A Peek into Japanese Mathematics

Nearly every country had its own special role in the development of mathematics. Many stem from one another, building off of past achievements to contribute to what we now would use in modern mathematics. There are few countries that were able to develop their own mathematical theories without being derived from past work. The Japanese in particular is one of the few that stands out, in such that it is distinguished from Western mathematics. During the 1870s Japanese mathematics was given the term “wasan”, which translates to “Japanese calculation”. This was the term that distinguished Japanese mathematics theory from Western mathematics (“yōsan”). The term was used after the Edo Period (1603 – 1867), when Japan was still isolated from the rest of the world. It wouldn’t be until the Meiji Era (1868 – 1912) when that isolation ended and Japan opened up to the West, leaving the ideas of wasan behind.

Yoshida Mitsuyoshi’s Jinkōki​, Image: Public domain, via Wikimedia Commons.

The first noted mathematician in Japanese history is Mori Kambei, the teacher of Japanese mathematics. (“Mori” is the family name, so he will be referred to by this.) As expected from one of the most prominent teachers in the country, Mori had started a school in Kyoto and also wrote several books that involved arithmetic and the use of an abacus. One of his well-known students had written the mathematical text Jinkōki, one of the oldest documents written on elementary mathematics for everyday use. This student was known as Yoshida Mitsuyoshi. (“Yoshida” is the family name, so he will be referred to by this.) Yoshida was an exceptional mathematician who published his work during the Edo Period. He and his fellow students Imamura Chishō and Takahara Kisshu were known as “The Three Arithmeticians”, primarily because they were Mori’s most prominent students. Yoshida’s Jinkōki dealt with soroban arithmetic (abacus arithmetic), including square and cube root operations.

Seki Takakazu, Image: uploaded by F. Lembrez to Wikimedia Commons.

Around the same time calculus was developed in Europe, Seki Takakazu founded what was known as “enri” (circle principles). (“Seki” is the family name, so he will be referred to by this.) These principles served the same purpose as Western calculus. This system was Japan’s foundation for the development of wasan. Seki was known as “Japan’s Newton”, who created a new algebraic notation system and worked on infinitesimal calculus and Diophantine equations. All of Seki’s work was independent, unlike his European counterparts (Gottfried Leibniz and Isaac Newton, just to name a few). However, much of his work paralleled European achievements; as an example, he was credited with the discovery of Bernoulli numbers (sequence of rational numbers that appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler-Maclaurin formula, and in expressions for certain values of the Riemann zeta function). Seki’s work were more or less based on or related to algebra with numerical methods, polynomial interpolation (and its applications), and indeterminate integer equations. He also worked on the development of general multi-variable algebraic equations and elimination theory– the equivalent of Gaussian elimination to solve linear equations. This timeline nearly reflects Western discovery of mathematical theories, just a few decades earlier.

To follow up on elimination theory, Seki developed the notion of determinant. Seki’s pupil, Takebe Katahiro, came up with the  resultant and Laplace’s formula of determinant for the nxn case. Seki’s first manuscript treated only up to the 3×3 case. A large part of the problems treated at the time became solvable in principle, and the elimination method would flounder under a very large computational complexity. When the elimination is completed, the real roots of a single variable equation had to be found numerically. Diverging from elimination theory, Seki also studied the properties of algebraic equations. The most prominent were the conditions for the existence of multiple roots based on the discriminant (the resultant of a polynomial and its derivative, which was the order (h) term in f (x + h) accessible through the binomial theorem). Seki had also contributed to the calculation of pi, with an approximation that was correct to the 10th decimal place. This approximation was found using what is now known as the Aitken’s Delta-Squared Process, a series acceleration method used for accelerating the rate of convergence of a sequence.

Many of Mori’s works were succeeded by Yoshida, whose work was succeeded by Seki, whose work was succeeded by his own students and so on. As each generation continued to work on wasan, the integration of yōsan progressively established a foundation in Japanese mathematics. European ideas helped develop Japanese arithmetic, which continued to produce work nearly identical to older Western discoveries. Aside from Mori, Yoshida, and Seki, there were several other Japanese mathematicians who significantly contributed to wasan. If you’re interested, these individuals included Takebe Kenko, Matsunaga Ryohitsu, Kurushima Kinai, Arima Raido, Fujita Sadasuke, Ajima Naonobu, Aida Yasuaki, Sakabe Kōhan, Fujita Kagen, Wada Nei, Shiraishi Chochu, Koide Shuki, Omura Isshu, and many more.