The History of Mathematics: A Brief Course

2. JAPAN 53

Shogun. He died at the age of 66, leaving no direct heirs. His tomb in the Buddhist

cemetery in Tokyo was rebuilt 80 years after his death by mathematicians of his

school. His pedagogical activity earned him the title of Sansei, meaning Arithmeti-

cal Sage, a title that was carved on his tomb. Although he published very little

during his lifetime, his work became known through his teaching activity, and he

is said to have left copious notebooks.

Seki Kowa made profound contributions to several areas of mathematics, in

some cases anticipating results that were being obtained independently in Europe

about this time. According to Mikami (1913, p. 160), he kept his technique a secret

from the world at large; but apparently he confided it to his pupil Takebe Kenko

(Takebe Katahiro, 1664-1739). Some scholars say that Takebe Kenko refused to

divulge the secret, saying, "I fear that one whose knowledge is so limited as mine

would tend to misrepresent its significance." However, other scholars claim that

Takebe Kenko did write an exposition of the latter method, and that it amounts to

the principles of cancellation and transposition. These two scholars, together with

Takebe Kenko's brother, compiled a 20-volume encyclopedia, the Taisei Sankyo

(Great Mathematical Treatise), containing all the mathematics known in their day.

Takebe Kenko also wrote a book that is unique in its time and place, bearing

the title Tetsujutsu Sankyo (roughly, The Art of Doing Mathematics, published in

1722), in which he speculated on the metaphysics of mathematical concepts and

the kind of psychology needed to solve different types of mathematical problems

(Murata, 1994, PP- 107-108).

In Japan, knowledge of the achievements of Western mathematicians became

widespread in the late nineteenth century, while the flow of knowledge in the op-

posite direction has taken longer. A book entitled The Theory of Determinants in

the Historical Order of Development, which is a catalog of papers on the subject

with commentaries, was written by the South African mathematician Thomas Muir

(1844-1934) in 1905. Although this book consists of four volumes totaling some

2000 pages, it does not mention Seki Kowa, the true discoverer of determinants!

Other treatises. The book Sampd Ketsugi-shd (Combination Book, but it contains

many results on areas and volumes that we now compute using calculus) was pub-

lished in 1661 by Isomura Yoshinori (Isomura Kittoku), a student of a student of

Mori Shigeyoshi. Although Isomura is known to have died in 1710, his birth date is

uncertain The book was revised in 1684. Sawaguchi Kazuyuki, whose exact dates

also are not known, wrote Kokon Sampd-ki (Mathematics Ancient and Modern) in

1671. This work is cited by Murata as the proof that wasan had developed beyond

its Chinese origins.

The modern era in Japan. In the seventeenth century the Tokugawa shoguns had

adopted a very strict policy vis-a-vis the West, one that could be enforced in an

island kingdom such as Japan. Commercial contacts with the Dutch, however,

resulted in some cultural penetration, and Western mathematical advances came

to be known little by little in Japan. By the time Japan was opened to the West in

the mid-nineteenth century, Japanese mathematicians were already aware of many

European topics of investigation. In joining the community of nations for trade

and politics, Japan also joined it intellectually. In the early nineteenth century,

Japanese mathematicians were writing about such questions as the rectification of

the ellipse, a subject of interest in Europe at the same period. By the end of the

nineteenth century there were several Japanese mathematical journals publishing