 賢弘), one of Seki's pupils, published Hatsubi-Sampo Genkai (発微算法諺解), notes on Hatsubi-Sampo, in which he in detail showed the process of elimination using algebraic symbols.

The effect of the introduction of the new symbolism was not restricted to algebra. With these, mathematicians at that time became able to express mathematical results in more general and abstract way. They concentrated on the study of elimination of variables.

Elimination theory

In 1683, Seki pushed ahead with elimination theory, based on resultants, in the Kai-fukudai-no-hō (解伏題之法). To express the resultant, he developed the notion of determinant. While in his manuscript the formula for 5×5 matrices is obviously wrong, being always 0, in his later publication, Taisei-sankei (大成算経), written in 1683-1710 with Katahiro Takebe (建部 賢弘) and his brothers, a correct and general formula (Laplace's formula for the determinant) appears.

Tanaka came up with the same idea independently. An indication appeared in his book of 1678: some of equations after elimination are the same as resultant. In Sampo-Funkai (算法紛解) (1690?), he explicitly described the resultant and applied it to several problems. In 1690, Izeki Tomotoki (井関 知辰), a mathematician active in Osaka but not in Hashimoto's school, published Sampo-Hakki (算法発揮), in which he gave resultant and Laplace's formula of determinant for the n×n case. The relationships between these works are not clear. But Seki developed his mathematics in serious competition with mathematicians in Osaka and Kyoto, at the cultural center of Japan.

In comparison with European mathematics, Seki's first manuscript was as early as Leibniz's first commentary on the subject, which treated only up to the 3X3 case. This subject was forgotten in the West until Gabriel Cramer in 1750 was driven to it by the same motivations. Elimination theory equivalent to the wasan form was rediscovered by Étienne Bézout in 1764. The Laplace's formula was established not earlier than 1750.

With elimination theory in hand, a large part of the problems treated in Seki's time became solvable in principle, given the Chinese tradition of geometry almost reduced to algebra. In practice, the method could flounder under huge computational complexity. Yet this theory had a significant influence on the direction of development of wasan. After the elimination is done, one has to find the real roots of a single variable equation numerically. Horner's method, though completely known in China, was not transmitted to Japan in its final form. So Seki had to work it out by himself independently—he is sometimes credited with Horner's method, which is not historically correct. He also suggested an improvement to Horner's method: to omit higher order terms after some iterations. This happens to be the same as the Newton-Raphson method, but in a completely different perspective. Neither he nor his pupils had the idea of derivative, strictly speaking.

He also studied the properties of algebraic equations, in the aim of assisting numerical work. The most notable of these are the conditions for the existence of multiple roots based on the discriminant, which is the resultant of a polynomial and its "derivative": His working definition of "derivative" was

the order(h) term in f(x + h),

accessible through the binomial theorem.

He obtained some evaluations of the number of real roots of an equation.

Calculation of pi

Another of Seki's contributions was the rectification of the circle, i.e., the calculation of pi; he obtained a value for π that was correct to the 10th decimal place, using what is now called "Aitken's delta-squared process," rediscovered in the 20th century by Alexander Aitken.

The asteroid 7483 Sekitakakazu is named after Seki Takakazu.

In a statistical overview derived from writings by and about Seki Takakazu, OCLC/WorldCat encompasses roughly 50+ works in 50+ publications in three languages and 100+ library holdings.

• 1683 — Kenpu no hō (驗符之法) OCLC 045626660
• 1712 — Katsuyō sanpō (括要算法) OCLC 049703813
• Seki Takakazu zenshū (關孝和全集) OCLC 006343391, collected works