Tag Archives: Japan

Rise and Fall of Wasan

Since most would not be able to read my report on Sangaku (Artfully done tablet of Geometry) I thought I would do a blog post on the rise and fall of wasan (Japanese Mathematics) for you all to enjoy. Never mind it’s a topic I know a lot about now and have done a bunch of research on the topic.

The Japanese didn’t really have their unique math until about the year 1627 when Jink ̄o-ki was published. This was the first Japanese mathematics book published. The Jink ̄o-ki was a Japanese publication that explained how to use the soroban (Japanese name for the abacus) to do things like calculate pi, and would provide other math instruction and problems. Until then, much of the learning and study in math came from the classics of China, with heavy emphasis on The Nine Chapters and Cheng’s Treatise. I’ll explain a bit as to why there was such a long delay in developing mathematics.

Leading up to the late 1500s, most uses for math in Japan was to levy taxes on the land and for basic arithmetic for business transactions. The government of the time actually created The Department of Arithmetic Intelligence to go to each landowner and measure the property so the owner would know how much tax to pay.

These math specialists only knew just enough geometry to get the area of the land and calculate the tax required. The government saw math as a means to an end for acquiring money. This meant that math was a tool used by the government and only a special few were educated in mathematics as deemed necessary. But since there was no one to teach them, they had to rely on the Nine Chapters to be their teacher.

Around the early 1600s things began to change. A new set of rulers named the Tokugawa family took over all of Japan, uniting all the land under one government. Taxes were no longer tied to the amount of land owned and The Department of Arithmetic Intelligence was no more. This, in turn, led the farmers to no longer know how much land they had and, as consequence, the amount of food they could produce.

The Tokugawas also brought about another important change, the closing of the Japanese boarder. Iemitsu Tokugawa outlawed Christianity and closed the boarders. The problem was that a growing number of converts started a community together and began to band together. At the same time the Spaniards attempted to compete for converts and in a bold attempt to be the only missionaries in Japan told the Tokugawa family that the other nation’s missionaries were trying to create an army to conquer Japan. The Spaniards’ plan backfired and all missionaries were put to death along with those that would not give up on Christianity.

With the closing of the boarders and all the enemies of the government crushed, a period of peace was created called the Edo period in Japan that lasted until 1868 when boarders were opened again. It was during this period that the Japanese culture became its own and flourished. Everything from haiku poems to flower arranging to tea ceremonies was created during this time. By the end of the Edo period a gentleman was expected to know “medicine, poetry, the tea ceremony, music, the hand drum, the noh dance, etiquette, the appreciation of craft work, arithmetic and calculation . . . not to mention literary composition, reading and writing.” (Hidetoshi)

During the time of Great Peace the samurai became the new noble men of Japan. No longer needed as warriors, many were given government jobs to help ease them into normal lives. As consequence, the men became some of the more educated citizens. That being said, the pay they received for working for the government was terrible. Most samurai had to pick up 2nd jobs; many of them become traveling schoolteachers.

The stage was now ripe for an explosion of learning. We had farmers that needed to learn math, we had samurai that needed second jobs, and a place for it all to happen, the local shrine or Buddhist temple. Since there were no school buildings, most lessons happened at the shrines and temples that dotted the land. This encouraged more people to gather together for religious, educational, social functions. Over the next century the Japanese people would have the highest literacy rate of all the nations and become one of the most educated.

During this time, the people began to make sangaku, which is basically an artistically made wooden tablet containing a geometric problem and most of the time the solution. These tablets would adorn the temples and shrines showing off the newest knowledge learned. However, these tablets also had a deeper meaning. These sangaku became a way of thanking the gods and spirits for the new knowledge.

Many of the sangaku that have been found focused on finding lengths, areas of various shapes, and even volumes. The sangaku found below is one example of finding a length. The problem asks to find the diameter of the north circle inside of the fan. The problem is setup so that the entire area of the fan is a third of a circle and you can assume you know the diameter of the south circle. The answer ends up being (sqrt(3072) + 62 )/193 times the diameter of the south circle.

Sadly, wasan (Japanese Mathematics) was one of the few things that didn’t survive the Japanese Renaissance, which is why many of the records of wasan and sangaku are only now being discovered. At the end of the Edo period a new government was formed that outlaws wasan from being taught. It turns out that wasan lacked Calculus but more importantly, was different than the rest of the world. With the opening of the boarders, the government needed to adopt Western Mathematics to be able to communicate with all the new trade partners that were being re-established. To that end, a law was created that outlawed wasan and Western math was forced in the schools. Anyone that still taught wasan had his teaching license stripped and imprisoned.


Hidetoshi, F., & Rothman, T. (2008). Sacred Mathematics. Princeton, New Jersey: Princeton University Press.


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.