- Seki Kōwa
Infobox_Scientist

name = Kōwa Seki (Takakazu Seki)

image_width = 200px

caption = Kōwa Seki (Takakazu Seki)

birth_date = March(?), 1642(?)

death_date =December 5 ,1708 (Gregorian calendar )

residence =

nationality = ese

birth_place =Edo orFujioka ,Japan

death_place =Japan

field =Mathematician **nihongo|Seki Kōwa|関孝和**or**nihongo|Seki Takakazu|関孝和|Seki Takakazu**(born 1637/1642? –October 24 ,1708 ) was aJapan esemathematician who created a new algebraic notation system and laid foundation for the later development ofwasan (Japanese traditional mathematics). He also, motivated by astronomical computations, had done some important works in calculus and integer indeterminate equations, which were to be developed by his successors. His successors later developed a school of mathematics (Seki's school) which was overwhelmingly dominant in Japanese mathematics until the end ofEdo era .He was a contemporary with

Gottfried Leibniz andIsaac Newton , although it is obvious that he could not have had contact with them. He discovered some of the theorems and theories that were being —or were shortly to be- discovered in the West. For example, discovery ofBernoulli numbers (published in 1712) anddeterminant (the first one in 1683, the complete version not later than 1710) are attributed to him. It is striking, since Japanese mathematics before his appearance was at such a primitive stage that comprehensive introduction of 13th century Chinese algebra was made as late as 1671, byKazuyuki Sawaguchi .However, it is not clear how much of the achievement under his name are his own contribution, since many of them appear only in the writings edited by/or co-authored with his pupils. Also, not much about his biography is known. His birth place can be either

Fujioka inGunma prefecture , orTokyo , and the birth year can be any one between 1635 and 1643 [*cite book|last=Sato|first=Kenichi|authorlink=Sato Kenichi Fujiwara|year=2005|title=Kinsei Nihon Suugakushi -Seki Takakazu no jitsuzou wo motomete|publisher=University of Tokyo|ISBN=4-13-061355-3*] . He was born to theUchiyama clan, a subject of Ko-shuhan , and later adopted into the Seki family, a subject ofShogun . While in Ko-shu han, he was involved in a project ofsurveying to edit the reliable map of the territory of his master. Also, he spent many years in studying 13th century Chinese calendar to replace the less accurate one used in Japan at that time.**Influence of Chinese mathematics**His mathematics (and wasan as a whole) is based on mathematics of the Chinese

Ming dynasty . [*和算の開祖 関孝和| 江戸の科学者列伝 | 大人の科学.net (publisher*] They are algebra with numerical method,Gakken ) [*http://otonanokagaku.net/issue/edo/vol5/index02.html*]polynomial interpolation and their applications, indeterminate integer equations.Seki's work is more or less based on/and related to them.Chinese algebra, first in the world, discovered numeric solution (

Horner's method , re-established by Horner in 19th century) of arbitrary degree algebraic equation with real coefficients.They reduced geometric problem to algebra systematically using Pythagorean theorem, which somewhat anticipatesDescartes .However, the number of unknowns in a equation was quite limited. They used array of numbers to represent a formula; for example, (a b c) for a+bx+cx^2. Later, they developed a method which uses two-dimensional arrays, representing four variables at most. However, obviously, there was a little room of further development in this way. Hence, a target of Seki and his contemporary Japanese mathematicians was the study of multi-variable algebraic equations, especially reduction to single variable algebraic equations.

Also, Chinese were the first who established

polynomial interpolation . The motivation was to predict the motion of celestial bodies from observed data (they never came up with least-square method.). They also applied the method to find various mathematical formulas. Seki learned this method most likely through his study of Chinese calenders.Chinese theory of indeterminate integer equations had overwhelmed west in the past (recall

Chinese remainder theorem , for example). They are motivated by computations required in making calendars. Naturally, Japanese mathematicians are attracted to the study of the field.**Algebra -- competition with mathematicians in**Osaka andKyoto In 1671, nihongo|Kazuyuki Sawaguchi|Kazuyuki Sawaguchi|沢口 一之, a pupil of Masakazu 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 successfully applied it to problems suggested by his contemporaries. Before him, these problems were solved using arithmetic method. 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 introducedkanji to represent unknowns andvariable s inequation s. Although it was possible to represent arbitrary degree equations (he even treated 1458th !) with negative coefficients, there was no symbol corresponding to '() ', '=', nor '/' (division). For example, ax+b could mean either ax+b=0 or ax+b. Later, the system was improved by other mathematicians, and in the end became as powerful as the one used in Europe.In his book in 1674, however, he only gave 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, nihongo|Yoshizane Tanaka|Yoshizane Tanaka| 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 Sawagushi's 15 problems, using his version of multi-variable algebra, similar to Seki's. To answer criticism, in 1685, nihongo|Takebe Kenko|Katahiro Takebe|建部 賢弘, one of Seki's pupil, published**Hatsubi-Sampo Genkai 発微算法諺解**, notes on**Hatsubi-Sampo**, in which he in detail showed process of elimination using algebraic symbols.Effect of introduction of such system of symbols is not restricted to algebra; with them, mathematicians at that time became able to express mathematical results in more general and abstract way.

Once being able to express the equations, they concentrated on the study of elimination of variables. In 1683 (解伏題之法, Kai-fukudai-no-hou), Seki came up with

elimination theory , based onresultant . To expressresultant , he developed the notion ofdeterminant Howard Eves : "An Introduction to the History of Mathematics", page 405, Saunders College Publishing, 1990. (ISBN 0030295580)] . However, in his manuscript, the formula for 5x5 matrices is obviously wrong, being always 0. Yet in in his later publication (大成算経 Taisei-sankei, written in 1683-1710, jointly with nihongo|Takebe Kenko|Katahiro Takebe|建部 賢弘 and his brothers), a correct and general formula (Laplace's formula ) appears.Tanaka also came up with the same idea independently.A sign already appeared in his book in 1678: some of equations after elimination are the same as

resultant . In**Sampo-Funkai 算法紛解**(1690?), he explicitly describedresultant , and applied to several problems. In 1690, nihongo|Tomotoki Izeki|Tomotoki Izeki|井関 知辰, a mathematician active inOsaka but not in Hashimoto's school, published**Sampo-Hakki算法発揮**, in which he gaveresultant and Laplace's formula ofdeterminant for nxn case.The relations between these works are not clear. But, one can see that Seki developed his mathematics in severe competition with mathematicians in

Osaka andKyoto , which were cultural center of Japan.In comparison with European mathematics, Seki'sfirst manuscript was as early as

Leibniz 's first commentary on the subject,which treated only up to 3x3 case. In addition, in Europe, this subject had been forgotten untilGabriel Cramer restarted it in 1750, driven by the same motivation aswasan mathematicians. Elimination theory which is equivalent to the one bywasan was rediscovered byBezout in 1764. So-calledLaplace's formula was established not earlier than 1750.Due to completion of

elimination theory , large part of problems treated in Seki's time became essentially solvable. (Recall in Chinese traditional mathematics, geometry almost had reduced to algebra. ) In practice, of course, the whole computation was not always able to carry out, due to huge computational complexity. Yet, this theory had significant influence on the direction of development ofwasan .After the elimination is done, one has to find out real root of a single variable equation by numerics. Honer's method, though completed in China, was not transmitted to Japan in its final form. So Seki had to work it out by himself independently. (Due to this, he is sometimes credited with Honer's method, which is not quite correct.) He also suggested an improvement to Honer's method: to omit higher order terms after some iterations. This happen to be the same as

Newton-Raphson method , but in completely different perspective.Note he (nor his pupils) had never come up with the idea of derivative in strict sense.He also studied properties of algebraic equations, in the aim of assisting numerics. The most notable of them is conditions for existence of multiple roots based on

discriminant (this also is earlier than west), which isresultant of a polynomial and its 'derivative': his definition of 'derivative' is o(h) term in f(x+h).He also obtained certain evaluation of the number of real roots of an equation.**Other works**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 byAlexander Aitken .**See also***

Sangaku

*wasan **References**

*

* [*http://www.questia.com/PM.qst?a=o&d=93950113# David Eugene Smith, Yoshio Mikami: A History of Japanese Mathematics. Open Court Publishing, Chicago,1914*]

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**Seki Kōwa**— Kōwa Seki (Takakazu Seki) Matemático Japonés del Wasan. Nacimiento Marzo(?), 1642 Edo o Fujioka, Japón Fallecimi … Wikipedia Español**Seki Kowa**— Kowa Seki Kowa Seki Seki Kowa (ou Seki Takakazu) est un mathématicien japonais, né en mars 1642 à Fujioka dans la préfecture Gunma, décédé le 24 octobre 1708 à Tokyo. Alors que le Japon était coupé de l Occident depuis quelques décennies, il… … Wikipédia en Français**Seki Kowa**— Zeichnung von Seki Takakazu in Tensai no Eikō to Zasetsu von Masahiko Fujiwara Zeichnung von Seki Takakazu Seki Takakazu (jap. 関 孝和 … Deutsch Wikipedia**Kowa Seki**— Seki Kowa (ou Seki Takakazu) est un mathématicien japonais, né en mars 1642 à Fujioka dans la préfecture Gunma, décédé le 24 octobre 1708 à Tokyo. Alors que le Japon était coupé de l Occident depuis quelques décennies, il établit un certain… … Wikipédia en Français**Seki**— ist: Seki (Go), eine spezielle Stellung im Spiel Go Seki (Gifu), eine Stadt in der japanischen Präfektur Gifu Seki (Han), ein Lehen mit Sitz in der Stadt Seki Seki (Sprache), eine Bantusprache Seki (Tavas), Dorf in der türkischen Provinz Denizli… … Deutsch Wikipedia**Seki Takakazu**— Zeichnung von Seki Takakazu in Tensai no Eikō to Zasetsu von Masahiko Fujiwara Zeichnung von Seki Takakazu … Deutsch Wikipedia**Seki Takakazu**— also called Seki Kōwa born с 1640, Fujioka, Japan died Oct. 24, 1708, Edo the most important figure of the wasan ( Japanese calculation ) tradition that flourished from the early 17th century until the opening of Japan to the West in the mid 19th … Universalium**Seki**— Cette page d’homonymie répertorie les différents sujets et articles partageant un même nom. Toponymes Seki est une ville du Japon située dans la Préfecture de Gifu. Şəki est une ville d Azerbaïdjan. Personnalités Kowa Seki est un mathématicien… … Wikipédia en Français**Determinante (matemática)**— En matemáticas se define el determinante como una forma multilineal alternada de un cuerpo. Esta definición indica una serie de propiedades matemáticas y generaliza el concepto de determinante haciéndolo aplicable en numerosos campos. Sin embargo … Wikipedia Español