Wasan (Japanese mathematics) (和算)

Wasan (Japanese mathematics) is a type of mathematics uniquely developed in Japan. In a strict sense, wasan refers to mathematics developed by Takakazu SEKI that flourished during the Edo period, but in a wider sense, it refers to mathematics in general practiced and studied in Japan before the introduction of mathematics from Europe.

Various and advanced mathematics, algebra, integration method, and geometry were developed independently of European influence.

Before the Edo period

Wasan was strongly influenced by Chinese mathematics. In China, a mathematical treatise titled "Kyusho Sanjutsu" (Art of Mathematics in Nine Chapters) appeared during the time of the Former Han Dynasty. In 'Kyusho Sanjutsu,' the topics including the methods of calculating dimensions, proportion, inverse proportion, and the Pythagorean theorem are explained. Since the seventh century, Japanese envoys to the Chinese Sui and Tang Dynasties were sent, and as a result, Chinese culture was introduced to Japan on a larger scale. In Taiho Code, based on the Chinese system of centralized governance, there were the governmental posts of San hakase (Professor of Mathematics) and Sanshi (a court official in charge of calculation). As well as contributing to the education of Sanshi, to be a San hakase, it was required to have extent knowledge of Chinese mathematical books such as "Kyusho Sanjutsu" (sando, or arithmetic). In the Manyoshu (Collection of Ten Thousand Leaves), the following poem was included.

Ever since I started to sleep on the hand of my new wife as a pillow, soft as young grass/ It is no longer possible to be separated from her over night/ How could I, when there is not a speck of hate in my heart (Wakakusa no niitamakura o makisomete yo o ya hedateru nikuku aranakuni: vol.11; no.2542). For the word 'kuku' (homonymous with '9x9'), the kanji characters of '八十一,' that denotes '81,' were used, indicating that multiplication tables were already known in Japan at that time.

It is not known what kind of mathematics was practiced during the medieval period and the early-modern period before the Edo period. It seems that Kyusho Sanjutsu was scattered and lost during this time, but a certain level of knowledge of mathematics must have been required for civil engineering, architecture, finance, and the calculation of the calendar. For there is a story about an old man who lived during the Edo period remembering using sangi (arithmetic blocks) around the time of the Taiko kenchi (Hideyoshi TOYOTOMI's nationwide land survey), and also for the fact that the method for calculating square roots recorded in "Jinkoki" is similar to calculations using sangi, it is thought that sangi was used as a primary tool for calculation up until just before the Edo period. Although it is not known when the abacus, used for everyday calculations until the time the electronic calculator became widely used, was introduced to Japan, there is an explanation of the method of calculating using an abacus in "Warizan sho" (the Book of Division) written by Shigeyoshi MORI in 1622.

Edo Period

During the Edo period mathematics greatly developed in Japan.

The beginning of the great development was triggered by "Jinkoki," written by Mitsuyoshi YOSHIDA in Kyoto during 1627. The idea for the book was taken from the Ming mathematical treatise called "Sanpo toso."
In it, there are explanations of practical mathematics such as how to use the abacus and the surveying method, as well as mathematical games such as 'mamakodate' and 'nezumisan.'
"Jinkoki" became a bestseller, and was used as a classic textbook for elementary mathematics throughout the Edo period. Many books that imitated "Jinkoki," as well as books including "jinhoki" in their titles, were published.

Although "Jinkoki" was an elementary textbook, some mathematical problems (idai) were presented at the end of certain editions without answers as a challenge to other mathematicians. Since then, the chain of solving an idai and presenting a new one in a new edition started (the succession of idai), and as such, the mathematical problems in wasan became increasingly technical and complicated beyond practical needs.

During the trends of that time, mathematical problems, not easily be solved using the basic method of calculation, started to crop up. Kazuyuki SAWAMURA introduced Tengen-jutsu (an algebra developed during 13th century China), which was an algebra equation for a variable number and its numerical method of solution, in his book "Kokon Sanpo-ki" (Old and New Mathematics), and he further presented a problem that required an algebra equation containing various numbers that could not be solved using only Tengen-jutsu as an idai to the book. As a response to this challenge, Takakazu SEKI from Edo and Yoshizane TANAKA from Kyoto created Tenzan-jutsu (also called Bosho-ho) in succession. This was how such a uniquely Japanese 'wasan' came into being that went beyond the level of merely studying Chinese mathematics, hitherto a model for everything with regard to mathematics.

At the same time, despite the fact that Kyoto had been the center of learning mathematics during the early Edo period, Seki school of mathematics, started by Takakazu SEKI of Edo, became overwhelmingly dominant around this time.
(Perhaps for this reason, the details of wasan mathematicians from the Kyoto and Osaka area are not widely known today.)

As a result of the succession of idai, the studies of algebra, the integer equation theory, mathematical analysis and geometry were developed to the advanced level beyond practical needs. Some of the mathematical discoveries as a result of such studies were known to be achieved around the same time as, or even earlier than, in Europe. Although there was a world of difference in the level of European mathematics and that of Japanese at that time, if we were to give a comprehensive evaluation, the speed in which wasan was developed was unprecedented.

The study concerning mathematical analysis in wasan is referred to as enri. Because, in the time of Takakazu SEKI or before, the main problem in mathematical analysis was the circumference ratio or the volume and surface area of a sphere, it was called the study of enri (circle principle). To give an example, Takakazu SEKI derived eleven digits of pi using the length of the sides of a regular polygon which touched the circle. Also, Katahiro TAKEBE, a disciple of SEKI, combined the method of Seki with the Richardson extrapolation, and correctly calculated up to fourty-two digits of pi. Using the result of the calculation, Takabe further calculated the power series expansion of (arcsin x)2 for the first time in the world.

In the same year, Toshikiyo KAMATA of Osaka also obtained the power series expansion of arcsin(x) and sin(x).

Since then, the study of mathematics was further developed by mathematicians who belonged to the Seki school, such as Yoshisuke MATSUNAGA, Kinai KURUSHIMA, Naonobu AJIMA, Nei WADA. Nei WADA's enri-hyo (enri table), in particular, enabled calculations equaling today's integration method, including the calculation of areas of complicated diagrams, volumes of solids, median points, and lengths of curves.

However, as the concept of integral signs did not exist, they were rendered by what can be classified as an infinite series.

On the other hand, the concept of differentials did not come to the fore in wasan.
This was partly due to the fact that wasan did not have the concept of a 'graph.'
The exception was that since the time of Takakazu SEKI, polynomial differentials had been considered in relation to repeated roots in algebraic equations. However, Seki's definition was a single term when f(x+e) was arranged for e, which did not consider its relation to the tangent. Katahiro TAKEBE applied this to the extreme value problem of polynomial function. He was probably aware of the fact that the principal term derived from the numerically miniscule difference and the derivative of polynomial equation defined by Seki was a same thing. Yoshihiro KURUSHIMA studied the extreme value problem from the point of view of series expansion, and came within an inch of understanding complex differentiation. Nei WADA also calculated Fermat's Principle, that is to say, (f(x + e/2) - f(x - e/2))/e, and delivered the method in which it was expressed as e 0.

Because the concept of differentials was not developed, there was no fundamental theorem of calculus in wasan. As such, in wasan it was not possible to calculate integration as an opposite of a differential nor was it possible to utilize partial integration. For calculating the integration of a complicated function, they skillfully used power series expansion and the formulae for the sum of series.

The study of integral equations and arithmetic was further developed by inheriting the tradition of Chinese mathematics. Takakazu SEKI, for example, gave a general solution of simultaneous linear diophantine equation, and Yoshihiro KURUSHIMA adopted Euler's function.

The main approach of wasan was characterized by numerical calculations of algebra. Especially during the period when Takakazu SEKI and Katahiro TAKEBE were active, problems concerning diagrams were represented and solved as algebra problems by applying simple relations such as Pythagorean theorem. The problems which solved the relationships between circles and ovals that touch each other, such as seen on Sangaku (Japanese votive tablets featuring mathematical puzzles), started to draw attention during the time of Yoshisuke MATSUNAGA. Naonobu Ajima, who belonged to the next generation, founded sansha sanen-jutsu (the method of three diagonals and three circles; Malfatti's Problem) and contributed to further solving these mathematical problems in a systematic manner. At the end of the Edo period, Zen HODOJI introduced a simplifying technique in which inverted circles are reflected in straight lines. In recent years, the beautiful geometric theorem discovered by wasan mathematicians is drawing the interests of people (at least on an avocational level), and is being introduced world-wide.

On the other hand, the tendency to rely on algebraic and numerical calculations for solving problems always remained with wasan. In wasan, construction problems were scarcely dealt with, not to mention Euclidean axiomatic geometry, which was entirely outside their interest. According to the memoir of a western teacher who taught at Kaigun Denshu-sho (the Naval School in Edo) at the end of the Edo period, Japanese people were strong in algebra, but they were slow to understand geometry.

Many of the findings in wasan were considered secret and protected within each school of mathematics.
(It is unclear, however, how much of such a secrecy had been kept intact.)
Yoriyuki ARIMA, lord of the Kurume Domain who learned arithmetic of the Seki school, revealed the mathematical secrets taught in Seki school in his "Shuki Sanpo," published in 1769, and contributed greatly to the advancement of wasan culture. "Sanpo-shinsho" (New Mathematics), supervised by Hiroshi HASEGAWA, edited by Tanehide CHIBA and published in 1830 (the end of Edo period), explained wasan in detail from the entry level up to the latest findings.

After the Meiji Period

During the Meiji period, western mathematics were introduced on a full scale (The word 'wasan,' meaning Japanese mathematics, was coined around this time as a term that counters 'yosan,' or western mathematics). As a result, wasan followed a course of decline. A symbolic event of such a decline was when the government at that time issued the proclamation of education in 1872 and declared that "wasan should be abolished and we should practice only yosan."
(The study of shuzan - calculation using an abacus was restarted the next year, however.)
However, the introduction of western mathematics at the beginning of the Meiji period emphasized the importation of technical tools from the west, and as such, the early 'yosan mathematicians' were technicians, rather than experts of mathematics. When Tokyo Sugaku Gaisha (Tokyo Mathematical Company), the former The Mathematical Society of Japan, was established in 1877, many wasan mathematicians, who were indeed more skillful and had better understanding of mathematics than yosan mathematicians, joined. Even then, new editions of books on wasan were published. One of the main criticisms against wasan was that it was "useless in terms of practical usage." In 1884, the full-scale transition to western mathematics took place when Tokyo Sugaku Gaisha was incorporated into the Japan Mathematics and Physics Society.

But until the fully-fledged introduction of western mathematics was under way, wasan (as well as wasan mathematicians) also played a part in the modernization of the country on the practical side. Itsumi UCHIDA, a famous wasan mathematician of Seki school, played a key role in the introduction of the solar calendar to Japan in 1873. There were others, such as Riken FUKUDA, who became successful in the field of surveying. Tomogoro ONO, a technician and a bureaucrat from the end of Edo period to the beginning of the Meiji period, was also a wasan mathematician, and it is said that he applied wasan for calculating the shipping route of Kanrin Maru (the first Japanese ship ever to cross the Pacific). Wasan was also applied to kiku-jutsu, which is a drafting technique for carpenters, in order to re-organize its theoretical aspects during the end of Edo period. The study on kiku-jutsu developed well into Meiji period, and the strong influence of wasan was palpable even as late as 1887. Wasan mathematicians continued to play important roles in elementary education during the Meiji period, and what is taught as tsurukamezan (solving a system of linear equations) for arithmetic classes today is said to be a vestige of wasan from earlier times.

In terms of the social background for producing specialists in mathematics, there were several contributing factors: the active money economy; the production of territorial maps of provinces; and demand for land surveying for the purpose of new field development and so on. The study of the calendar also required an advanced level of mathematics. Takakazu SEKI joined a group producing a territorial map of the Koshu Domain, and he also studied Juji-reki Calendar in preparation for the changing of a calendar (which did not come to fruition in the end). Some suggest that the studying calendars was an important incentive for Seki's research in mathematics.

Sangi and the Abacus

Sangi (arithmetic blocks) and abacus are tools used when practicing wasan. "Sanpo toso" explains how to use both sangi and the abacus. "Jinkoki" also contains an explanation of how to use the abacus with detailed illustrations.

While the abacus was used for a wider purpose such as accounting, sangi was used exclusively by wasan mathematicians for calculating with Tengen-jutsu (the theory of algebraic equations developed in China) and so on. In Tengen-jutsu, a combination of a chart and sangi called sanban was used. Sanban is a grid chart with each column expressing numbers such as one, ten, one hundred, one thousand, ten thousand, and each row expressing answers to algebraic equations and their coefficients, such as "sho" (answer), "jitsu" (constant term), "ho" (x), "ren" (x2), "gu" (x3), "sanjo" (x4) and so on. Algebraic equations were solved by replacing a block of sangi with others in each cell. Takakazu SEKI formulated a method of calculation for solving algebra equations with a pen and paper alone without using sangi. This method was later called Tenzan-jutsu, and became the principal method in wasan.


Sangaku is a framed picture or Ema (a votive picture tablet) with mathematical problems and answers written on it, and dedicated to shrines and temples. Many of sangaku deal with problems concerning plain figures. Sangaku were dedicated not only by mathematicians, but also by math-lovers and practitioners in general.

It is said that sangaku was dedicated as a sign of gratitude to gods for being able to solve mathematical problems and to promise them to devote oneself further to studying. Such shrines and temples became a place where people gathered to present and solve mathematical problems; some dedicated sangaku with difficult problems without answers written on them, and others who solved such problems dedicated their own sangaku with answers to problems dedicated earlier. The dedication of sangaku is a practice unknown to other countries, and is unique to Japanese culture. When yosan, or western mathematics, was introduced to Japan after opening the country during the Meiji period, this practice of dedicating sangaku also contributed to the smooth introduction of a new type of mathematics.

According to the survey carried out in 1997, there are 975 pieces of existing sangaku all over Japan ("Reidai de shiru nihon no sugaku to sangaku," published by Morikita Publishing Co., Ltd.). The oldest existing sangaku is the one known to be dedicated in 1657 in Hoshinomiya-jinja Shrine in Sano City, Tochigi Prefecture. Sangaku is the theme of the novel "Sanpo shojo," written by the novelist Hiroko ENDO.

People who Contributed to the Development of Wasan
Shigeyoshi MORI
Mitsuyoshi YOSHIDA
He wrote "Jugairoku" (1639). "Jugairoku" was a collection of formulae regarding surveying and quadrature. It was written in kanbun (Sino-Japanese) targeting the specialists. An approximate formula regarding the relationship between the arc and chord was included.

Yoshizane TANAKA
He was a wasan mathematician from Kyoto. He practiced multivariable algebra equations by writing them down on paper. He studied small-sized determinants and resultants. Tanaka became independent from Takakazu SEKI. He researched on mahojin (magic square) and sugaku-yugi (mathematical tricks). He wrote "Sanpo meikai" (1679) and "Sangaku funkai" (circa 1690).

Tomotoki IZEKI
He was a wasan mathematician from Osaka. He studied the theory of determinants and resultants. Izeki became independent from Seki. He wrote "Sanpo hakki" (1690).

Toshikiyo KAMATA
He was a wasan mathematician from Osaka. He evaluated the least upper bound and the greatest lower bound of the circumference ratio by calculating the circumference of the inscribed polygon and the circumscribed polygon. In his "Takuma-ryu Enri" (1722), he presented the infinite series expansion such as arcsin and sin. Alongside with that of Katahiro TAKEBE, this was the first time infinite series expansion was defined in Japan. He wrote "Enri hijutsu."

Takakazu SEKI
Katahiro TAKEBE
Murahide ARAKI: he reached the first level of learning at the Seki school. After the death of Takakazu SEKI, he organized the posthumous writings of his master and edited "Katsuyo sanpo." He did not achieve a great deal in terms of mathematics, and it is said that the fact that "Katsuyo sanpo" is full of spelling mistakes is an indication of Araki's lack of skill.

He was a wasan mathematician. He presented a way to arrive at an equal temperament of twelve degrees by opening one octave into the twelfth root in his "Ritsugen hakki" (1692). He was also well-versed with the study of the calendar, and together with Takebe, he appealed to Shogun Yoshimune TOKUGAWA the need to import western books on astronomy translated into Chinese. Besides mathematics, Nakane had great knowledge of many fields. He was originally from Kyoto, and studied mathematics under Yoshizane TANAKA at first, but later became a disciple of Katahiro TAKEBE.

Date of birth unknown – 1757: he was also famous as the originator of tsume shogi (tume shogi problems).

He studied infinite series, especially double series for the first time in wasan. He furthered the study of enri (circle principle), which was started in earnest by Takebe. He presented lots of the achievements of his friend Kurushima in his own writings. He reached the second level of learning at the Seki school.

Nushizumi YAMAJI
He reached the third level of learning at the Seki school. He reorganized the system of teaching at the Seki school, and taught many disciples.

Naonobu AJIMA
Yasuaki AIDA
Sadasuke FUJITA
1734 – 1807: he was an excellent teacher and contributed greatly to the dissemination of wasan. He avoided unnecessarily complicated mathematical problems, and placed an emphasis on systematic and general solutions. He wrote "Seiyo Sanpo" (1781). He also became famous from debating with Yasuaki AIDA. He reached the fourth level of learning at the Seki school.

1787 - 1840: he is renowned for Enri-hyo (the enri chart that charts the results of the definite integral of 0.1 intervals of various functions), which he completed. He also generalized the theory of the double series of Ajima. As a result, complex quadrature problems became easier to solve. He presented Fermat's Principle of differentials, and applied it to the extreme value problem. He was originally a feudal retainer of the Mikazuki Domain of Harima Province and later became terazamurai (samurai who performed administrative functions at temples) at Zojo-ji Temple, but was expelled as a result of his bad behavior, and ended up teaching mathematics and calligraphy, as well as practicing fortune-telling. He was a high spender, and as such, his wife and children had to live on the street after his death. Wada's originality and talent, however, were truly distinguished, and many famous wasan mathematicians of the day became his disciples in order to see his enri-hyo.

Shingen TAKEDA
1820 - 1868: he was active at the end of Edo period. At that time, people were concerned with problems concerning the relationship of the radii of several circles that touch each other. In order to simplify the problem, he introduced Sanpenho, that simplified the calculation by converting a circle into a straight line. This method equals inversion today. He also dealt with problems regarding the center of gravity of diagrams as well as cycloids.


[Original Japanese]