关孝和Takakazu Seki

关孝和出生于一个武士家庭。然而在幼年时，他被一个名为关孝和五郎左卫门的贵族家庭收养。他现在为人所知的名字关孝和，来源于收养他的家庭，而非他的亲生父母。

关孝和在数学方面是个神童。他自学数学，最初是由家中的一名仆人引入这一领域的，当关孝和九岁时，这名仆人意识到了这个男孩的天赋。

关孝和很快建立了一个关于数学的日文和中文书籍的图书馆，并被公认为专家。他被称为“算术圣人”，这个词刻在他的墓碑上，并很快有了许多学生。他的社会地位在[18]中描述如下：-

适时地，作为武士阶级的后裔，他以公职身份服务，他的职务是甲州藩主的账目审查官，正如艾萨克·牛顿在安妮女王手下成为铸币厂长一样。当他的领主成为将军的继承人时，关孝和成为幕府武士，并于1704年被授予将军府中典礼官的荣誉职位。

1674年，关孝和出版了Hatsubi Sampo，其中他解决了四年前提出的十五个问题。这部作品因关孝和对问题的仔细分析而著称，这无疑是他作为教师取得巨大成功的原因之一。

关孝和预见了西方数学的许多发现。
关孝和是1683年第一个研究行列式的人。十年后，哥特弗里德·威廉·莱布尼茨独立地使用行列式来解联立方程，尽管关孝和的版本更为一般。

关孝和也在雅各布·伯努利之前发现了伯努利数。他研究方程，同时处理正根和负根，但没有复数的概念。他在1683年的著作中再次写到了magic squares，此前他在1661年研究了杨辉关于该主题的一部中国著作。这是日本对该主题的首次处理。

1685年，他使用与一百年后的威廉·乔治·霍纳相同的方法解出了三次方程$30 + 14x - 5x^{2} - x^{3} = 0$。

他发现了求解方程的艾萨克·牛顿或艾萨克·牛顿-约瑟夫·拉弗森方法，并且也有一个艾萨克·牛顿插值公式的版本。

在关孝和研究过的其他问题中有丢番图方程。例如，在1683年，他考虑了$ax - by = 1$的整数解，其中$a, b$是整数。

日本各学派笼罩在保密之中，因此很难确定关孝和所作的贡献，但他也被认为在微积分中做出了重大发现，并将其传授给了他的学生。

Takakazu Seki was born into a samurai warrior family. However at an early age he was adopted by a noble family named Seki Gorōzaemon. The name by which he is now known, Seki, derives from the family who adopted him rather than from his natural parents.

Seki was an infant prodigy in mathematics. He was self-educated in mathematics having been introduced to the topic by a servant in the household who, when Seki was nine years old, realised the talent of the young boy.

Seki soon built up a library of Japanese and Chinese books on mathematics and became acknowledged as an expert. He was known as 'The Arithmetical Sage', a term which is carved on his tombstone, and soon had many pupils. His position in life is described in [18] as follows:-

In due time he, as a descendant of the samurai class, served in public capacity, his office being that of examiner of accounts to the Lord of Koshu, just as Newton became master of the mint under Queen Anne. When his lord became heir to the Shogun, Seki became Shogunate samurai and in 1704 was given a position of honour as master of ceremonies in the Shogun's household.

In 1674 Seki published Hatsubi Sampo in which he solved fifteen problems which had been posed four years earlier. The work is remarkable for the careful analysis of the problems which Seki made and this certainly was one of the reasons for his great success as a teacher.

Seki anticipated many of the discoveries of Western mathematics.
Seki was the first person to study determinants in 1683. Ten years later Leibniz, independently, used determinants to solve simultaneous equations although Seki's version was the more general.

Seki also discovered Bernoulli numbers before Jacob Bernoulli. He studied equations treating both positive and negative roots but had no concept of complex numbers. He wrote on magic squares, again in his work of 1683, having studied a Chinese work by Yank Hui on the topic in 1661. This was the first treatment of the topic in Japan.

In 1685, he solved the cubic equation $30 + 14x - 5x^{2} - x^{3} = 0$ using the same method as Horner a hundred years later.

He discovered the Newton or Newton-Raphson method for solving equations and also had a version of the Newton interpolation formula.

Among other problems studied by Seki were Diophantine equations. For example, in 1683, he considered integer solutions of $ax - by = 1$ where $a, b$ are integers.

Secrecy surrounded the schools in Japan so it is hard to determine the contributions made by Seki, but he is also credited with major discoveries in the calculus which he passed on to his pupils.