,
[ Selin, Helaine. (1997). ''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures,'' p. 890] also known as ,
[Selin, ] was a Japanese
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
and author of the
Edo period
The or is the period between 1603 and 1867 in the history of Japan, when Japan was under the rule of the Tokugawa shogunate and the country's 300 regional '' daimyo''. Emerging from the chaos of the Sengoku period, the Edo period was characte ...
.
Seki laid foundations for the subsequent development of
Japanese mathematics
denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term ''wasan'', from ''wa'' ("Japanese") and ''san'' ("calculation"), was coined in the 1870s and employed to distinguish native Japanese ...
, known as ''wasan''.
He has been described as "Japan's Newton".
He created a new algebraic notation system and, motivated by astronomical computations, did work on
infinitesimal calculus and
Diophantine equations
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
. Although he was a contemporary of German polymath mathematician and philosopher
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
and British polymath physicist and mathematician
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
, Seki's work was independent. His successors later developed a school dominant in Japanese mathematics until the end of the
Edo period
The or is the period between 1603 and 1867 in the history of Japan, when Japan was under the rule of the Tokugawa shogunate and the country's 300 regional '' daimyo''. Emerging from the chaos of the Sengoku period, the Edo period was characte ...
.
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. For example, he is credited with the discovery of
Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
. The
resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ove ...
and
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
(the first in 1683, the complete version no later than 1710) are attributed to him.
Biography
Not much is known about Seki's personal life. His birthplace has been indicated as either
Fujioka in
Gunma Prefecture
is a Prefectures of Japan, prefecture of Japan located in the Kantō region of Honshu. Gunma Prefecture has a population of 1,937,626 (1 October 2019) and has a geographic area of 6,362 Square kilometre, km2 (2,456 Square mile, sq mi). Gunma P ...
, or
Edo. His birth date ranges from 1635 to 1643.
He was born to the
Uchiyama Uchiyama (written: ) is a Japanese surname. Notable people with the surname include:
* Akira Uchiyama (born 1954), Japanese politician of the Democratic Party of Japan
* Atsushi Uchiyama (born 1959), former Japanese football player
* Uchiyama Gudō ...
clan, a subject of Ko-shu ''
han
Han may refer to:
Ethnic groups
* Han Chinese, or Han People (): the name for the largest ethnic group in China, which also constitutes the world's largest ethnic group.
** Han Taiwanese (): the name for the ethnic group of the Taiwanese p ...
'', and adopted into the Seki family, a subject of the ''
shōgun
, officially , was the title of the military dictators of Japan during most of the period spanning from 1185 to 1868. Nominally appointed by the Emperor, shoguns were usually the de facto rulers of the country, though during part of the Kamak ...
''. 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
Chinese mathematical roots
His mathematics (and ''wasan'' as a whole) was based on mathematical knowledge accumulated from the 13th to 15th centuries. The material in these works consisted of algebra with numerical methods,
polynomial interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset.
Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no ...
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
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horn ...
, re-established by
William George Horner
William George Horner (9 June 1786 – 22 September 1837) was a British mathematician. Proficient in classics; mathematics, he was a schoolmaster, headmaster and schoolkeeper, who wrote extensively on functional equations, number theory and app ...
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
Elimination may refer to:
Science and medicine
*Elimination reaction, an organic reaction in which two functional groups split to form an organic product
*Bodily waste elimination, discharging feces, urine, or foreign substances from the body ...
.
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
![Hatsubi Sanpou](https://upload.wikimedia.org/wikipedia/commons/5/56/Hatsubi_Sanpou.jpg)
In 1671, , a pupil of in
Osaka
is a designated city in the Kansai region of Honshu in Japan. It is the capital of and most populous city in Osaka Prefecture, and the third most populous city in Japan, following Special wards of Tokyo and Yokohama. With a population of ...
, published ''Kokon Sanpō Ki'' (古今算法記), in which he gave the first comprehensive account of Chinese algebra in Japan. 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.
In 1674, Seki published ''Hatsubi Sanpō'' (発微算法), giving solutions to all the 15 problems. The method he used is called ''bōsho-hō''. He introduced the use of ''
kanji
are the logographic Chinese characters taken from the Chinese family of scripts, Chinese script and used in the writing of Japanese language, Japanese. They were made a major part of the Japanese writing system during the time of Old Japanese ...
'' to represent unknowns and
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
s in
equations. 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
parentheses
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
,
equality
Equality may refer to:
Society
* Political equality, in which all members of a society are of equal standing
** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...
, or
division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...
. For example,
could also mean
. Later, the system was improved by other mathematicians, and in the end it became as expressive as the ones developed in Europe.
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, , who was from Hashimoto's school and was active in
Kyoto
Kyoto (; Japanese language, Japanese: , ''Kyōto'' ), officially , is the capital city of Kyoto Prefecture in Japan. Located in the Kansai region on the island of Honshu, Kyoto forms a part of the Keihanshin, Keihanshin metropolitan area along wi ...
, authored ''Sanpō Meiki'' (算法明記), and gave new solutions to Sawaguchi's 15 problems, using his version of multivariable algebra, similar to Seki's. To answer criticism, in 1685, , one of Seki's pupils, published ''Hatsubi Sanpō Genkai'' (発微算法諺解), notes on ''Hatsubi Sanpō'', in which he showed in detail the process of elimination using algebraic symbols.
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
In 1683, Seki pushed ahead with
elimination theory
Elimination may refer to:
Science and medicine
*Elimination reaction, an organic reaction in which two functional groups split to form an organic product
*Bodily waste elimination, discharging feces, urine, or foreign substances from the body ...
, based on
resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (ove ...
s, in the ''Kaifukudai no Hō'' (解伏題之法). To express the resultant, he developed the notion of the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
. 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 with Katahiro Takebe (建部 賢弘) and his brothers, a correct and general formula (
Laplace's formula for the determinant) appears.
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'' (算法紛解) (1690?), he explicitly described the resultant and applied it to several problems. In 1690, , a mathematician active in Osaka but not in Hashimoto's school, published ''Sanpō 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. Seki developed his mathematics in competition with mathematicians in Osaka and Kyoto, at the 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 matrices only up to the 3x3 case. The subject was forgotten in the West until
Gabriel Cramer
Gabriel Cramer (; 31 July 1704 – 4 January 1752) was a Genevan mathematician. He was the son of physician Jean Cramer and Anne Mallet Cramer.
Biography
Cramer showed promise in mathematics from an early age. At 18 he received his doctorate ...
in 1750 was brought to it by the same motivations. Elimination theory equivalent to the ''wasan'' form was rediscovered by
Étienne Bézout
Étienne Bézout (; 31 March 1730 – 27 September 1783) was a French mathematician who was born in Nemours, Seine-et-Marne, France, and died in Avon (near Fontainebleau), France.
Work
In 1758 Bézout was elected an adjoint in mechanics of the ...
in 1764. The
Laplace's formula was established not 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
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-va ...
, but with a completely different perspective. Neither he nor his pupils had, strictly speaking, the idea of
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
.
Seki also studied the properties of
algebraic equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
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
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
.
He obtained some evaluations of the number of real roots of a polynomial equation.
Calculation of pi
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 the
Aitken's delta-squared process
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926.Alexa ...
, rediscovered in the 20th century by
Alexander Aitken.
Legacy
The asteroid
7483 Sekitakakazu is named after Seki Takakazu.
Selected works
In a statistical overview derived from writings by and about Seki Takakazu,
OCLC/
WorldCat
WorldCat is a union catalog that itemizes the collections of tens of thousands of institutions (mostly libraries), in many countries, that are current or past members of the OCLC global cooperative. It is operated by OCLC, Inc. Many of the O ...
encompasses roughly 50+ works in 50+ publications in three languages and 100+ library holdings.
WorldCat Identities
関孝和 ca. 1642-1708
/ref>
* 1683 –
OCLC 045626660
* 1712 –
OCLC 049703813
*
OCLC 006343391
collected works
Gallery
Seki Takakazu.jpg, Seki on a 1992 stamp, taken from an Edo era ink drawing
Seki Takakazu monument.jpg, Memorial to Seki, with stele and statue
Seki Takakazu grave.jpg
都史跡「関孝和」墓.jpg, Seki's grave marker outside Jyōrin-ji temple in Tokyo
See also
* Sangaku
Sangaku or San Gaku ( ja, 算額, lit=calculation tablet) are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes ...
, the custom of presenting mathematical problems, carved in wood tablets, to the public in Shinto shrines
A is a structure whose main purpose is to house ("enshrine") one or more ''kami'', the deities of the Shinto religion.
Overview
Structurally, a Shinto shrine typically comprises several buildings.
The '' honden''Also called (本殿, meani ...
* Soroban
The is an abacus developed in Japan. It is derived from the ancient Chinese suanpan, imported to Japan in the 14th century. Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocket electr ...
, a Japanese abacus
The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hi ...
* Japanese mathematics
denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term ''wasan'', from ''wa'' ("Japanese") and ''san'' ("calculation"), was coined in the 1870s and employed to distinguish native Japanese ...
* Napkin ring problem
In geometry, the napkin-ring problem involves finding the volume of a "band" of specified height around a sphere, i.e. the part that remains after a hole in the shape of a circular cylinder is drilled through the center of the sphere. It is a co ...
Notes
References
* Endō Toshisada (1896). . 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.
OCLC 318334322
* Howard Whitley, Eves. (1990)
''An Introduction to the History of Mathematics.''
Philadelphia: Saunders.
OCLC 20842510
* Poole, David. (2005)
''Linear algebra: a Modern Introduction.''
Belmont, California: Thomson Brooks/Cole.
OCLC 67379937
* Restivo, Sal P. (1992)
''Mathematics in Society and History: Sociological Inquiries.''
Dordrecht: Kluwer Academic Publishers.
OCLC 25709270
* Sato, Kenichi. (2005), ''Kinsei Nihon Suugakushi -Seki Takakazu no jitsuzou wo motomete.'' Tokyo: University of Tokyo Press.
* Selin, Helaine. (1997)
''Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures.''
Dordrecht: Kluwer
Wolters Kluwer N.V. () is a Dutch information services company. The company is headquartered in Alphen aan den Rijn, Netherlands (Global) and Philadelphia, United States (corporate). Wolters Kluwer in its current form was founded in 1987 with a m ...
/Springer
Springer or springers may refer to:
Publishers
* Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag.
** Springer Nature, a multinationa ...
.
OCLC 186451909
* David Eugene Smith
David Eugene Smith (January 21, 1860 – July 29, 1944) was an American mathematician, educator, and editor.
Education and career
David Eugene Smith is considered one of the founders of the field of mathematics education. Smith was born in Cortl ...
and Yoshio Mikami
was a Japanese mathematician and historian of '' Japanese mathematics''. He was born February 16, 1875, in Kotachi, Hiroshima prefecture. He attended the High School of Tohoku University, and in 1911 was admitted to the Imperial University of To ...
. (1914)
''A History of Japanese Mathematics.''
Chicago: Open Court Publishing
OCLC 1515528Alternate online, full-text copy at archive.org
External links
Sugaku-bunka
*
{{DEFAULTSORT:Seki, Takakazu
1642 births
1708 deaths
17th-century Japanese mathematicians
18th-century Japanese mathematicians
Linear algebraists
Hatamoto
Japanese writers of the Edo period
Samurai