Seki Takakazu

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Drawing by Seki Takakazu

Seki Takakazu ( Japanese 関 孝 和 ; * between 1640 and 1644 probably in Edo (now Tokyo ); † December 5, 1708 ; traditional date: Hōei 5/10/24), also Seki Kōwa (the Sino-Japanese reading of his name), was a Japanese mathematician. He discovered many theorems and theories that were discovered shortly before or only shortly after independently in Europe, and is considered the most important mathematician of Wasan .

His original family name was Uchiyama ( 内 山 ), his nickname ( tsūshō ) Shinsuke ( 新 助 ).

Drawing by Seki Takakazu in Tensai no Eikō to Zasetsu by Masahiko Fujiwara

Life

Seki's father Uchiyama Nagaakira ( 内 山 ​​永明 ) was a samurai and at the time of Seki's birth he was a guard in the main tower of Edo Castle . Seki was adopted by a member of the Seki family who was something of an accountant. He was possibly a student of the mathematician Takahara Yoshitane , who in turn was a student of the mathematician Mōri Shigeyoshi , who probably published the first book on mathematics written by a Japanese (Book of Division 1622). Or maybe he was a self-taught mathematician. He studied Chinese mathematical works such as that of Yang Hui and Japanese. In 1695 he lived in front of the Tenryu-ji temple, probably in that of Yotsuya . It is not known exactly where he was at other times in his life. In 1684/85 there are some documents with his signature, which suggest that he was employed as a surveyor.

Like his adoptive father, he was a vassal of Tokugawa Tsunashige , daimyo of the fiefdom of Kofu , and his son Tokugawa Tsunatoyo , who later became the 6th Tokugawa Shogun . In 1665 he followed his adoptive father as head of the Seki family and began to work for the Kofu fief. Before 1695 he became head of the supply department there. A document from 1698 shows that he supervised the demarcation of a neighboring fief. In 1701 he is mentioned as an examiner with the head of accounting. In 1704 he became the master of ceremonies of the Shogun in the western part of Edo Castle when his master Tokugawa Tsunatoyo became a Shogun. In 1706 he retired due to illness.

He was married and had two daughters who died young and two adopted sons, including his nephew.

Mathematical work

His reputation comes mainly from the Seki school established by his work , which was the most influential in Japan until the end of the Edo period . Takebe Katahiro was one of the most important members of his school .

Replica of Hatsubi sanpō in the National Museum of Natural Sciences

For his studies, Seki Takakazu resorted to Chinese works, in particular the Suanxue qimeng by Zhu Shijie . It introduces a method (called by the Chinese method of the heavenly element in Chinese  天元 術  /  天元 术 , pinyin tiān yuán shù , Japanese tengenjutsu ) to solve problems by reducing them to an algebraic equation in a variable. The first Japanese to use this method successfully was Sawaguchi Kazuyuki (Kokon Sanpoki 1670), who solved 150 problems from Satō Masaoki and posed 15 new problems that, in his opinion, could not be solved with this method. Seki solved these problems and published this in 1674 in his work Hatsubi sanpō ( 発 微 算法 ), but without a precise explanation of how he came to it. Seki's pupil Takebe Katahiro made up for this in a commentary on it in 1685 ( 発 微 算法 演 段 諺 解 Hatsubi sanpō endan genkai ). However, Seki kept essential sentences and methods a secret; they were only published by his students after his death. Most of the scriptures were passed on from student to student within his school. The already mentioned Hatsubi sanpō was published only in 1674 and in 1709 in Tokyo and Kyoto by his students after his death Katsuyō sanpō ( 括 要 算法 ; with further sentences and methods from him). Another collection of his writings, which were passed on within the school, did not appear until 1907 ( 関 流 算法 七 部 書 Seki-ryū sanpō shichibusho , "Seven books on mathematics from the Seki school", Tokyo 1907).

Seki developed a method to expand the representation of algebraic equations in a variable from China to several variables, called bōshohō ( 傍 書法 , "method of writing on the side") and the basis of his Endan method ( Endanjutsu ). In equation theory, he describes an old Chinese method for calculating the zeros of polynomials, which essentially corresponded to the Horner scheme , and extends it to find all real zeros. He also independently discovered the discriminant . He also discovered the Bernoulli numbers before Bernoulli and knew a variant of Newton's method (although he knew no derivative, but an algebraic expression that was equivalent to it, since he only worked with polynomials). He also studied Diophantine equations and was the first to study Magic Squares in Japan (using a Chinese source).

He made an important contribution to the discovery of determinants (1683, in the scripture method for solving hidden problems ). Although he only dealt with 2 × 2 and 3 × 3 matrices and did not manage to extend the calculation to the general case, it was still more general than Leibniz ten years later. The general case was treated by his students who discovered the evolutionary theorem before Laplace . They used determinants to eliminate variables in systems of equations.

Also of importance is, the procedure of Seki, he is Enri ( 円理 , dt. "Circle principle") mentioned, and that is the determination of circumference of the circle arcs of a circle and sphere volume by as in the exhaustion was approximated circle by regular polygons. This also provides approximations for the circle number Pi. By making the sides of the polygons ever smaller, this also led to an infinite series for the square of arcsine (found in the west by Euler in 1737) in the further development of the Enri method by Seki's pupil Takebe. .

He initiated a series of books on mathematics Taisei sankei in 20 volumes, which his students, the brothers Takebe Katahiro ( 建 部 賢 弘 ) and Kataaki ( 建 部 賢明 ), published from 1683 to 1710 and mostly wrote themselves (although it is not always very clear what from Seki).

Influence from the west

A controversial question is to what extent the Japanese mathematicians of the Seki School were influenced by knowledge from the West. The country had actually been isolated from the west since 1639 and only Dutch dealers had restricted access (a limited opening for foreign books did not come until 1720). Even the acquisition of mutual language skills was systematically prevented. Smith and Mikami therefore saw in their 1914 book no evidence that the Seki school, and in particular Seki itself, had its methods from the West, but rather found that they were inspired by Chinese sources. Seki himself also made some discoveries such as determinants and Bernoulli numbers before the corresponding publications in the West and even exceeded them in part. Mikami and Smith also admitted in 1914 that the research was just beginning. For example, there was a Japanese student of mathematics and later medicine in Leiden in the 1650s and 1660s who was a student of Frans van Schooten and was praised by him. He was called Petrus Hartzingius (probably after the merchant with whom he arrived). Such a stay abroad was forbidden to the Japanese on the death penalty and it is not known whether he will return; but there was at least one example of a doctor who had studied in Europe at the time and was returning. Smith and Mikami thought most likely an influence in the treatment of infinite series, especially the arcsine series by Seki's pupil Takebe. The influence could have come from the Jesuit missionary in China Pierre Jartoux (1668–1721), who made some infinite series developments known in China without explaining the methods behind them, which in China, for example, inspired research by Ming Antu . There were also two translators from the Dutch in Nagasaki who knew about astronomy, among other things. One (Hayashi Kichizaemon) was executed in 1646, the other (Kobayashi Yoshinobu) imprisoned at the same time and not released until 1667 (both were suspected of being Christians). Horiuchi and Silke Wimmer-Zagier (who examined the records of the Dutch East India Company) see no evidence of a scientific influence from the West on the Seki school, neither in documents nor from the content analysis of their writings. They also doubt any Jartoux influence on Seki and Takebe in terms of infinite series.

literature

Web links

Individual evidence

  1. ^ Hideyuki Majima: Seki Takakazu, His Life and Bibliography . In: E. Knobloch u. a .: Seki, Founder of Modern Mathematics in Japan . Springer 2013, p. 4. There, the year of birth is given between 1640 and 1644 and the place of birth is very likely Edo. Fujioka is also given as the place of birth in the literature . According to Hideyuki Majima, the date of death is October 24, 1708, which corresponds to December 5, 1708 according to the Gregorian calendar.
  2. ^ Website of the Japanese Mathematical Society with the correct date of death
  3. ^ Dictionary of Scientific Biography
  4. ↑ a b 平 山 諦 : 関 孝 和 . In: 日本 大 百科全書 . Retrieved October 21, 2017 (Japanese).
  5. ↑ Hideyuki Majima. In: Knobloch u. a .: Seki Takakazu . Springer 2013, p. 4
  6. ^ Kobori, Dictionary of Scientific Biography
  7. ^ Dictionary of Scientific Biography
  8. ↑ Hideyuki Majima. In: garlic u. a. Seki . 2013, p. 14
  9. ↑ Hideyuki Majima. In: garlic u. a. Seki . 2013, p. 8. There were two temples of that name, one in Yotsuya and one in Ushigome.
  10. ↑ Today four copies are known (Japan Academy, Kansai University, Wasan Institute, private property). There are two versions as Seki corrected the solution to the 7th problem.
  11. ↑ More detailed in Volume 17 of the collective work Taisei sankei. Goto, Komatsu: Seki's theory of elimination as compared to the others . In: Knobloch u. a .: Seki . Springer 2013, p. 555
  12. ^ H. Eves: An Introduction to the History of Mathematics . Saunders College Publishing, 1990, p. 405.
  13. ^ David Eugene Smith, Yoshio Mikami: A History of Japanese Mathematics. Open Court Publishing, Chicago 1914, p. 124.
  14. ↑ Mikami Smith: A History of Japanese Mathematics . Chicago 1914, Chapter 7, pp. 132ff
  15. ↑ Mikami Smith, p. 154
  16. ^ Horiuchi: Japanese Mathematics in the Edo Period . Birkhäuser 2010
  17. ↑ Silke Wimmer-Zagier, Don Zagier: Some questions and observations around the mathematics of Seki Takakazu . In: Knobloch u. a .: Seki . Springer 2013, especially p. 290 ff. A catalog of questions from Takebe to the Dutch from 1727 has been preserved, but it did not concern any scientific questions. The Japanese tried to ask such questions on the same day, but the Dutch captain indicated that he had no scientific knowledge.
Japanese name: As is common in Japan, the family name comes before the first name in this article . Thus Seki is the family name, Takakazu the first name.
personal data
SURNAME Seki, Takakazu
ALTERNATIVE NAMES 関 孝 和 (Japanese); Seki Kōwa (on reading); Seki Kowa
BRIEF DESCRIPTION Japanese mathematician
DATE OF BIRTH between 1640 and 1644
PLACE OF BIRTH Tokyo
DATE OF DEATH December 5, 1708