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2025/10/11‗初稿投稿
2025/11/09_改訂投稿

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In 17th-century Japan, Takakazu Seki (Seki Takakazu) was an innovator who brought wasan—traditional Japanese mathematics—to unprecedented heights.
He created a unique symbolic notation called the bōsho-hō (“marginal notation”) and developed tenzan-jutsu, a method of symbolic manipulation inspired by written arithmetic. These techniques enabled him to solve higher-degree equations that had previously been intractable. His innovations opened a new mathematical horizon capable of addressing systems of equations, determinants, and even problems equivalent to those of calculus.

Thanks to his extraordinary contributions, later Japanese mathematicians referred to his school as the Seki-ryū (“Seki School”) and honored him with the title “Sansei” (Mathematical Saint).
This article explores Seki’s life and the core of his research, explaining its significance from a modern perspective.

Life of Takakazu Seki and the Rise of Wasan

Origins and Background
Takakazu Seki was born in the early Edo period into a samurai family and is said to have served as a kanjō-yaku (official accountant) for the shogunate.
Although details about his birth and early life remain uncertain, what is clear is that he demonstrated remarkable mathematical ability and greatly advanced wasan.
While wasan had its roots in Chinese mathematics, it developed its own uniquely Japanese character. Seki’s emergence marked the beginning of wasan’s mature period, symbolizing a high point in the intellectual culture of the Edo era.

Influence of Chinese Mathematics
At the time, Japanese mathematics was heavily influenced by Chinese classics such as the Suan Shu Shu and the Tian Yuan Shu (method of the celestial element).
However, the traditional Chinese method could only handle a single unknown variable, which limited its range of problems. Seki sought to overcome this limitation, and through his bōsho-hō and tenzan-jutsu, he developed a revolutionary approach capable of treating multiple unknowns simultaneously.

The Path to Becoming the “Mathematical Saint”
Seki’s achievements were passed down through his disciples and successors, and by the 18th century he was revered as the “Sansei”, or “Saint of Mathematics.”
He became a cultural giant comparable to the poet Matsuo Bashō or the tea master Sen no Rikyū, securing an enduring place in Japan’s intellectual history.

Mathematical Innovations — Bōsho-hō and Tenzan-jutsu

The Birth and Significance of Bōsho-hō
The bōsho-hō (“marginal notation”) was a unique system in which symbols and numbers were written along the margins of the page.
This allowed mathematicians to handle multiple unknowns simultaneously and greatly simplified the organization of complex formulas.
It was a groundbreaking innovation, a forerunner of modern algebraic notation, and significantly enhanced the ability to manipulate abstract mathematical symbols.

Tenzan-jutsu and the Revolution in Calculation
Tenzan-jutsu was a symbolic computational method similar to written arithmetic, used to solve higher-order polynomial equations.
It made it possible to handle complex problems involving multiple variables systematically, representing a true computational revolution in wasan.
This method contained the seeds of modern elimination theory and determinants, marking an important step toward later algebraic concepts.

Expanding the Application of Tian Yuan Shu
The original Tian Yuan Shu method had been limited to one-dimensional problems, but Seki’s introduction of bōsho-hō and tenzan-jutsu extended its application to equations with multiple variables and higher degrees.
For example, Seki systematized the calculation of the areas of regular polygons ranging from equilateral triangles to regular 20-gons, advancing mathematics in both algebraic and geometric directions.

The Development of Wasan and the Formation of the Seki School

Influence on Later Wasan Scholars
Seki’s technical innovations allowed wasan to expand into many fields. His methods streamlined computation and provided a foundation upon which later mathematicians could develop new formulas.
This influence spread throughout the 18th century, supporting the flourishing of Japan’s unique mathematical culture.

The Birth of the Seki-ryū (Seki School)
By the late 18th century, the lineage of mathematicians who followed Seki’s methods became known as the Seki-ryū.
Members of this school took pride in their intellectual heritage and adopted Seki’s notational and computational techniques as their standard.
The Seki-ryū became a major force in spreading wasan across Japan, ensuring its survival and growth for generations.

Cultural Status as the “Mathematical Saint”
Takakazu Seki transcended the role of mathematician to become a symbol of Japanese intellectual culture.
Wasan came to be regarded not only as a scholarly pursuit but as a form of artistic and cultural expression, alongside poetry and the tea ceremony.
In this context, Seki’s name was enshrined in history as the “Sansei,” the Saint of Mathematics.

Conclusion

Takakazu Seki was an innovator who played a decisive role in the history of Japanese mathematics.
Through his bōsho-hō and tenzan-jutsu, he opened a new era in which wasan could handle multiple unknowns and systematically solve systems of linear and higher-degree equations.
These achievements were carried on by later mathematicians, spreading throughout Japan as the Seki-ryū, and elevating wasan both as a scholarly discipline and a cultural tradition.

Like Bashō in poetry and Rikyū in tea, Seki Takakazu stands as Japan’s “Mathematical Saint”, his name still shining brightly in the annals of Japanese mathematical history.