Seki Takakazu (
kizaru | |
---|---|
![]() Ink painting of Seki Takakazu, from the Japan Academy archives in Tokyo. | |
Born | 1642(?) |
Died | December 5, 1708 (Gregorian calendar) |
Nationality | Japanese |
Other names | Seki Kōwa |
Scientific career | |
Fields | Mathematics |
Seki laid foundations for the subsequent development of Japanese mathematics, known as wasan.[2] He has been described as "Japan's Newton".[4]
He created a new algebraic notation system and, motivated by astronomical computations, did work on infinitesimal calculus and Diophantine equations. Although he was a contemporary of German polymath mathematician and philosopher Gottfried Leibniz and British polymath physicist and mathematician Isaac Newton, Seki's work was independent. His successors later developed a school dominant in Japanese mathematics until the end of the Edo period.
While it is not clear how much of the achievements of wasan are Seki's, since many of them appear only in writings of his pupils, some of the results parallel or anticipate those discovered in Europe.[5] For example, he is credited with the discovery of Bernoulli numbers.[6] The resultant and determinant (the first in 1683, the complete version no later than 1710) are attributed to him.
Seki also calculated the value of pi correct to the 10th decimal place, having used what is now called the Aitken's delta-squared process, rediscovered later by Alexander Aitken.
Seki was influenced by Japanese mathematics books such as the Jinkōki.[7]
Biography
editNot much is known about Seki'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 shōgun. 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.
Career
editChinese mathematical roots
editHis mathematics (and wasan as a whole) was based on mathematical knowledge accumulated from the 13th to 15th centuries.[8] The material in these works 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 algebraists 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, for .
Later, they developed a method that uses two-dimensional arrays, representing four variables at most, but the scope of this method was limited. Accordingly, a target of Seki and his contemporary Japanese mathematicians was the development of general multivariable 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
editIn 1671, Sawaguchi Kazuyuki [ja] (
In 1674, Seki published Hatsubi Sanpō (
In his book of 1674, however, Seki gave only single-variable equations resulting from elimination, but no account of the process at all, nor his new system of algebraic symbols. There were a few errors in the first edition. A mathematician in Hashimoto's school criticized the work, saying "only three out of 15 are correct." In 1678, Tanaka Yoshizane (
The effect of the introduction of the new symbolism was not restricted to algebra. With it, 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
editIn 1683, Seki pushed ahead with elimination theory, based on resultants, in the Kaifukudai no Hō (
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 Sanpō Funkai (
In comparison with European mathematics, Seki's first manuscript was as early as Leibniz's first commentary on the subject, which treated matrices only up to the 3x3 case. The subject was forgotten in the West until Gabriel Cramer in 1750 was brought to it by the same motivations. Elimination theory equivalent to the wasan form was rediscovered by Étienne Bézout in 1764. Laplace's formula was established no 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 founder under huge computational complexity. Yet this theory had a significant influence on the direction of development of wasan. After the elimination is complete, one is left to find numerically the real roots of a single-variable equation. Horner's method, though well 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 practice happens to be the same as that of Newton–Raphson method, but with a completely different perspective. Neither he nor his pupils had, strictly speaking, the idea of derivative.
Seki also studied the properties of algebraic equations for assisting in numerical solution. 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 O(h) -term in f(x + h), which was computed by the binomial theorem.
He obtained some evaluations of the number of real roots of a polynomial equation.
Calculation of pi
editAnother of Seki's contributions was the rectification of the circle, i.e., the calculation of pi; he obtained a value for
Legacy
editThe asteroid 7483 Sekitakakazu is named after Seki Takakazu.
Selected works
editIn 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.[10]
- 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
Gallery
edit-
Seki on a 1992 stamp, taken from an Edo era ink drawing
-
Memorial to Seki, with stele and statue
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Seki's grave marker outside Jyōrin-ji temple in Tokyo
See also
edit- Sangaku, the custom of presenting mathematical problems, carved in wood tablets, to the public in Shinto shrines
- Soroban, a Japanese abacus
- Japanese mathematics
- Napkin ring problem
Notes
edit- ^ Selin, Helaine. (1997). Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, p. 890
- ^ a b Selin, p. 641., p. 641, at Google Books
- ^ Smith, David. (1914) A History of Japanese Mathematics, pp. 91-127. , p. 91, at Google Books
- ^ Restivo, Sal P. (1992). Mathematics in Society and History: Sociological Inquiries,, p. 56, at Google Books
- ^ Smith, pp. 128-142. , p. 128, at Google Books
- ^ Poole, David. (2005). Linear algebra: a Modern Introduction, p. 279. , p. 279, at Google Books; Selin, p. 891.
- ^
鳴海 風 「和算 」『東京 人 』第 321号 、都市 出版 、52-56頁 、2013年 2月 3日 。 - ^
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。) - ^ Eves, Howard. (1990). An Introduction to the History of Mathematics, p. 405.
- ^ WorldCat Identities:
関 孝和 ca. 1642-1708
References
edit- 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. ISBN 9782711612130; OCLC 318334322
- Howard Whitley, Eves. (1990). An Introduction to the History of Mathematics. Philadelphia: Saunders. ISBN 9780030295584; OCLC 20842510
- Poole, David. (2005). Linear algebra: a Modern Introduction. Belmont, California: Thomson Brooks/Cole. ISBN 9780534998455; OCLC 67379937
- Restivo, Sal P. (1992). Mathematics in Society and History: Sociological Inquiries. Dordrecht: Kluwer Academic Publishers. 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. ISBN 9780792340669; OCLC 186451909
- David Eugene Smith and Yoshio Mikami. (1914). A History of Japanese Mathematics. Chicago: Open Court Publishing. OCLC 1515528 Alternate online, full-text copy at archive.org
External links
edit- Sugaku-bunka
- O'Connor, John J.; Robertson, Edmund F., "Takakazu Shinsuke Seki", MacTutor History of Mathematics Archive, University of St Andrews