Sando (算道)

Sando (the study of mathematics) was a department for the study of arithmetic and mathematics in the Daigaku-ryo (Bureau of Education) under the Ritsuryo system (a system of centralized government based on the Ritsuryo Code) in ancient Japan.

The formative period

Sando had existed as a department since the Daigaku-ryo was established based on the Taiho Ritsuryo (Taiho Code). Initially there were no clearly defined departments, but the course established to educate general government officials consisted mainly of Confucian lectures (later Myogyo-do (the study of Confucian classics)), complemented by Shodo (calligraphy), which involved writing down the original Confucian scriptures verbatim (Daigaku-ryo), and Ondo, learning to pronounce them correctly. In contrast, the fields covered by Sando had no direct relevance to Confucianism and aimed at educating technical government officials in calculation, survey, and other similar subjects; in this way, Sando was the only existence independent from the regular course.

The Daigaku-ryo consisted of two San hakase (Doctor of Numbers, equivalent to Jushichiijo (Junior Seventh Rank, Upper Grade)) who taught Sando, and thirty Sansho (students studying arithmetic). These numbers are the same as those of the Sangaku (study of mathematics) in the Kokushikan (educational ministry) of Tang Dynasty, China, which had one doctor as professor, two associate professors, and a few chokko (lecturers); while the Sando was smaller than the regular course, which had four hundred students, it is thought not to have been categorically in a low position during its early days, considering that the Tang's Kokushikan was larger than the Japanese Daigaku-ryo, and that the Tang's San hakase was put in the low position of Jukuhonge (Junior Ninth Rank, Lower Grade). Later, beginning in 730, two San tokugosho (Distinguished Scholars of Numbers) were selected from among the Sansho as special fellowship students to be candidates for San hakase. Furthermore, starting in 757, rekisansho, a combination of calendar study and Sando, was set up as another division so that Reki no sho (students of calendar) who were learning rekido (calendar study) at the Onmyo-ryo (Bureau of Divination) could learn the mathematics required for rekido.

Sandoin, a facility used for Sando lectures and to board students, was established in the Daigaku-ryo. After the national capital was transferred to the city of Heian-kyo, it is said to have been located in the south of myogyodoin, the Myogyo-do (study of Confucian classics) facilities, and the north of Myobodoin, the Myobodo (study of Code) facilities, in the premises of Daigaku-ryo. As far as can be told from existing Sansho-related records, there were many people who belonged to the clans of toraijin (people from overseas, especially from China and Korea, who settled in early Japan and introduced Continental culture to the Japanese).

Contents of Sando

The following nine types of texts were defined as Sando textbooks by the Ritsuryo law and other laws.

Sonshisangyo: A classic textbook on mathematics written under the name of Sonshi (a tactician) at the time of Western and Eastern Jin Dynasties in Three Kingdoms period China. Famous for the commentary by Chunfeng LI of the Tang Dynasty.

Gososangyo: Written by Chon Luan in the Northern Zhou period.
The document explained the basic mathematical knowledge deemed necessary for government officials in five chapters: 'Den, Hyo, Shu, So and Kin.'

Kyushosanjutsu: The oldest existing Chinese book on mathematics
A systematic document consisting of 9 chapters. Famous for the commentary by Liu Hui of the Wei Dynasty (Three Kingdoms period).

Kaitosankei: A document related to survey calculations, written by Liu Hui.

Rokusho: Details are unknown, but it is said to have applied and expanded the "Kyushosanjutsu."

Tenjutsu: A document on advanced mathematics written by Chong-Zhi ZU in which it is said that calculation methods for the circumference ratio and the volume of a sphere were written; it no longer exists.

Sankaijusa: A document related to advanced mathematics such as series expansion.

Shuhisankei: A document related to astronomical calendrical calculation. It was established in 2B.C. and is regarded as the fundamental document of calendrical calculation.

Kyushi: Details are unknown, but it is said to have been a document on mathematics geared for government officials, similar to "Gososangyo."

Furthermore, of the textbooks known as "Sankei ten documents" which were adopted in the Tang Dynasty at that time, four types; "Chokyukensankei," "Kakoyosankei," "Gokyosanjutsu," and "Shukosankei," were excluded and "Rokusho," "Sankaijusa" and "Kyushi" were adopted instead. While the details are unknown, it is assumed that this was related to redundancy in the contents, the circumstances of their introduction to Japan, or other factors.

The Hoshi Kanjintoyoshiken (appointment examination for the bureaucracy) for students was defined by the school administration, but the rules were subtly different between the Taiho and Yoro Codes. In the Taiho Code, a total of nine questions were delivered, three from one of the three basic documents ("Kyushosanjutsu," "Rokusho" and "Tenjutsu"), and one from each of the other six documents; students passed if they answered six correctly, but failed if they gave wrong answers to any of the three questions drawn from the basic documents, even if they had answered the other eight correctly. There were two methods in the Yoro Code, each of which delivered a total of nine questions: in the first, three questions were drawn from the "Kyushosanjutsu" with each of the remaining documents other than "Rokusho" and "Tenjutsu" providing one of the other six; in the second, three questions were drawn from "Rokusho" and the remaining six from "Tenjutsu," so that the former method tested knowledge of basic arithmetic while the latter tested knowledge of advanced mathematics. Students who answered all questions correctly were automatically appointed to Daishoijo (Greater Initial Rank, Upper Grade), while those who answered six to eight questions correctly, including all questions from "Kyoshosanjutsu" or "Rokusho," were appointed to Daishoige (Greater Initial Rank, Lower Grade). After 731, students could not pass the examination if they gave wrong answers to any one of four questions: one from the "Shuhisankei," or any of the three from the basic documents ("Kyushosanjutsu," "Rokusho" and "Tenjutsu").

After the establishment of San tokugosho, those who learned for seven years and passed Hoshi were appointed as San hakase or Sanshi (a court official in charge of calculation) and placed in the Shukeiryo (account office), Shuzeiryo (bureau of taxation), Dazai-fu (local government office in Kyushu region), Zogusho (ministry of making and mending palaces) (later Shurishiki (The Palace Repairs Office) and Mokuryo (Bureau of Carpentry), and were also made lower provincial governors who handled land tax accountancy and other such matters.

At that time, arithmetic was not widespread and few people had such knowledge; therefore, although the official court rank of the Sansho was low, once they passed the examination and became government officials, various government officials held great expectations for their mathematical processing abilities. However, this was arithmetic and mathematics as skill: mathematical thinking, cultivation, and sensibility were not sought after, and it seems that the advanced mathematics seen in China was not required by the Japanese society of that time, so this was not an opportunity for unique progress in the Japanese history of mathematics.

The Decline of Sando

In the early days of the Daigaku-ryo, the regular course (Myogyodo) for the education of general government officials and Sando, for the education of technical government officials, were virtually the only ones in existence, but somewhere between 724 and 729, Ritsugaku hakase (later Myoho hakase) (professor of law, for the teaching of the Ritsuryo Code) and Monjo hakase (professor of literature. for the teaching of history) were established by separating from Myogyodo, subsequently becoming independent departments between 729 and 748; they would later evolve into Myobodo (study of Code) and Kidendo (study of the histories).

During all this, Sando's position did not change greatly from what it had been before, and, as Myobodo and Kidendo were on the rise, it decreased by comparison. It began with the 802 decrease of regular Sansho students from thirty to twenty in order to increase the number of Myobosho (students of law studies). Then, the demand for Sando decreased due to the corruption of local politics posed by the decline of the Ritsuryo system. It was the same in the other departments, and gradually Sando-kyo and Sando-nenkyo began to recommend those who were unlikely to become Jogo (a student who completed a course) (Tokugosho (Distinguished Scholars of Numbers) and Hoshikyudai (examination passed)) as provincial governors and low-level functionaries of Capital officials without the examination recommended by San hakase; additionally, it was made possible for general Sansho to be qualified according to Tokugosho by satisfying certain conditions and taking an examination (Juntokugyoshoshi (Associate Distinguished Scholars of Numbers)). Under these circumstances, Sando fell to the bottom among the four studies of Kiden (biographical history), Myogyo (study of classic Confucian writings), Myobo (law), and San (arithmetic); these had been established to replace Shodo and Ondo, which were actually absorbed into Myogyo-do.

Furthermore, San hakase always had additional duties as heads or assistants in the Shuzeiryo (Bureau of Taxation) or Syukeiryo (Bureau of Popular Affairs), and one in two also worked as Goinosakan (the lowest class official in four class officials) ("Kanshoku hisho" (a text describing rules of ceremony and etiquette to be observed in the Imperial Court and by samurai that was written during the early Heian period in around 1200)). This reflected the gradual shift of the official duties of San hakase to those of central government officials of finance and accounting, rather than those of Sansho educators. In 871, in stark contrast to Sando's waning prosperity, the official court rank equivalent to San hakase was raised to Shoshichiinoge (Senior Seventh Rank, Lower Grade), a result of the high regard given to the duties of the San hakase other than Sando. Around this time, the heredity of San hakase grew in importance, and being part of a hereditary family joined passing Hoshi as a requirement for assuming the role of San hakase; initially, in the early Heian period, the Iehara and Okura clans began to make it into the heredity, but this did not last; then, after the ruin of the preceding clans, the Ozuki and Miyoshi clans began to utilize heredity, made their own Sando into hereditary learning and secret teachings, and excluded other clans by adopting capable disciples to inherit the family name.

As a result, heredity made Sando a closed discipline, and the Japanese history of mathematics entered a time of stagnation. In "Konjaku Monogatari (Shu)" (The Tale of Times Now Past) Volume 24, Chapter 22 and "Uji Shui Monogatari" (a collection of the Tales from Uji) Episode 185, a younger brother of Nyudo Toshihira TAKASHINA controlled human life and death with Sando, and people feared it, calling it 'terrible Sando,' which mirrored the actual condition that arithmetic and mathematics were no longer considered learning, much less science, and were feared and detested as magic (spells).

Only after the days of "Jinkoki" (a mathematical book) in the early modern period were Japanese mathematics again resurrected as learning in the shape of 'Wasan' (Japanese mathematics).