Trigonometry and Its Acceptance
in the 18^{th} 19^{th}
Centuries Japan
Tatsuhiko Kobayashi
Name: Tatsuhiko Kobayashi, Historian (b. Sukumo,
Kochi Prefecture, Japan, 1947).
Address:
Department of Civil Engineering, Maebashi Institute of Technology, 4601
Kamisadori, MaebashiCity, Gunma. 3710816, Japan. Email: koba@maebashiit.ac.jp
Fields of
interest: Wasan (Premodern Japanese mathematics), the History of
Science in East Asia.
Awards: Kuwabara Award, 1987.
Publications: What was known about the polyhedra
in ancient China and Edo Japan?, In: Hargittai, I. and Laurent, T.
C., eds., Symmetry 2000, Part 1, London: Portland Press,
2002; What kind of mathematics
and terminology was transmitted into 18thcentury Japan from China?, Historia
Scientiarum, Vol. 12, No.1, 2002.

Abstract: The
mathematics developed in Japan during the Edo Period (16031867) is called
Wasan (Japanese mathematics). Wasan has its roots ancient Chinese mathematics.
The concept of angle, however, did not grow up either of ancient Chinese
and Japanese mathematics. In the sixteenth century Jesuit missionaries
as a part of their propagation activity bought Western scientific knowledge
and technology into Ming China which still had been maintaining traditional
academic system since ancient time. At that time Western trigonometry was
introduced into China, and encounter of different mathematics thought made
to mean an opening of new intelligence activities in the history of East
Asian mathematics. In 1720 the eighth Shogun Tokugawa Yoshimune permitted
the import of Chinese books on Western Calendrical Calculations from Qing
China. From this time, openly, Japanese scientists could make to study
the Western scientific and technology. It means indirect acceptance of
Western knowledge via Chinese works. And they were able to know
element of trigonometry and its application through the study of Chinese
books on Western Calendrical Calculations. During the 17^{th} and
19^{th} centuries Western scientific knowledge was also introduced
into Japan from Netherlands. Dutch interpreter Shiduki Tadao was one of
men who studied directly trigonometry from Dutch book. He comprehended
Napier’s rules in plane trigonometry. In this paper we discuss the study
of trigonometry by Wasanka and its acceptance of trigonometry from China
and Netherlands in the 18^{th}19^{th} centuries Japan.

1 TRANSMISSION OF TRIGONOMETRY
INTO CHINA
In seventh August, 1582, Italian Jesuit missionary
Matteo Ricci (15521610) arrived at Macao, China. His arrival meant not
only the reopening of Christian propagation in China but also transmission
of Western science and technology in East Asia. M. Ricci was born in 1552
at around Macerata, Italy. In his young age he left for Rome to learn law,
but in 1572 he entered to Collegio Romano, which was one of Jesuit missionary
school in Europe. There he took the lectures of the natural science; mathematics,
astronomy and geography etc. from Christoph Clavius (15381612) and other
teachers.
His main purpose came to China was, of course,
Christianity propagation. As a part of propagation activity M. Ricci strove
to introduce Western scientific knowledge and technology. For that purpose
he translated many Western academic books into Chinese in cooperation with
the Chinese intelligencers. In those days especially Chinese astronomers
wanted to have uptodate astronomical information or calculation for calendar
reform, therefore they had strongly interest in Western scientific knowledge
or mathematics and astronomical observation technique.
One of the new mathematical information, which
was brought by Jesuit missionaries, was trigonometry. As soon as it was
introduced, Chinese astronomers comprehended that trigonometry is very
useful as mathematics technique on calendar calculation. Moreover they
devoted it to apply to astronomical observation or land surveying and so
forth.
2 TRIGONOMETRIC STUDIES IN WASAN
The mathematics developed in Japan during the
Edo Period (16031867) is called Wasan (Japanese mathematics). Wasan has
its roots in Chinese ancient mathematics. Unfortunately the
concept of angle did not grow up neither Chinese nor Japanese mathematics.
But it was incomplete, Japanese mathematicians were able to have idea of
trigonometry or trigonometric function in their original study. Here we
will introduce particular example by two Japanese mathematicians.
In 1627 an elementary mathematical textbook was
published by Mitsuyoshi Yoshida (15981672). The book title was Jinkoki,
means "From the Large Number to the Small Number". In the twelfth chapter
of volume two in this book we find a problem, was named "Kobai no nobi
no koto". The author M. Yoshida asks there that how to calculate the length
of pitch of a roof and used the Pythagorean Theorem for the answer. In
our modern sense we regard meaning of "nobi" as secant, that is;
sec q= (r +
nobi) / r.
Katahiro Takebe (16641739), who was a disciple
of Takakazu Seki (?1708) and an excellent mathematician, devoted to study
that how to find the length of an arc, the chord and the arrow of a circle.
Finally he fund an equation for the chord which expresses in a series the
square of arc sin x; (sin^1x)^2. Relating to these study K. Takebe
made a numerical table for the length of the chord and the arrow in his
manuscript, Sanreki Zakko (Calculation of the Circular Arc). We
can recognize it as a kind of trigonometric function table. But he did
not use any trigonometric formulae in this calculation. His method was
quite based on geometry.
3 TRANSMISSION OF TRIGONOMETRY
INTO JAPAN
In 1726, the second edition of Mei Wending‘s works
Li
suan quan shu (the first edition was compiled in 1723) was imported
into Japan from China. Soon after of transmission of, K. Takebe and Genkei
Nakane begun to translate Mei Wending‘s works into Japanese. Then, in 1733,
K. Takebe presented the Japanese version of Li suan quan shu under
the title of Shinsha yakuhon rekisanzensho to the Shogun Yoshimune
Tokugawa, containing his preface. In this preface of Shinsha yakuhon
rekisanzensho he stresses the importance of a careful study of trigonometry.
About this time the Tokugawa government ordered
the import of trigonometric function tables from Chinese traders because
the editors of Li suan quan shu did not print the trigonometric
function tables in Li suan quan shu. One year from the arrival of
Mei Wending’s works, three books of trigonometric function tables arrived
at Nagasaki port.
Based on our research, we can confirm that one
of the three books of trigonometric function tables is reproduced
from Chong Zhen li shu (Chong Zhen Reign Treatise on Astronomy and
Calendrical Science) which were compiled by Jesuit missionaries Jacqaes
Rho (15931638) and Adam Shall von Bell (15911666) under the support of
Chinese high rank bureaucrat Xu Guang Qi (15621633) and were presented
to Emperor Chong Zhen.
After this, the use of trigonometric function
tables began to spread among Japanese scientists. The first Japanese mathematicians
to make use of trigonometry were G. Nakane and his disciples. G. Nakane
has left two manuscripts; Nichi getsu Kosoku (The Measurement of
Altitude of Sun and Moon) written in 1732 and Hassenhyo sanpo kaigi
(The Mathematical Solution by Using Trigonometric Table). These are the
earliest examples of the use of trigonometry by Japanese mathematicians.
4 THE DUTCH INTERPRETER SHIDUKI
TADAO AND TRIGONOMETRY
Toward the end of the eighteenth century, the
Dutch interpreter Tadao Shiduki (17601806) began to translate a scientific
book entitled Rekisho shinsho (The Guide Book of New Astronomy and
Natural Science). This manuscript was composed of three volumes and completed
in 1802, which was never printed. Description concerned with astronomy
and physics was quite based on Johan Lulofs (17111768)’s Inleidinge
tot de ware Natuuren Sterrekunde of de Natuuren Sterrekundige Lessen, which
was published in Leiden in 1741.
Here, we trace the process of the introduction
of Napier’s rule into Japan. Baron John Napier(15501617) Scottish mathematician,
who is wellknown as discover of logarithms and inventor of Napier’s rod,
was first referred by T. Shiduki in Sankaku teiyo hisan. T. Shiduki
describes Napier’s homeland and his name using Chinese characters. The
Chinese characters were used to mimic as closely as positive the Dutch
pronunciation equivalent. Referring to Napier’s rule for the right spherical
triangles are wrote as follows:
1. cos c r = cot A cot B.
2. cos c r = sin a sin b.
(r = 1)
These rules can be easily memorized by the expressions
cotan.
ad.
and sin. op. Now we compare his terminology and expressions
with the contents of the J. Lulofs’ book as he translates, all of them
are obviously leaded from the chapter of Over de ontbindinge der regthoekige
klootsche Driehoeken door vyf delen eens Cirkels.
5 REMARKS
Japanese mathematicians completely comprehended
the use of trigonometry by the nineteenth century. Then they knew that
a side of inscribed regular npolygon of a circle can be expressed
by trigonometric function. Those particular examples will be fund in Akiyuki
Kenmochi’s study Kakujutsu shokei and so forth.
Chinese books on Western calendrical calculations
played an important role in bringing Western knowledge indirectly, via
Chinese works, into Edo Japan. Eventually these books and those who read
them served to form a bridge between the East and the West.
References
Kobayashi,
T. (2002) What kind of mathematics and terminology was transmitted into
18thcentury Japan from China?, Historia Scientiarum, Vol.12, No.1.
Kobayashi, T. (2002) On Mei Wending’s
works in the Momijiyama Bunko Library, [in Japanese], Journal of History
of Science, [Japan], Vol. 41, No. 221.
Wasan Institute (2000) Jinkoki,
[Reprint ed.], Tokyo: Tokyo Shoseki Printing Co., Ltd.
Yan, Li and Shiran, Du (1987) Chinese
Mathematics: A Concise History, Oxford: Clarendon Press, [translated
by John N. Crossley and Anthony W.C. Lun].
*This study was conducted under
the financial support from the Science Research Fund, the Ministry of Education
and Science, Japan, GrantinAid for Scientifics Research (1).
