Table Of ContentA
ク、-
ADVANCED STUDIES
IN PUREM ATHEMATICS 79
Chief Editor of the Series: ,Ke n'ichi Ohshika (Osaka University)
Mathematics of Takeb e Katahiro and
History of Mathematics in East Asia
Edited by
Tsukane Ogawa (Yokkaichi University)
Mitsuo Morimoto (Seki Kowa Institute of Mathematics, Yokkaichi
University)
岱
MathematicalSociety of Japan
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Advanced Studies in Pure Mathematics 79
ISBN 978-4-86497-057-0
PRINTED IN JAPAN
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2010 Mathematics Subject Classification.
Primary 01A27.
Secondary 01A13, 01A25.
Advanced Studies in Pure Mathematics
Chief Editor
Ken'ichi Ohshika (Osaka Univ.)
Editorial Board of the Series
SHIGEKI AIDA KENGO HIRACHI YUKARI ITO
(Univ. of Tokyo) (Univ. of Tokyo) (Nagoya Univ./Kavli IPMU, Univ. of Tokyo)
SHINICHI KOBAYASHI KAYO MASUDA KENJI NAKANISHI
(Kyushu Univ.) (Kwansei Gakuin Univ.) (Kyoto Univ.)
SHIN NAYATANI KEN'ICHI 0HSHIKA WAYNE ROSSMAN
(Nagoya Univ.) (Osaka Univ.) (Kobe Univ.)
YOSHIHIRO TONEGAWA KEN-ICHI YOSHIKAWA
(Tokyo Inst. of Tech.) (Kyoto Univ.)
The Kyoho Map of Japan
Originallyo wnedb yM atsura Seizan
Now conservedi n theM oriya Hisashi Collection
Part of them aps howingt hea rea around Mt. Fuji
TheM oriyaH i畑aヽhil守犀，.~n collection,o neo fth el argeヽtprivatec ollec-
tionso fo ld mapsi n Japan,d epo5iteda tth eH iroshima PrefecturalM useumo f
Hiヽtory[広島県立歴史拇物館].containsa m ap ofJ apan madeb yt heT okugawa
Shogunated uringt heK y6h6y ears( 1716-1736): namelyt heK y6h6 Mapo f
r
Japan 訂呆I□1本図]whoヽedrawingw aヽsupervisedbyT akebe Katahiro.
Theヽhogunatecommiヽsionedaiヽ eriesof "mapso fJ apan .. .I「I本図Jduring
theT okugawa era (1603-1867). Oneo fth ese, theS h6h6M apo fJ apanl 1If 呆
H本図],was producedd uringt hem id-17th centuryb yc ombiningp rovincial
maps[I 可絵図]whicht hes hogunateh ad speciallyo rdered localf eudalol rds
tom ake. Thep rocess itself involvedr educingt hes caleo fe achm ap and then
patchingt hem togetherto f orm al argerw hole.
Later,in 1702 duringt heG enroku years( 1688-1704), another map, known
as theG enrokuM ap ofJ apan[ 冗禄H本似IJ,waヽproduced,again by patching
togethera s eto fp rovincial maps[ 元禄国絵図].WhenT okugawa Yoshimune
l徳川占宗]becamet hee ighth shogun in 1716, hef ound theret o be manyd is-
crepancies in theG enroku Map and thusd ecidedt hata n ew, more accurate
map ofJ apann eeded tob ep roduced.F ollowingt hem odels etb yt he Sh6h6
and Genroku maps, hed ecidedt hatth eG enroku provincialm aps:ヽh ouldagain
form theb asiso ft hen ew map ofJ apan,b ut thata n ew method of patch-
ingt hese togethers hould bee mployed toa djustt hei naccuracies of theo ld
map. Hea ssignedt hist askt oH 6j6 Ujisuke [北条氏如],w hoo rdered feudal
lordst o conduct surveys and to report thed irectionso fla ndmark-objectss uch
as prominent mountainsl ike Mt.F ujifr om designated spots. However,hi ヽ
method provedt oo complicateda nd H6j6 thusf ailedt op roduce thed esired
map.A t thatp oint,Y oshimuneo rderedT akebet ot akeo vera s supervisoro f
thep roject. Takebei nventeda s implea nd reasonablem ethod forc ompiling
theK y6h6M apo f Japan (1723), whicht urnedo ut tob em orea ccurate thant he
Genroku Map.O n thism ap,w ec ans ee moret han 200 survey reports which
werer eceiveda nd recordedb yT akebei n ano fficiald ocument.
If we turn tol ooka t them ap ont heo ppositep age, we cans ee thatth ere
arem anyr ed lines and place names writtena tb oth endso f each line. It is
ac haracteristicf eatuero f thism apt hat thep lace names area lmosti dentical
tot hose on thed ocument thatw as compiled from thes urvey reports. The
mapd isplays thes hapeo ft hel and in thes amew ay as ah alf-sized copyo f
theK y6h6 Mapc onserveda tt heN ationalM useum ofJ apanese History. In
addition,o n thec ornero ft he map we cans ee aw ritten statementa nd the
signaturea nd sealo f Matsura Seizan[ 松iiiJi'j和I1J, feudallo rdo f Hirado. who
―
acquirdet hem ap in 1785: the:ヽ ta tementincludes thew ords" Ky6h6M apo f
Yoshimune." Based ont hise vidence.we canc oncludet hatth ism ap is indeed
ana uthenticc opy of theK y6h6M ap of Japan.
Kuge Minoni [久下実J
ChiefC urator. HiroshimaP refectural Museumo f History
336.00 cm in length and 152.00 cm in width
三 」二三二苫
1 Rosalie Hosking 9 Andreas M. Hinz 17 Ying Jiaming 25 Kota Osamu 33 Lee Haemoon (Mrs.Kim) 41 Harikae Tcishio
2 Tamura Makoto 10 Ozone Jun 18 Morimoto Mitsuo 26 N皿 ikawaYukio 34 Harald Kfunmerle 42 Kim Dohan
3J i Zhigang 11 Ueno Kenji 19 Kitagawa Issei 27 Tanabe Sumie 35 David Mumford 43 Tang Qicui (Mrs.Xu)
4 Annick Horiuchi 12 Guo Shlrong 20 Albrecht Heeffer 28 XuZ elin 36 Sarina
5K obay邸hiTatsuhiko 13 Wate-Mizuno Mitsuko 21 Li Wenlin 29 S祖akiChikara 31 Kim Young Wook
6 Christine Hinz 14 E.H. Knobloch 22 Osada Naoki 30 Hong Young Hee 38 Koh Sung-Eun
7M ats:μmoto Takao 15 Feng Lisheng 23 Oda Tadao 31 Hong Sung Sa 39 Majima Hideyuki
8 Ogawa Tsukane 16 TanakaNoriko 24 Karine Chemla 32 Nakamura Yukio 40 WangL ifang (Mrs.Ji)
建部賢弘生誕350周年
Takebe Conference 2014
A Satellite Conference of SEOUL ICM2014
"Traditional Mathematics ofE ast Asia andR elated Topics"
Ochanomizu University, Tokyo, Japan
August 25-30, 2014
Preface
This monograph is ac ollection of papers contributed by participants at
the "International Conference on Traditional Mathematics in East Asia and
Related Topics" held August 25—30, 2014 at Ochanornizu University, Tokyo,
Japan, which was one of the satellite conferences of the Seoul ICM2 014. The
year 2014 also coincides with the three hundred and fiftieth anniversary of the
birth of Takeb e Katahiro (1664—1739), one of the great mathematicians of the
Edo period. In his honor, the conference was called the "Takebe Conference
2014."
In pre-modern Japan, from the 17th to the middle of the 19th century, Japan
was shut off from the outer world and av ernacular culture was flourishing. An
original mathematics, sometimes called wasan, was developed on the basis
of af ew volumes of ancient Chinese mathematics. One of the founders of
Japanese mathematics was Seki Takakazu (? —1708), who invented am ethod
to deal with polynomials of several variables. He considered simultaneous
systems of algebraic equations, gave an algorithm to solve them when correctly
posed, and established at heory of resultants and determinants. These algebraic
tools allowed Japanese mathematics to produce af ruitful crop of achievements.
Takebe Katahiro was the most brilliant of Seki Takakazu's disciples. He
improved on his master's method for calculating the circle constant 1r and ob-
tained the value of 1r to 43 digits. Takebe's method turns out to be equivalent
to Aitken'sぶ accelerationmethod. Going on the same idea, Takebe was able
to find an infinite term expansion of the power of the back arc with respect to
the sagitta. His discovery is nothing less than the Taylor expansion formula of
an inverse trigonometric function.
Takebe had ag reat respect for his master Seki but also felt an inferiority
complex which he had to struggle to overcome. After Seki passed away in
1708, Takebe resumed his research in mathematics and at last found his own
path of research, ap ainstaking way of numerical calculation with which he
achieved remarkable results in his studies of the circular principle, enri.
In sum, it may be said that Seki and Takebe laid the algebraic and the ana-
lytic foundations of the mathematics of the early Edo period. Their results can
be regarded as the culmination of efforts of pre-modern Japanese mathematics.
This volume is divided into four parts and an appendix. Part one is con-
cerned with Takebe Katahiro, his mathematics and his times. The editors be-
lieve that wasan represents one phase of the mathematics of East Asia, es-
pecially of China, Japan and Korea, for which ancient Chinese mathematics
served as ab asis, and Chinese characters as al ingua franca. Part two concerns
the old mathematics of Korea in the Joseon Dynasty. Weh ope that this part
will fill ag ap in the knowledge of Japanese scholars about their neighbor's tra-
ditional mathematics. Part three treats the mathematics of Ancient China. It is
regrettable that there is not sufficient content here to cover the long time span
of this topic field. Part four, finally, follows the subsequent development of
Japanese mathematics and mathematical education after the Meiji Restoration
(1868).
The very substantial appendix contains English translations of three vol-
umes (12, 17 and 19) of the Taisei Sankei (twenty volumes, 1711), am ono-
graph written by the master mathematician Seki Takakazu and the two Takebe
brothers, Kata'akira and Katahiro. Their ambition was to compile the whole of
the available Chinese and Japanese mathematics and all of their own achieve-
ments. Volume 12 treats among other things calculations of circular lengths
and arc length; Volume 17 contains the theory of determinants; Volume 19
presents 15 problems on algebraic equations.
Although the mathematics of East Asia was developed using Chinese char—
acters, we have tried to present its several aspects in English, in the hope that
this volume will open up research in the mathematics of East Asia to am ore
international readership.
The Takebe Conference 2014 was prepared for by the Organizing Com-
rnittee consisting of Karine Chemla (CNRS, France), Sung Sa Hong (Sogang
U., Korea), Dohan Kim (Seoul National U., Korea), Eberhard Knobloch (T. U.
Berlin, Germany), Tatsuhiko Kobayashi (SKIM, Japan), Hikosaburo Komatsu
(Honorary Chair; U. of Tokyo, Japan), Wenlin Li (CAS, China), Hideyuki
Majima (Ochanornizu U., Japan), Mitsuo Morimoto (Chair; SKIM, Japan),
Tsukane Ogawa (Yokkaichi U., Japan), Kenji Ueno (SKIM, Japan) and by the
Local Organizing Committee consisting of Hideyuki Majima (Chair; Ocha-
nornizu U., Japan), Tatsuhiko Kobayashi (SKIM, Japan), Mitsuo Morimoto
(SKIM, Japan), Tsukane Ogawa (Yokkaichi U., Japan), Tomohiro Takata
(SKIM, Japan), Makoto Tamura (Osaka Sangyo U., Japan), Kenji Ueno (SKIM,
Japan) and Shoko Higuchi (Secretary; Yokkaichi U., Japan).
The conference was co-sponsored by the Faculty of Science, Ochanomizu
University and received financial or moral support from the following organi-
zations: The Mathematical Society of Japan (MSJ), the History of Mathemat-
ics Society of Japan, Seki Kowa Institute of Mathematics (SKIM), Yokkaichi
University, and the Japan Society for the Promotion of Science, Grant-in-
Aid for Scientific Research (C) 22540155, (C) 23501204, (C) 23540126, (C)
23540169, and (C) 24540045; generous donation were also recieved from
Nan'o Munemura, Masayasu Ueno, Yuji Ueno and others.
Wew ould like to thank all of the above for their generous assistance.
Tokyo, March, 2017
Tsukane Ogawa and Mitsuo Morimoto
All papers in this volume have been refereed and are in final form. No version
of any of them will be submitted for publication elsewhere.
Program
August 25, 2014 (Monday)
• 12:00—17:00 Registration at Faculty of Science, Ochanomizu Uni-
vers1ty
Public Lectures in Japanese
• 13:00—13: 10 Opening Addresses [開会の辞］
• Session 1C hair: Ueno Kenji [上野健爾］
-13:10-13:50 Lecture [講演］
Imanishi Yuichiro [今西祐一郎]:Academicism and Enlighten-
ment from the Viewpoint of Japanese Notations [学術と啓蒙—
日本語表記の観点から—]
-14:00—14:40 Lecture [講演］
Kuge Minoru [久下実]:A bout aJ apanese Map made by Takeb e
Katahiro [建部賢弘の日本地図について］
• Session 2C hairs: Feng Lisheng [i: 馬立昇],Ogawa Tsukane [小川束］
-15:40—16:20 Lecture [講演］
XuZ elin [徐澤林]:T akeb e Katahiro and Shoushi li calendar [建
部賢弘と『授時暦』］
-16:30—16:55 Lecture [講演］
Fujii Yasuo [藤井康生]:T he Jujireki (Shoushili) and the Method
of Finding Differences by Seki Takakazu and Takebe Katahiro
versus the Jokyoreki and the Method of Finding Differences by
Shibukawa Harumi [授時暦と関孝和・建部賢弘の招差法 対 貞
享暦と渋川春海の招差法］
August 26, 2014 (Tuesday)
• 9:00—9:45 Opening Ceremonies
• Session 3C hairs: Hong Sung Sa [洪性士],H oriuchi, Annick
-9:50—10:30 Lecture
Ueno Kenji [上野健爾]:S eki Takakazu and Takeb e Katahiro -
Two Different Types of Mathematicians -
