Takakazu Seki

A mathematician from the mid-Edo period. Later, he was called "Saint Saint." He was commonly known as Shinsuke, and his pen name was Shihyo and he was called Jiyutei. According to the Kansei Choshu Shokafu and other sources, Takakazu (also read as "Kowa") was the second or third child of Uchiyama Nagaakira. The Uchiyama clan was one of the Ashida Gojuki, and initially served Tokugawa Tadanaga, who was called Suruga Dainagon, but when Tadanaga was confined to Takasaki, the Uchiyama clan established their residence in Fujioka (Gunma Prefecture). Later, he was summoned by the third shogun, Tokugawa Iemitsu, and became the castle guard. Takakazu was adopted into the Seki family, but it is unclear which Seki family he was from. He was a member of the Ashida Gojuki. Takakazu's mother was the daughter of Yuasa Yoemon, and the Yuasa clan was a vassal of Ando Tsushima no Kami. After being adopted into the Seki family, Takakazu served Tokugawa Tsunashige and his son Tsunatoyo (later the sixth shogun Ienobu) in Kofu. He served as Kanjo Ginmiyaku (accounting auditor) in Kofu. In 1704 (the first year of the Hoei period), Tsunatoyo was adopted by the fifth shogun Tsunayoshi and entered the Nishinomaru of Edo Castle, and Takakazu also became a samurai directly under the shogunate. He was head of the storehouse and received a salary of 250 bales of rice and a stipend for ten people, which later increased to 300 bales. In 1706, he resigned from his post due to illness and died on October 24, 5th year of the Hoei era. Takakazu had no children and adopted his elder brother's son, Shinshichi (or Shinshichiro), but he was banished for misconduct while on duty in Kofu, and the Seki clan became extinct. Takakazu's grave is at Jorin-ji Temple in Bentencho, Shinjuku Ward, Tokyo, which is the family temple of the Uchiyama clan.

It is not known at all where or by whom Seki Takakazu learned mathematics. It is said that he gained his mathematical ability by reading the Jingoki on his own. When Takakazu was around 20 years old, excellent mathematical books such as Sanpo Ketsugisho and Sanso were published one after another, so he had no difficulty in finding examples to study on his own. This was the heyday of the succession of posthumous mathematical themes, which began with the Jingoki, and the dedication of mathematical plaques was also becoming more and more popular. It is believed that Takakazu was stimulated by these problems and became engrossed in studying mathematics. He also sought out and read as many old Chinese mathematical books as he could. This becomes clear when we summarize Takakazu's achievements. Seki Takakazu's first work to be presented to the public was the solution to a posthumous problem in Sawaguchi Kazuyuki's Kokin sanpo-ki, which he published under the title Hatsubisanpo (1674). This work is a calculation book that improves on Tianyuanshu, an instrumental algebra invented in China, to allow simultaneous multivariate higher-order equations to be calculated by hand, and explains the calculation method, which he calls Engdanshu. This Engdanshu was further expounded on by his disciple Takebe Katahiro, and published as Hatsubisanpo Engdangenkai (1685).

Seki Takakazu's achievements can be summarized as follows: (1) the creation of endojutsu, (2) Horner's approximate solution method, (3) interpolation, (4) discriminants for equations, (5) formulas equivalent to derivatives, (6) extreme values, (7) transformation of solutions to equations, (8) various series, (9) Bernoulli numbers, (10) equations relating the sides and diagonals of a regular n-gon, (11) the method of subtraction, (12) number theory, (13) magic squares, ensan (circles), (14) extrapolation, (15) various curves, (16) the method of Pappus-Guldin, (17) many studies on astronomy and calendars, etc.

Most of the problems that Seki Takakazu dealt with were old problems for which he provided solutions, but his achievements made mathematics significantly more advanced. Takakazu was blessed with successors, and his achievements were compiled by his disciples, the brothers Takebe Kataaki and Katahiro, who passed them on to Nakane Genkei, and then on to Matsunaga Yoshisuke and Kurushima Yoshihiro, allowing Japanese mathematics to progress to ever more advanced content. In later generations, mathematics (Japanese mathematics) came to be referred to as the Seki school.

[Shimohira Kazuo]

"Seki Takakazu by Hirayama Akira (1981, Koseisha Kouseikaku)"

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

江戸中期の数学者。後世「算聖」と称される。通称は新助、字(あざな)は子豹(しひょう)、自由亭と号した。『寛政重修諸家譜(かんせいちょうしゅうしょかふ)』その他によると、孝和（「こうわ」とも読む）は、内山永明(ながあきら)の第2子（または第3子）に生まれる。内山氏は芦田(あしだ)五十騎の一つで、初め駿河大納言(するがだいなごん)と称された徳川忠長に仕えたが、忠長が高崎へ幽閉されたとき、内山氏は藤岡（群馬県）に居を構えた。のちに第3代将軍徳川家光に召し出され、天守番となる。孝和は関家へ養子に出たが、どの関家か未詳。芦田五十騎のなかの一家である。孝和の母は湯浅与右衛門の娘で、湯浅氏は安藤対馬守(つしまのかみ)の家来である。孝和は関家に養子に入ったのち、甲府の徳川綱重(つなしげ)とその子綱豊(つなとよ)（後の第6代将軍家宣(いえのぶ)）に仕えた。甲府では勘定吟味役(かんじょうぎんみやく)（会計監査役）を務めた。1704年（宝永1）綱豊が第5代将軍綱吉(つなよし)の養子となり、江戸城西の丸へ入ったため、孝和も幕府直属の侍となった。御納戸(おなんど)組頭で、俸禄(ほうろく)は御蔵米250俵および十人扶持(ぶち)で、のちに300俵となった。1706年に病のため職を辞し、宝永(ほうえい)5年10月24日没す。孝和には子がなく、兄の子新七（または新七郎）を養子としたが、甲府勤番中、不行跡のため追放され、関家は絶えた。孝和の墓は、内山家の菩提(ぼだい)寺である東京都新宿区弁天町の浄輪寺にある。

　関孝和がどこでだれに数学を教わったかは、まったくわかっていない。『塵劫記(じんごうき)』を独学で読破し、数学の力を得たと伝えられる。孝和の20歳前後は、『算法闕疑抄(けつぎしょう)』や『算俎(さんそ)』などりっぱな数学書が次々と出版されたころであり、独学のための手本に困ることはなかった。『塵劫記』から始まる遺題継承の最盛期であり、算額の奉掲もいよいよ盛んになろうとしている時期である。孝和はこれらの問題に刺激され、夢中になって数学を勉強したものと思われる。また、中国の古算書もできる限り探して読破した。孝和の業績を整理してみれば、このことは明らかである。関孝和が最初に世間に発表したのは、沢口一之(さわぐちかずゆき)の『古今算法記』にある遺題の解答で、『発微算法(はつびさんぽう)』（1674）と題して刊行した。本書は、中国で発明された器具代数である天元術を、連立多元高次方程式が筆算でできるように改良し、その計算を演段術と称して説明した算書である。この演段術は、弟子の建部賢弘(たけべかたひろ)によって詳しく解説され、『発微算法演段諺解(えんだんげんかい)』（1685）として世に出た。

　関孝和の業績をまとめると次のようになる。(1)演段術の創始、(2)ホーナーの近似解法、(3)補間法、(4)方程式の判別式、(5)導関数に相当する式、(6)極値、(7)方程式の解の変換、(8)各種の級数、(9)ベルヌーイ数、(10)正n角形の辺と対角線の関係式、(11)招差法、(12)整数論、(13)魔方陣、円攅(えんさん)（円陣のこと）、(14)エクストラポレーション、(15)各種の曲線、(16)パップス・ギュルダンの方法、(17)天文、暦についての多くの研究、などである。

　関孝和の取り扱った問題のほとんどは従来の問題で、それに解法を与えたのであるが、孝和の業績により数学が著しく高度になった。孝和は後継者にも恵まれ、弟子の建部賢明(かたあき)・賢弘兄弟により孝和の業績はまとめられ、中根元圭(げんけい)に伝えられ、さらに松永良弼(よしすけ)や久留島義太(くるしまよしひろ)に伝わって、日本の数学はますます高度な内容へ進歩することができたのである。後世、数学（和算）といえば、関流とまで称せられるようになった。

［下平和夫］

『平山諦著『関孝和』（1981・恒星社厚生閣）』

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