Mathematics of Seki Takakzu and His
Influence on Japanese Mathematics
Kenji Ueno
Seki Kowa Institute of Mathematics

関孝和（SEKI Takakzu）
Born around 1640〜1645
There are no records SEKI was born in1642．
Recently it was reported that there is adocument showing he was born in 1645, but
the document is not open to the public.
Died 24th of October, 1708. (5th of December)

SEKI Takazu• Born as the second son of 内山 永明（UCHIYAMA Nagaakira）• Adopted by the family SEKI
• Adoptive father 関 十郎右衛門 (SEKIJuroemon) died on 9th of August 1665.
• 小十人組 A member of a team of defense ofthe lord

Theory of Equationsin Jin, Song, Yuan Dynasty（12th century〜14th century）• Numerical solution by synthetic division(counting rods) Representation of equations
by counting rods

Tanabata(たなばた)天元術(Tianyuan Shu)
• Equation• No equal sign ！

李治(Li Zhi)
「測円海鏡」(Ceyuan Haijing1248

朱世傑(Zhu Shijie)『算学啓蒙』(Suanxue qimeng) 1299

『算学啓蒙』(Suanxue qimeng)
Last Chapter 開方釋鎖門中根彦循(NAKANE Genjun)「数学小成」(Sugaku shosei)東福寺（Tofukuji)Tofukuji• Collection of printed books published in Yuan
Dynasty (元)• Many books were published in Yuan
Dynasty(元) by the Yuan government東福寺 霊雲院(不二庵）
Tofukuji Reiun-in(Fuji-an)• Founded 1390

Kokon sanpoki『古今算法記』1671
澤口一之

古今算法記遺題

古今算法記遺題

Kokon Sapoki Prob. 4
古今算法記遺題

SEKI Takakzu• Very few were known how SEKI studied
mathematics.
• In 1665 he succeeded his father’s positiona member of a team of defense of the lord(小十人組)• He studied 『楊輝算法』（Yang Hui suanfa)

Mathematical Works of
SEKI Takakzu• 延宝元年(1673) 『楊輝算法』(Yang Hui suanfa)の改訂版を作る (revised edition)• 延宝2年(1674) 『発微算法』（Hatsubi sanpou)• 天和3年(1683) 「解伏題之法」(Kaifukudai noho)• 貞享2年(1685) 「解隠題之法」(Kaiinndai no ho)「開方飜変之法」(Kaiho-honpen no ho)• 宝永6年(1709) 『括要算法』(Katsuyo sanpo)• 宝永7年(1710) 「大成算経」(Taisei sankei)

関孝和編「揚輝算法」
SEKI’s revised edition of
Yang Hui suanfa 1673

Numerical solution by counting rods

Seki’s correction

Mathematics of SEKI Takakazu• New development beyond the classicalChinese mathematics• SEKI invented a method to write equations
with many unknowns ( side writing method(傍書法)• Elimination theory by introducing the notionof determinants.
• 1683 [解伏題之法」, E. Bézout 1764

『発微算法』(Hatsubi sanpo)1674• Inside the frameworkof Chinese mathematics
Change the meaning『当世改算記』 傍書法 1847
Tosei Kaisanki, side writing system
KaisankiMathematics of SEKI
• Establish a general theory of equations of one
variable• Establish a general theory of elimination
theoryHatsubi sanpo endan genkai『発微算法演段諺解』跋 1685

『発微算法演段諺解』跋• 算学は何の為ぞや。難題、易題、盡(ことごと)く明かにせずと云うこと無くの術を学ぶなり。
理を説くこと高尚なりと雖も、術を解くこと迂闊(うかつ)なるものは、乃れ算斈(がく)の異端なり。一日、門人建部氏三子、相具に来たりて謂て曰く、発微算法演段諺解既に成れり、本書に附して、これを刊せんと欲す、可ならんか。

跋文読み下し• 余が曰く、いまだ釈鎖(しやくさ)の奥妙を竭(つ)くさずと雖も、世人の昏蒙(こんもう)を啓くにおいては、是の如くのものも亦可なり、唯、流傳して真を訛(あやま)らんことを恐るるのみ、後学、忽緒(こつしよ)すること莫くんば幸甚からん。
貞享乙丑孟穐(もうしゆう)関氏孝和筆す藤印 孝和之印
貞享2年(1685年)秋Afterword by SEKI• For what purpose one studies mathematics?
One learns the art of solving all problems, notonly difficult but easy ones. Only tellingsophistic things without solving all theproblems, he is not a true mathematician.
• One day three brothers of TAKEBE visited measking a permission of publishing Hatsubi
sanpo endann genkai.

Afterword of SEKI
• I answered that even though the book does
not contain the most important part of thetheory, such a book may be useful for thepublic. But I am afraid that only methods tosolve a few problems would become popularso that they would misunderstood the truetheory.
• Young scholars should not forget the point!

Elimination theory• Method to find simultaneous equations• Simplify the simultaneous equations• Algorithm to find n equations of degree n-1

Elimination theory

Both equations are of degree n in y .

Elimination theory• Elimination of y, resolvent

SEKI’s slant product (斜乗)Theory of equations• Kai-indai no ho (解隠題之法)• Addition and subtraction of polynomials• Multiplication of polynomials• A=B  A – B =0
• Systematic method to find numerical solutionsof polynomials (before SEKI they used amethod to find a solution from the highest
digit to lower digits.

Theory of equations• Use synthetic divisions• Use negative numbers to perform thesynthetic divisions• Approximated numercal solutions (the same
result by Newton’s method)

After SEKI
• SEKI’s mathematics was accepted widely• At the end of Edo period, many people could
use the side writing system.
• But SEKI’S view to mathematics was neverunderstood『当世改算記』 傍書法 1847
Tosei Kaisanki, side writing system
Tosei KaisankiTaisei sankei (大成算経)
Project was started on 1683
TAKEBE Kata-akira 1710
20 volumes, written systematic ways, but SEKI
seemed unsatisfactory with the book.
It was not written in logical order.
e.g. vol.3 A part of the theory of equations
vol. 17 The first part of the theory ofequations.( Kai indai no ho (解隠題之法))Myojorinji Temple in Ogaki
大垣市明星輪寺算額 1865
• Problem 3 河合沢女十六歳(16, girl)• Problem 6 奥田津女
• Problem 10 田辺重利十五歳 (15, boy)
• 58×224cm

Sozume Hachimangu, Okayama
惣爪八幡宮 岡山市 1861

SATO Noriyoshi
佐藤 則義

Miki Scholl 三木流『和洋一覧』