Seki Takakazu (関孝和?, 1642 – December 5, 1708), also known as Seki Kōwa (関孝和?), was a Japanese mathematician in the Edo period.
Seki laid foundations for the subsequent development of Japanese mathematics known as wasan; and he has been described as Japan's "Newton." 
He created a new algebraic notation system, and also, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations. A contemporary of Gottfried Leibniz and Isaac Newton, Seki's work was independent. His successors later developed a school dominant in Japanese mathematics until the end of the Edo era.
While it is not clear how much of the achievements of wasan are actually Seki's, since many of them appear only in writings with his pupils, some of the results parallel or anticipate those discovered in Europe. For example, he is credited with the discovery of Bernoulli numbers. The resultant, and determinant (the first in 1683, the complete version no later than 1710) are also attributed to him. This work was a substantial advance on, for example, the comprehensive introduction of 13th century Chinese algebra made as late as 1671, by Kazuyuki Sawaguchi.
Not much is known about Kōwa's personal life. His birth place has been indicated as either Fujioka in Gunma prefecture, or Edo, and his birth date ranging anywhere from 1635 to 1643.
He was born to the Uchiyama clan, a subject of Ko-shu han, and later 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 solution (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,
(a b c) for ax2 + bx + c.
Later, they developed a method which uses two-dimensional arrays, representing four variables at most, but the scope was still 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.
Competiting with contemporaries
In 1671, Sawaguchi Kazuyuki (沢口 一之?), a pupil of Hashimoto Masakazu (橋本 正数?) in Osaka, published Kokin-Sanpo-Ki (古今算法之記), in which he gave the first comprehensive account of Chinese algebra in Japan, and 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 kanji to represent unknowns and variables in equations. Although it was possible to represent arbitrary degree equations (he once treated degree 1458th) with negative coefficients, there were no symbols corresponding to parentheses, equality, or division. For example, ax + b could also mean ax + b = 0. Later, the system was improved by other mathematicians, and in the end became as expressive as the one used in Europe.
A page from Seki Kōwa's Katsuyo Sampo (1712), tabulating binomial coefficients and Bernoulli numbers
In his book in 1674, however, he 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 3 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 Kenko (Katahiro Takebe, 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 them, 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.
In 1683, Seki pushed ahead with elimination theory, based on resultants, in the Kai-fukudai-no-hō (解伏題之法,); and 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, jointly with Katahiro Takebe (建部 賢弘) and his brothers, a correct and general formula (Laplace's formula for the determinant) appears.
Tanaka also came up with the same idea independently. An indication already 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, aat 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 3×3 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 so-called 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, of course, 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' is
the order(h) term in f(x + h),
accessible through the binomial theorem.
He also 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.
Seki's published writings encompass 52 works in 56 publications in 3 languages and 118 library holdings.
This is an incomplete list, which may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries.
* 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
* Sangaku, the custom of presenting mathematical problems, carved in wood tablets, to the public in shinto shrines
* Soroban, a Japanese abacus
* Japanese mathematics (wasan)
* Napkin ring problem
1. ^ Selin, Helaine. (1997). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, p. 890. at Google Books
2. ^ a b Selin, p. 641. at Google Books
3. ^ Smith, David. (1914) A History of Japanese Mathematics, pp. 91-127. at Google Books
4. ^ Restivo, Sal P. (1992). Mathematics in Society and History: Sociological Inquiries, p. 56. at Google Books
5. ^ Smith, pp. 128-142. at Google Books
6. ^ Poole, David. (2005). Linear algebra: a Modern Introduction, p. 279. at Google Books; Selin, p. 891.
7. ^ 和算の開祖 関孝和 ("Seki Takakazu, founder of Japanese mathematics"), Otonanokagaku. June 25, 2008. -- Seki was greatly influenced by Chinese mathematical books Introduction to Computational Studies (1299) by Zhu Shijie and Yang Hui suan fa (1274-75) by Yang Hui. (とくに大きな影響を受けたのは、中国から伝わった数学書『算学啓蒙』(1299年)と『楊輝算法』(1274-75年)だった。)
8. ^ Eves, Howard. (1990). An Introduction to the History of Mathematics, p. 405.
9. ^ WorldCat Identities: 関孝和 ca. 1642-1708
* Endō Toshisada (1896). History of mathematics in Japan (日本數學史史 , Dai Nihon sūgakush?). Tōkyō: _____. OCLC 122770600
* Horiuchi, Annick. (1994). Les Mathematiques Japonaises a L'Epoque d'Edo (1600–1868): Une Etude des Travaux de Seki Takakazu (?-1708) et de Takebe Katahiro (1664–1739). Paris: Librairie Philosophique J. Vrin. 10-ISBN 2711612139/13-ISBN 9782711612130; OCLC 318334322
* Howard Whitley, Eves. (1990). An Introduction to the History of Mathematics. Philadelphia: Saunders. 10-ISBN 0030295580/13-ISBN 9780030295584; OCLC 20842510
* Poole, David. (2005). Linear algebra: a Modern Introduction. Belmont, California: Thomson Brooks/Cole. 10-ISBN 0534998453/13-ISBN 9780534998455; OCLC 67379937
* Restivo, Sal P. (1992). Mathematics in Society and History: Sociological Inquiries. Dordrecht: Kluwer Academic Publishers. 10-ISBN 0792317653/13-ISBN 9780792317654; OCLC 25709270
* Sato, Kenichi. (2005), Kinsei Nihon Suugakushi -Seki Takakazu no jitsuzou wo motomete. Tokyo:University of Tokyo Press. ISBN 4-13-061355-3
* Selin, Helaine. (1997). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Dordrecht: Kluwer/Springer. 10-ISBN 0792340663/13-ISBN 9780792340669; OCLC 186451909
* David Eugene Smith and Yoshio Mikami. (1914). A History of Japanese Mathematics. Chicago: Open Court Publishing. OCLC 1515528 -- note alternate online, full-text copy at archive.org
* O'Connor, John J.; Robertson, Edmund F., "Takakazu Shinsuke Seki", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Seki.html .
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