# Seki Kōwa: Wikis

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# Encyclopedia

In this Japanese name, the family name is Seki.
Kōwa Seki (Takakazu Seki)

Kōwa Seki (Takakazu Seki)
Born March(?), 1642(?)
Edo or Fujioka, Japan
Died December 5, 1708 (Gregorian calendar)
Japan
Residence Japan
Nationality Japanese
Fields Mathematician

Seki Kōwa (関孝和?) or Seki Takakazu (関孝和 Seki Takakazu?) (born 1642 – December 5, 1708[1]) was a Japanese mathematician who created a new algebraic notation system and laid foundation for the later development of early Japanese mathematics (wasan). 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, which was overwhelmingly dominant in Japanese mathematics until the end of Edo era.

He was a contemporary with Gottfried Leibniz and Isaac 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, he is credited with the discovery of Bernoulli numbers in 1712.[2][3][4] The resultant, and determinant (the first in 1683, the complete version no later than 1710) are also attributed to him.[4] These achievements are astonishing, considering that Japanese mathematics before his appearance was at such a primitive stage—for example, comprehensive introduction of 13th century Chinese algebra was made as late as 1671, by Kazuyuki 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.

## Biography

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 [5].

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.

## Influence of Chinese mathematics

Seki Takakazu, from Tensai no Eikō to Zasetsu

His mathematics (and wasan as a whole) is based on mathematics from the 13th to 15th centuries.[6] They are algebra with numerical method, polynomial interpolation and their applications, indeterminate integer equations. Seki's work is more or less based on and related to them.

Chinese algebra discovered numeric solution (Horner's method, re-established by Horner in 19th century) of arbitrary degree algebraic equation with real coefficients. By using the Pythagorean theorem, they reduced geometric problems to algebra systematically.

However, the number of unknowns in an equation was quite limited. They used 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. Obviously, there was a little room of further development in this way. Hence, a target of Seki and his contemporary Japanese mathematicians was the development of general multi-variable algebraic equations, and elimination theory.

Also, the Chinese 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 close examination of Chinese calendars.

## Elimination theory: competition with other mathematicians

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 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 introduced kanji to represent unknowns and variables in equations. Although it was possible to represent arbitrary degree equations (he even treated 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 powerful 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 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 Sawagushi'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 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, Seki came up with elimination theory, based on resultant in the Kai-fukudai-no-hō (解伏題之法,). To express resultant, he developed the notion of determinant[7] . However, in his manuscript, the formula for 5×5 matrices is obviously wrong, being always 0. Yet 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) 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 described the resultant, and applied 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 n×n 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 and Kyoto, which were 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 3×3 case. In addition, in Europe, this subject had been forgotten until Gabriel Cramer restarted it in 1750, driven by the same motivation as wasan mathematicians. Elimination theory which is equivalent to the one by wasan was rediscovered by Bezout in 1764. So-called Laplace'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 of wasan.

After the elimination is done, one has to find out real root of a single variable equation numerically. 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 the 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 the properties of algebraic equations, in the aim of assisting numerics. The most notable of these are the conditions for the existence of multiple roots based on discriminant, which is resultant 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 by Alexander Aitken.

## Notes

1. ^ http://www.sugaku-bunka.org/#830
2. ^ Selin, H. (1997), p. 891
3. ^ Poole, D. (2005), p. 279
4. ^ a b Styan & Trenkler. (2007), p. 2
5. ^ Sato, Kenichi (2005). Kinsei Nihon Suugakushi -Seki Takakazu no jitsuzou wo motomete. University of Tokyo. ISBN 4-13-061355-3.
6. ^ 和算の開祖　関孝和| 江戸の科学者列伝 | 大人の科学.net (publisher　Gakken) [1]
7. ^ Eves, H. (1990), p. 405