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| {{Japanese name|Seki}}
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| {{Infobox scientist
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| | name = Seki Takakazu (Seki Kōwa)
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| | image = Seki.jpeg
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| | image_size = 200px
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| | caption = Seki Takakazu (Seki Kōwa)
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| | birth_date = March(?), 1642(?)
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| | death_date = December 5, 1708 ([[Gregorian calendar]])
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| | residence = [[Image:Flag of Japan.svg|20px|]] [[Japan]]
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| | nationality = [[Image:Flag of Japan.svg|20px|]] [[Japan]]ese
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| | birth_place = [[Edo]] or [[Fujioka, Gunma|Fujioka]], [[Japan]]
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| | death_place = [[Japan]]
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| | field = [[Mathematics]]
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| }}
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| {{nihongo|'''Seki Takakazu'''|関 孝和||1642 – December 5, 1708}},<ref>[[Helaine Selin|Selin, Helaine]]. (1997). ''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures,'' p. 890</ref> also known as {{nihongo|'''Seki Kōwa'''|関 孝和}},<ref name="selin641">Selin, {{Google books|raKRY3KQspsC&dq|p. 641.|page=641}}</ref> was a [[Japan]]ese [[mathematician]] in the [[Edo period]].<ref>Smith, David. (1914) {{Google books|J1YNAAAAYAAJ|''A History of Japanese Mathematics,'' pp. 91-127. |page=91}}</ref>
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| Seki laid foundations for the subsequent development of Japanese mathematics known as ''[[wasan]]'';<ref name="selin641"/> and he has been described as Japan's "Newton." <ref>Restivo, Sal P. (1992). {{Google books|gvMm0jv-xPIC|''Mathematics in Society and History: Sociological Inquiries,'' p. 56. |page=56}}</ref>
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| 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 period]].
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| While it is not clear how much of the achievements of ''wasan'' are actually Seki's, since many of them appear only in writings of his pupils, some of the results parallel or anticipate those discovered in Europe.<ref>Smith, {{Google books|J1YNAAAAYAAJ|pp. 128-142. |page=128}}</ref> For example, he is credited with the discovery of [[Bernoulli numbers]].<ref>Poole, David. (2005). {{Google books|oBk3u2fDFc8C|''Linear algebra: a Modern Introduction,'' p. 279. |page=279}}; Selin, p. 891.</ref> 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]].
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| == Biography ==
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| Not much is known about Kōwa's personal life. His birthplace has been indicated as either [[Fujioka, Gunma|Fujioka]] in [[Gunma prefecture]], or [[Edo]], and his birth date ranging anywhere from 1635 to 1643.
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| He was born to the [[Uchiyama]] clan, a subject of Ko-shu ''[[Han (administrative division)|han]]'', and later adopted into the Seki family, a subject of the [[Shogun]].
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| 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.
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| ==Career==
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| ===Chinese mathematical roots ===
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| [[Image:Seki Kowa.jpg|thumb|Seki Takakazu, from ''Tensai no Eikō to Zasetsu'']]
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| His mathematics (and ''wasan'' as a whole) was based on mathematical knowledge from the 13th to 15th centuries.<ref>[http://otonanokagaku.net/issue/edo/vol3/index02.html 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。)</ref> 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.
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| 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,
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| :<math>(a\ b\ c)</math> for <math>ax^2 + bx + c.</math>
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| 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]].
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| 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.
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| === Competing with contemporaries ===
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| [[File:Hatsubi Sanpou.jpg|thumb|240px|Replica of ''Hatsubi-Sampo'' exhibited in the [[National Museum of Nature and Science]], [[Tokyo]], [[Japan]].]]
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| In 1671, {{nihongo|Sawaguchi Kazuyuki|沢口 一之}}, a pupil of {{nihongo|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 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.
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| 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 [[variable (mathematics)|variable]]s in [[equation]]s. 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 [[bracket|parentheses]], [[equals sign|equality]], or [[division (mathematics)|division]]. For example, <math>ax+b</math> could also mean <math>ax+b=0</math>. Later, the system was improved by other mathematicians, and in the end it became as expressive as the ones developed in Europe.
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| [[Image:Seki Kowa Katsuyo Sampo Bernoulli numbers.png|thumb|A page from Seki Kōwa's ''Katsuyo Sampo'' (1712), tabulating binomial coefficients and Bernoulli numbers]]
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| 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, {{nihongo|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, {{nihongo|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.
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| 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.
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| ===Elimination theory===
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| In 1683, Seki pushed ahead with [[elimination theory]], based on [[resultant]]s, in the ''Kai-fukudai-no-hō'' (解伏題之法,); and to express the resultant, he developed the notion of [[determinant]].<ref>Eves, Howard. (1990). [http://books.google.com/books?id=PXvwAAAAMAAJ&pg=PA405&sig=ACfU3U194N23V2YZt5NDGa_jLiqaZCw0LQ&q=seki#search_anchor ''An Introduction to the History of Mathematics,'' p. 405.]</ref> 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 expansion|Laplace's formula]] for the determinant) appears.
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| 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, {{nihongo|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.
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| 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 so-called [[Laplace expansion|Laplace's formula]] was established not earlier than 1750.
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| 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's method|Newton-Raphson method]], but in a completely different perspective. Neither he nor his pupils had the idea of [[derivative]], strictly speaking.
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| 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
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| :the order(''h'') term in ''f''(''x'' + ''h''),
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| accessible through the [[binomial theorem]].
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| He also obtained some evaluations of the number of real roots of an equation.
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| ===Calculation of Pi===
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| 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]].
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| ==Selected works==
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| In a statistical overview derived from writings by and about Seki Takakau, [[OCLC]]/[[WorldCat]] encompasses roughly 50+ works in 50+ publications in 3 languages and 100+ library holdings.<ref>[http://www.oclc.org/research/activities/identities/default.htm WorldCat Identities]: [http://www.worldcat.org/identities/lccn-n84-106242 関孝和 ca. 1642-1708]</ref>
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| {{dynamic list}}
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| * 1683 — {{nihongo|''Kenpu no hō''|驗符之法}} [http://www.worldcat.org/search?q=no:%22045626660%22 OCLC 045626660]
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| * 1712 — {{nihongo|''Katsuyō sanpō''|括要算法}} [http://www.worldcat.org/search?q=no:%22049703813%22 OCLC 049703813]
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| * {{nihongo|''Seki Takakazu zenshū''|關孝和全集}} [http://www.worldcat.org/search?q=no:%22006343391%22 OCLC 006343391], collected works
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| == See also ==
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| * [[Sangaku]], the custom of presenting mathematical problems, carved in wood tablets, to the public in [[shinto shrines]]
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| * [[Soroban]], a Japanese [[abacus]]
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| * [[Japanese mathematics]] (''[[wasan]]'')
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| * [[Napkin ring problem]]
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| == Notes ==
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| {{reflist|2}}
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| ==References ==
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| * Endō Toshisada (1896). {{nihongo|''History of mathematics in Japan''|日本數學史史 |Dai Nihon sūgakush}}. Tōkyō: _____. [http://www.worldcat.org/title/dai-nihon-sugakushi-history-of-mathematics-in-japan-by-endo-toshisada/oclc/122770600&referer=brief_results OCLC 122770600]
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| * Horiuchi, Annick. (1994). [http://books.google.com/books?id=qMnZHUSAYzMC&dq=History+of+Mathematics+in+Japan+1896&lr=lang_ja&as_brr=0&source=gbs_navlinks_s ''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; [http://www.worldcat.org/title/mathematiques-japonaises-a-lepoque-dedo-1600-1868-une-etude-des-travaux-de-seki-takakazu-1708-et-de-takebe-katahiro-1664-1739/oclc/318334322 OCLC 318334322]
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| * Howard Whitley, Eves. (1990). [http://books.google.com/books?id=PXvwAAAAMAAJ&dq=An+Introduction+to+the+History+of+Mathematics&q=Seki#search_anchor ''An Introduction to the History of Mathematics.''] Philadelphia: Saunders. 10-ISBN 0030295580/13-ISBN 9780030295584; [http://www.worldcat.org/title/introduction-to-the-history-of-mathematics/oclc/20842510 OCLC 20842510]
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| * Poole, David. (2005). [http://books.google.com/books?id=oBk3u2fDFc8C&printsec=frontcover&dq=Linear+algebra:+a+Modern+Introduction&cd=1#v=onepage&q=Seki&f=false ''Linear algebra: a Modern Introduction.''] Belmont, California: Thomson Brooks/Cole. 10-ISBN 0534998453/13-ISBN 9780534998455; [http://www.worldcat.org/title/linear-algebra-a-modern-introduction/oclc/67379937?referer=di&ht=edition OCLC 67379937]
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| * Restivo, Sal P. (1992). [http://books.google.com/books?id=gvMm0jv-xPIC&dq=Yoshida+Koyu+arithmetic&source=gbs_navlinks_s ''Mathematics in Society and History: Sociological Inquiries.''] Dordrecht: Kluwer Academic Publishers. 10-ISBN 0792317653/13-ISBN 9780792317654; [http://www.worldcat.org/title/mathematics-in-society-and-history-sociological-inquiries/oclc/25709270 OCLC 25709270]
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| * Sato, Kenichi. (2005), ''Kinsei Nihon Suugakushi -Seki Takakazu no jitsuzou wo motomete.'' Tokyo:University of Tokyo Press. ISBN 4-13-061355-3
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| * Selin, Helaine. (1997). [http://books.google.com/books?id=raKRY3KQspsC&dq=Aida+Yasuaki&source=gbs_navlinks_s ''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures.''] Dordrecht: [[Kluwer]]/[[Springer Science+Business Media|Springer]]. 10-ISBN 0792340663/13-ISBN 9780792340669; [http://www.worldcat.org/title/encyclopaedia-of-the-history-of-science-technology-and-medicine-in-non-western-cultures/oclc/186451909 OCLC 186451909]
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| * [[David Eugene Smith]] and [[Yoshio Mikami]]. (1914). [http://books.google.com/books?id=J1YNAAAAYAAJ&dq=Shiraishi+Chochu&source=gbs_navlinks_s ''A History of Japanese Mathematics.''] Chicago: Open Court Publishing. [http://www.worldcat.org/title/history-of-japanese-mathematics/oclc/1515528 OCLC 1515528] [http://www.archive.org/details/historyofjapanes00smitiala -- note alternate online, full-text copy at archive.org]
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| ==External links==
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| * [http://www.sugaku-bunka.org/#830 Sugaku-bunka]
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| * {{MacTutor Biography|id=Seki|title=Takakazu Shinsuke Seki}}
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| <!-- last name is Seki -->
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| {{Persondata
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| | NAME =Seki Takakazu
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = Japanese mathematician
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| | DATE OF BIRTH = 1642
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| | PLACE OF BIRTH =[[Edo]] or [[Fujioka, Gunma|Fujioka]], [[Japan]]
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| | DATE OF DEATH =December 5, 1708
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| | PLACE OF DEATH =[[Japan]]
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| }}
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| {{DEFAULTSORT:Seki, Takakazu}}
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| [[Category:Japanese mathematics]]
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| [[Category:1708 deaths]]
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| [[Category:Japanese mathematicians]]
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| [[Category:17th-century mathematicians]]
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| [[Category:18th-century mathematicians]]
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| [[Category:Samurai]]
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| [[Category:Hatamoto]]
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| [[Category:1642 births]]
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| [[Category:Japanese writers of the Edo period]]
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| [[Category:17th-century Japanese people]]
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