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Introduction of Seki Takakazu (Kowa)

Shigeru JOCHI


INTRODUCTION@OF SEKI TAKAKAZU

    (1)  The influence of Chinese mathematics on Wasan (Japanese mathematics)

    The influence of Chinese mathematics was felt in most East Asian countries.
In the case of Japan, it was introduced into the country two times.  The first
time was in the eighth century, when many mathematical arts were introduced
and taught in the University, but Japanese mathematicians only imitated the
work of Chinese authorities, and its level was limited.  The second time was at
the end of the sixteenth century|at which point, Japanese mathematicians applied
Chinese mathematics, and were to produce brilliant achievements.
    It is difficult to make a comparative study of the mathematics of two
completely different civilizations, because they do not have the same
intellectual background.  But Chinese mathematics and Japanese mathematics used
the same language, rather than just the same Chinese characters, thus
mathematicians could understand mathematical notions easily.  That is to say,
if Japanese mathematicians had Chinese mathematical books, they could have had
the same background as Chinese mathematicians.  I wonder whether it is possible
to make a comparative study of them using historical method.  Instead, through
studying Japanese mathematics in the 17th century, we may be able to understand
strong and weak points of Chinese mathematics in the 13th century.
    I would like to consider the case of Seki Takakazu 萍F˜a (1642?-1708), the
best mathematician in Japan.  He very probably studied Chinese mathematics, but
it is difficult to demonstrate precisely how he studied Chinese mathematics
from his biography.  Because his son-in-law gambled away his post, nobody could
hand down Seki Takakazu's biography.  Thus we cannot know exactly the nature of his
education.  Therefore, I will have to try, in this thesis, to consider his
education in light of the similarity between his works and Chinese mathematics.

    (2)  Biographical study of Seki Takakazu

    Because nobody could hand down Seki Takakazu's biography, we cannot even fix
his birth year exactly.  He became revered as a "Sansei" ŽZ¹ (mathematical
sage), and the scientific studies of his biograpy was neglected.  However,
since Mikami Yoshio's ŽOã‹`•v (1875-1950) study (*1) , there are some good
studies (*2), and his biographical table(*3) is as follows

                              Seki Takakazu's biography                             

1637? - 1642?
      Born in Edo ? (now Tokyo) or Fujioka ? (now Fujioka-shi, Gumma prefecture
      ŒQ”nãp“¡‰ªŽs).
      The Second Son of Uchiyama Nagaakira “àŽR‰i–¾ (?-1662?), and of the daughter
      (?-1682, name unknown) of Yuasa Yoemon “’ŸÇäo‰E‰q–å. His  other names are
      Sinsuke V•, Shihyo Žq•^ and Jiyute Ž©—R’à.
year unknown
      Became a son-in-law of Seki Gorozaemon  萌ܘY¶‰q–å(?-1665).
1661
      Transcribed the Yang Hui Suan Fa  —k‹PŽZ–@ (Yang Hui's Method of
      Computation) at Nara “Þ—Ç.
      Became a member of Load Kofu (after became 6th Shogun of Ienobu “¿ì‰Æé).?
1663
      Wrote the Kiku Yomei Sampo  ‹K‹é—v–¾ŽZ–@ (Essential Mathematical Methods
      of Measures)
1665
      Seki Gorozaemon was died.
1672?
      Wrote the Ketsugi-Sho To-jutsu  荋^´“šp (Answers and Methods of the
      Sampo Ketsugi-Sho).
1674
      wrote the Huttan Kai To-jutsu –ܜ݉ü“šp (Answers and Methods of the
      Sampo Huttan Kai).
      In Dec., published the Hatsubi Sampo  ᢔ÷ŽZ–@ (Mathematical Methods for
      Finding Details).

1680
      In Mar. wrote the Juji Hatsumei ŽöŽžá¢–¾ (Comments of the Works and Days
      Calendar).
      In July, wrote the Happo Ryakketsu ”ª–@—ªŒ (Short Explanations of Eight
      Items).
1681
      In Apr., wrote the Jujireki-kyo Rissei no Ho ŽöŽž—ïãS—§¬”V–@ (Methods of
      Manual Tables of the Works and Days Calendar).
1683
      In June, wrote the Shoyaku no Ho  ”–ñ”V–@ (Methods of Reduction),
      Sandatsu Kempu no Ho   ŽZ’E鄕„”V–@ (Methods of Solving Josephus Question)
      , Hojin Ensan no Ho  •ûwš¢·”V–@ (Methods of Magic Squares and Magic
      Circles).
      In Aug., wrote the Kaku Ho Šp–@•À‰‰’iš¤ (Methods of Angles and Figures of
       Japanese Algebra).
      On 9th Sep., wrote the Kai Hukudai no Ho ‰ð•š‘è”V–@ (Methods of Solving
      Secret Questions).
1685
      In Aug., wrote the Kai Indai no Ho ‰ð誑è”V–@ (Methods of Solving
      Concealed  Questions).
      Wrote the Byodai Meichi no Ho •a‘è–¾’v”V–@(Methods of Correcting Failures 
      as Questions).
      In Nov., wrote the Kaiho Hompen no Ho ŠJ•û–|Ì”V–@ (Overturn Methods of
      Solving Higher Degree Equations).
      In Dec., wrote the Daijutsu Bengi no Ho ‘èp™ž‹c”V–@(Methods of
      Discriminant).
      Wrote the Kai Kendai no Ho  ‰ðŒ©‘è”V–@ (Methods of Solving Findable
      Questions), Ky#seki ‹Ï (Computations of Area and Volume), Ky#ketstu
      Henk$ So Ÿ{荝̌`‘ (Manuscript of Transformation of Spheres) and Kaiho
      Sanshiki ŠJ•ûŽZŽ® (Formulae of Solving Higher Degree Equations).
1686
      In Jan., wrote the Seki Teisho 萒ù‘ (Seki Kowa's Amendments).
1695
      Became Makanai-gashira ˜d“ªichief treasurer, 200-pyo •U and 10-nin buchi
      •}Ž (ten retainers' salary)
1697
      In May, wrote the Shiyo Sampo ŽléPŽZ–@ (Mathematical Methods of Computing
      Four Points on the Lunar Orbit).
1699
      In Jan., wrote Temmon Sugaku Zatcho “V•¶É›{趒˜ (Notes of Astronomy and
      Mathematics).
      Became Kanji-gata Yoyaku Š¨’è•û—p–ðiauditor, 300-pyojH
1701?
      Was Kanji-gata Yoyakui300-pyoj
@@@Uchiyama Nagayoshi (Seki's 3rd brother) “àŽR‰iÍ became Kanjo Š¨’èiAccouter,
      100-pyojof Load Kofu.
1702
      Was Kanji-gata Yoyakui300-pyoj
1704
      In Nov., gave a "Sampo Kyojo" ŽZ–@‹–ó (Licence of Mathematics) to Miyaji
      Shingoro  ‹{’nVŒÜ˜Y.?
      In Dec., when Tokugawa Ienobu became Shogun, Seki Kowa became a Nando
      Kumigashira ”[ŒË‘g“ª (chief treasurer, 250 Pyo and  10nin-buchi, then 300 Pyo.   
1706
      In Nov., retired, and became a member of Kobushin-gumi  ¬•¿‘g (lit.
      small builders group).
1708
      On 24th Oct., died from a disease.
1710
      The Taisei Sampo  ‘听ŽZ–@ (Complete Works of Seki Kowa) was edited.
1712
      The Katsuyo Sampo Š‡—vŽZ–@ (Essential Mathematics) was published.
1714
      On 30th Mar., the Shukuyo Sampo h—jŽZ–@ (Mathematical Methods of
      Constellations).
1724
      Seki Kowa's son-in-law, Hisayuki ‹v”V (1690-?) (or Shinshichiro VŽµ˜Y) became
      Kofu Kimban b•{‹Î”Ô (member of Kofu city office).
1737
      Seki Hisayuki lost his position owing to his gambling activities.

    As above, Seki Kowa's works are quite huge, however, there are few
evidences about his education and his personal infomation.  Even his exact name
is notknown to foreign scholars.  Now, let us consider this problem.
    In Eastern Asian countries, an adult had some names, which were called "Zi"
(or "Azana" in Japanese)  Žš (alias) or "Hao" (or "Go" in Japanese) åj (penname)
and so on.  Their real name was used only in the family, "Zi" (alias) was used
officialy, thus the real name was not known to unrelated person.
    In the case of Seki Kowa, his alias ("Azana") was Shinsuke V•, and his
styles or pen-names ("Go") was Shihyo Žq•^ and Jiyute Ž©—R’à, and real name was
Takakazu  F˜a.  He used his real name when he published books, it was a custom
of Japanese mathematicians in that age.  Thus readers did not know how "F˜a"
was read.
    It does not mean that the education of readers was inferior.  It was a
characteristic of Japanese names.  Ancient Japan had no written characters of
its own; Chinese characters were introduced in the Nara  “Þ—Ç period, and not
before.  Chinese characters have no element of pronunciation, i.e., they are
not phonetic signs.  Therefore Japanese could read one Chinese character in
several ways of pronunciation.
    For example; the character of É (numbers) is read "Shu" by Chinese.
Japanese imitated this sound, but the system of Japanese utterance was not the
same as that of the Chinese, thus Japanese spoke with an accent to read this
character to "Su".  It is called "On" ‰¹ (Chinese pronunciation) reading.  And
this character means "number", In Japanese it is read "Kazu", which means
"number".  This is the translation, it is called "Kun" ŒP (Japanese
pronunciation) reading.
    Moreover Chinese culture was introduced for many years and from many
localities in China, sometimes the pronunciations were different.  Thus there
are three "On" readings in Japan, basically.

                                          Kan-on  Š¿‰¹ (Han dynasty's)\\1)  
    On ‰¹ (Chinese pronunciation)         Go-on   Œà‰¹ (Wu dynasty's) \\2)  
    b                                    So-on   ‘v‰¹ (Song dynasty's) \3)  
    Kun  ŒP (Japanese pronunciation)    \\\\\\\\\\\\\\\\\4)  
                                                                                
    Usually, Confucian terms are "Kan-on" reading and Buddhist's terms are "Go-
on" or "So-on" reading.
    The two characters "F" and  "˜a" can be read in the followings ways:

              1) Ko                         1) Ka                               
       "F"   2) Kyo                 "˜a"   2) Wa                               
              3) -                          3) -                                
              4) Takashi (adv.)             4) Kazofu (v.)
                 (Tsukaeru (v.))               (Yawaragu (v.))
                                               (Nagomu (v.))

    Therefore it is possible to read "F˜a" in nine ways.
    There are some rules to read a set phrase, which is consisted of two
Chinese characters. If "On" reading is used for the first character, the same
would apply in reading the second character; if "Kun" reading is used for the
first character, it is also used in reading the second character.  Usually
"On" and "Kun" readings are not used together, but there is no absolute rule.
Japanese personal names are often too difficult to read.
    In Japan, when there is uncertainty concerning how a name is read it is
customary to use the "On" reading to avoid serious mistake.  Seki Kowa became
too famous, most Japanese mathematicians knew him only through his mathematical
arts.  Thus Seki Takakazu became known as Seki Kowa.
    For Japanese mathematicians I give the original pronunciation of the real
name if possible, but sometimes I cannot read the name.  Thus all nemes of
Oriental persons are given in Chinese characters, together with birth year and
death year in the text.


    (3)  Similarity between Chinese mathematics and Japanese mathematics

    As we considered in section 2, Seki Kowa's works are  in many fields.
According to Hirayama Akira's •½ŽR’ú (fl.1959) studies(*4), these are classified
under 15 categories, as follows:

                   Seki Takakazu's works                                            
    1) "Bosho-ho" –T‘–@ and "Endan-jutsu"  ‰‰’ip, Japanese algebra
    2)  solution of higher degree equation
    3)  properties (e.g., number of solutions) of higher degree equation
    4)  infinite series
    5)  "Reyaku-jutsu"  —ë–ñp, approximate value of fractions
    6)  "Senkan-jutsu"  ãƊǏp, indeterminate equations
    7)  "Shosa-ho"  µ·–@, the method of interpolation
    8)  Obtained Bernoulli numbers by "Ruisai Shosa-ho" —ݍُµ·–@
    9)  computing area of polygons
    10)  "Enri"  š¢—, principle of the circle
    11)  Newton's formula by "Kyusho"  ‹‡¤ method                                  
    12)  computing area of rings
    13)  conic curves line
    14)  "Hojin" •ûw, magic squares, and "Enjin"  š¢w, magic circle
    15)  "Mamakodate"  ㋎q—§, Josephus question
        "Metuke-ji"   –Ú•tŽš, game of finding a Chinese character
                                                                                
           Table  1     Hirayama's classification of Seki Kowa's works

    Each of Seki Kowa's works was a great one, but most of the subjects he took
up were not his original ideas, being typical works of Chinese mathematics in
the Song  ‘v and Yuan Œ³ dynasties.  These are, according to Li Yan's —›™V
(1892-1963) studies(*5), as follows:

        a) "Cheng-Chu Ke-Jue" ˜©œ‰ÌŒ (verses for multiplication and division) 
               Yang Hui Suan Fa —k‹PŽZ–@ (Yang Hui's Method of Computation)
               Suan Fa Tong Zong  ŽZ–@“@ (Systematic Treatise on Arithmetic)

        b) "Cong-Huang-Tu Shuo" c‰¡š¤à (magic squares)
               Yang Hui Suan Fa (Yang Hui's Method of Computation)
               Suan Fa Tong Zong (Systematic Treatise on Arithmetic)
        c) "Shu Run"  É˜_ (numbers theorems)
               Shu Shu Jiu Zhang  É‘‹ãÍ (Mathematical Treatise in Nine
               Sections)
        d) "Ji-Shu Run" ‹‰É˜_ (series)
               Meng Xi Bi Duan  –²Ÿâ•M’k (Dream Pool Essays)
               Yang Hui Suan Fa (Yang Hui's Method of Computation)
               Si Yuan Yu Jian  ŽlŒ³‹ÊŠÓ (Precious Mirror of the Four Elements) 
        e) "Fang-Cheng Run" •û’ö˜_ (higher degree equations)
               Ce Yuan Hai Jing ‘ªš¢ŠC‹¾ (Sea Mirror of Circle Measurements)
               Shu Shu Jiu Zhang (Mathematical Treatise in Nine Sections)
               Suan Xue Qi Meng ŽZ›{Œ[–Ö (Introduction to Mathematical Studies) 
        f) "He Yuan Shu"  Š„š¢p (the method of dividing the circle)
               Shou Shi Li  ŽöŽž—ï (Works and Days Calendar)

                                    Table  2                                    
          Classification of Chinese mathematics works and important books
          in the Song and Yuan dynasties

    Considering to which categories each Seki Kowa's work belongs in table 2,
we see that only  1) and  14) are Seki Kowa's original subjects, the others
belong to inherited Chinese subjects in the Song and Yuan dynasties, as
illustrated in table 3.
                  {\\\\\\\\\\\{\\\\\\\{
                  b      categories      b Seki's work  b                    
                  {\\\\\\\\\\\{\\\\\\\{                    
                  b a) verse             b-             b                    
                  {\\\\\\\\\\\{\\\\\\\{                    
                  b b) magic squares     b14)           b                    
                  {\\\\\\\\\\\{\\\\\\\{                    
                  b c) indeterminate eqs.b5)6)15)       b                    
                  {\\\\\\\\\\\{\\\\\\\{                    
                  b d) series            b4)7)8)11)     b                    
                  {\\\\\\\\\\\{\\\\\\\{                    
                  b e) higher degree eqs.b2)3)          b                    
                  {\\\\\\\\\\\{\\\\\\\{                    
                  b f) circles           b9)10)12)      b                    
                  {\\\\\\\\\\\{\\\\\\\{                    
                                    Table  3                                    
          Similarity between Chinese Mathematics and Seki Kowa's Works          

    In table 3, category f) describes one of the most popular subjects in
Eastern mathematics.  Since the Jiu Zhang Suan Shu  ‹ãÍŽZp (Nine Chapters on
the Mathematical Arts), most mathematical books described this subject.  Thus
it is not worth our while considering this origin.  While category a) is one of
the most important subjects for primary pupils of mathematics, it is not
necessary to consider it in this thesis.  Therefore we will consider categories
b) to e) in the next section.

    (4)  Opinions of how Chinese mathematics influenced on Seki Kowa

    As we considered above, there is no doubt that Seki Kowa was influenced by
Chinese mathematics.  Of course, he, as the other Japanese mathematicians,
studied two popular texts, the Suan Fa Tong Zong and the Suan Xue Qi Meng.  But
the main works of the former fall into categories a) and b) of table 2, and the
latter falls entirely into category e).  Moreover, these are only introductions
to these subjects.  Thus  it is not only these two books  that influenced  Seki
Kowa. 
 We must consider the more important books.
    But his biography is unreliable so much so that, we cannot know even his
birth year.  We have some indirect evidence about which Chinese mathematical
arts influenced Seki Kowa.  We will consider this first.


         (a)  Opinions that Seki Kowa studied Chinese mathematics

    According to chapter 5 of the Burin Inkenroku •—Ñ誌©˜^ (Anecdotes of
Mathematicians) by Sai To Ya Jin   âV“Œ–ìl (18c), written in 1738, Seki Kowa
discovered a difficult mathematics book in Nara  “Þ—Ç, and studied it.

               There was a Chinese book with Buddhistic books in Nanto  “ì“s
          (lit. northern capital, Nara “Þ—Ç), but nobody had been able to
          understand it.  That book was not a Buddhistic book, a Confucian book
          nor a medical book.  It was not known what kind of book it was, so it
          was only mended and given a summer airing.  Shinsuke V• (Seki Kowa)
          knew it, and he guessed that it might be a mathematical book.  He
          took vacations and went to Nanto to borrow it.  He stayed there and
          sat up all night to copy it.  Then he brought this hand-copied book
          back to Edo  ]ŒË (Tokyo “Œ‹ž).  He studied it day and night for
          three years, at last he mastered the secrets.  He became the best
          mathematician in Japan (*6) .

   And Aida Yasuaki ˜ð“cˆÀ–¾ (1747-1817) criticized the alleged fact that Seki
Kowa had burned this text-book in the Toyoshima Sankyo Hyorin  沓‡ŽZãS•]—Ñ
(Toyoshima's Comments about Mathematical Manual), written in 1804:

              Toyoshima Masami  沓‡³”ü (?-?)  said, "Seki Kowa was a good
          mathematician but his manner of study was poor.  He burned his
          mathematical text book which he found."  It seems probably that he
          burned the book because he had plagiarized Chinese mathematical
          methods and then wrote about them as if they were his own work. (*7)

    It had been known that Seki Kowa referred to Chinese mathematical texts
since that time.  Very probably he had studied some Chinese mathematical books
that were not popularly known, and obtained certain ideas for his own works.
    Many scholars concluded what Seki Kowa's text book was.  Here I introduce
the former opinions and would like to comment on them.

         (b)  Opinion concerning the Suan Xue Qi Mengg (Introductionn to
              Mathematical Studies)

    In a memorandum which is kept in Mito Shokokan  …ŒË²lŠÙ (Mito private
school), Honda Toshiaki  –{‘½—˜–¾ (1744-1821) concluded that the book Seki Kowa
copied was the Suan Xue Qi Meng.

              Seki Kowa was self-educated.  First, he studied three
          mathematicians' books, Imamura Chisho ¡‘º’m¤ (?-?), Yoshida
          Mitsuyoshi  ‹g“cŒõ—R (1598-1672) and Takahara Yoshitane ‚Œ´‹gŽí
          (?-?), and he mastered these mathematicians' strong points.  He
          became the best mathematician. 
              Then he borrowed Suan Xue Qi Meng (Introduction to Mathematical
          Studies) at Kofuku-ji ‹»•ŸŽ› (Kofuku temple) in Nanto, hand-copied it
          and mastered "Tengen-jutsu" “VŒ³p ("Tian Yuan Shu" in Chinese,
          technique of the celestial element).  Then he continued to study
          mathematics, and completed the great works.(*8)

    The Suan Xue Qi Meng is the introduction to "Tian Yuan Shu" (Technique of
the Celestial Element); this view is directed at Seki Kowa's work in higher
degree equations.  However, the Suan Xue Qi Meng was already republished in
1658 in Japan.|Hisada Gentetsu ‹v“cŒº“N (?-?) translated it to Japanese(*9).
So, of course, Seki Kowa studied the Suan Xue Qi Meng, but it would not have
been necessary to go to Nara to find this text book.  Mikami Yoshio ŽOã‹`•v
(1875-1950) suggested that it was Hisada Gentetsu that found the Suan Xue Qi
Meng at Tofuku-ji “Œ•ŸŽ› (Tofuku temple) in Raku  —Œ (Kyoto ‹ž“s) (10), not at
Kofuku-ji (Kofuku temple) in Nara.  Moreover the works of higher degree
equations had been realized by Sawaguchi Kazuyuki  àVŒûˆê”V (17c)(11).
Therefore, we cannot accept this evidence.

         (c)  Opinion concerning the Ce Yuan Hai Jing (Sea Mirror of Circle
              Measurement)

    Kano Ryokichi Žë–ì‹œ‹g (1865-1942) also pointed to the work of higher
degree equation,  but he concluded the textbook in question was the  Ce Yuan
Hai Jing (12) , which is the speciality book about "Tian Yuan Shu" (Technique
of the Celestial Element).
    It, however, was introduced into Japan for the first time in 1726 (13), so
Seki Kowa could not have read it.  The fact that it had not been introduced
before 1726 is supported by the evidence as follows: the letter which Mukai
Motonari ŒüˆäŒ³¬ (?-?) sent to Hosoi Kotaku ×ˆäœAàV (?-?), published in the
Sokuryo Higen  ‘ª—ʔ錾 (Secret Comments of Surveying) by Hosoi Kotaku, states

             The chapter of Choken-jutsu  ’¬Œ©p (Surveying);  Some methods
             were described in the Ce Yuan Hai Jing Len Rei Shi Shu  ‘ªš¢ŠC‹¾•ª
             —Þç׏p (Classified Methods of the Ce Yuan Hai Jing )(14), Li Suan
             Quan Shu  —ïŽZ‘S‘ (Complete Works on Calendar and Mathematics),
             Gou Gu Yin Meng ‹åŒÒˆø–Ö (Introduction to Sides of Triangle) and
             Shu Du Yan   É“xŸ¥ (Generalisation on Numbers) which are thereby
             introduced into Japan (15).
    This evidence is very reliable because the Li Suan Quan Shu was introduced
into Japan this same year(16).  Therefore, we can conclude that the Ce Yuan Hai
Jing was first known to Japanese mathematicians in 1726.  Therefore we cannot
accept Kano Ryokichi's opinion.

         (d)  Opinion concerning the Zhui Shu ’ԏp (Bound Methods)
    Uchida Itsumi “à“cŒÜæV (1805-1882) told Okamoto Noriyoshi ‰ª–{‘¥˜^ (1847-
1931) that Seki Kowa's text book was Zhui Shu(17), which is Zu Chongzhi's  ‘c‰«
”V (429-500) work, however, this work was lost in the Northern Song –k‘v
dynasty.
    If the Zhui Shu was extant in Seki Kowa's time, it would be the biggest
discovery in the history of mathematics in Eastern Asia.  A handwritten
manuscript entitled  Zhui Shu is kept at  Tokyo University Library  “Œ‹ž‘å›{š¤
‘ŠÙ.  It is an 1897 copy from a manuscript of Okamoto's collections, so it
must be Uchida's " Zhui Shu ".  It, however, describes series, which was a very
popular subject among Japanese mathematicians only after Seki Kowa developed
Japanese algebra.  Probably this MS. was forged much after Seki Kowa's time.
Thus we can not agree with Uchida's claim.

         (e)  Opinions concerning the Yang Hui Suan Fa (Yang Hui's Method of
              Computation)
    Fujiwara Shozaburo  “¡Œ´¼ŽO˜Y (1881-1946) and Shimodaira Kazuo ‰º•½˜a•v (b.
1928) concluded that the book Seki kowa used was Yang Hui Suan Fa(18).
    There are two pieces of evidence.  The first is that Seki Kowa hand-copied 
it in his youth (see section   1-2-a).  The other is that the term Seki Kowa
used to refer to his treatment of indeterminate equations is the same as
"Senkan-jutsu"  ãƊǏp (technique of cutting tube) of Yang Hui Suan Fa.
    Seki Kowa, however, did not burn the text book he used, rather, he retained
it.  And the explanation of solving indeterminate equations of this book is too
 simple to complete Seki Kowa's works on indeterminate equations (I will
discuss in chapter  3).  Thus I think that he used other text books, as well as
Yang Hui Suan Fa, as sources in his development of method for solving
indeterminate equations.
    Seki Kowa did not open his manuscript to other mathematicians, thus it is
worth to consider the influence of Yang Hui Suan Fa for analysing his original
works.  Yang Hui Suan Fa is one of the best works about magic squares.  Magic
squares were a very popular subject for Japanese mathematicians, and most of
them worked on this subject.  Therefore, we will have to consider the influence
of Yang Hui Suan Fa with respect to magic squares in chapter 2, in order to
determinate how Seki Kowa and Japanese mathematicians influenced the design of
magic squares beyond the treatment found in the Yang Hui Suan Fa .

         (f)  New Opinion: Opinion concerning the Shu Shu Jiu Zhang 
              (Mathematical Treatise in Nine Sections)

    We introduced some opinions, but these cannot perfectly explain the
influence of Chinese mathematics (see table 4).  In particular, these opinions
do not explain Seki Kowa's treatment of indeterminate equations.  Some
mathematical books we have discussed above describe indeterminate equations,
but they are too simple.  Therefore we will consider the subject of whether
Japanese mathematicians were influenced by Chinese mathematical arts.
    Because the best and the only work of indeterminate equations in China is
 Qin Jiushao's work, we will consider whether Seki Kowa's text book was Shu Shu
Jiu Zhang.
    {\\\\\\\\\\\{\\\\\\\\{\\\\\\\\\\\\\{ 
    b      categories      b   Seki's work  b  Opinions of Seki's text b 
    {\\\\\\\\\\\{\\\\\\\\{\\\\\\\\\\\\\{ 
    b a) verse             b  -             b   (Suan Fa Tong Zong )   b 
    {\\\\\\\\\\\{\\\\\\\\{\\\\\\\\\\\\\{ 
    b b) magic squares     b  14)           b     Yang Hui Suan Fa?    b 
    {\\\\\\\\\\\{\\\\\\\\{\\\\\\\\\\\\\{ 
    b c) indeterminate eqs.b  5)6)15)       b             -            b 
    {\\\\\\\\\\\{\\\\\\\\{\\\\\\\\\\\\\{ 
    b d) series            b  4)7)8)11)     b         Zhui Shu?        b 
    {\\\\\\\\\\\{\\\\\\\\{\\\\\\\\\\\\\{ 
    b e) higher degree eqs.b  2)3)          b     Suan Xue Qi Meng     b 
    b                      b                b     Ce Yuan Hai Jing     b 
    {\\\\\\\\\\\{\\\\\\\\{\\\\\\\\\\\\\{ 
    b f) circles           b  9)10)12)      b   (Jiu Zhang Suan Shu)   b 
    {\\\\\\\\\\\{\\\\\\\\{\\\\\\\\\\\\\{ 
                                                                             
                      Table 4  Opinions of Seki Kowa's text

                                      Notes

(*1): Mikami Yoshio, 1932.
(*2): Nihon Gakushiin, 1954, vol.2: 133-46.  Hirayama Akira, 1959.  Hirayama 
      Akira et al (eds.), 1974.
(*3): See Hirayama Akira et al (eds.), 1974: 14-24.
(*4): See Hirayama Akira, 1959. 
(*5): Li Yan, 1937.
(*6): See Nihon Gakushiin, 1954, vol.2: 142-3.
(*7):
(*8): Hosoi Sosogu, 1941: 93.
(*9): Chinese characters are also used in Japanese, so with the help of some
      "Kaeriten" •Ô‚èêy (reading order symbol) and "Okurigana" ‘—‚艼–¼
      (traditional Japanese pronunciation to support reading Chinese), Japanese
      scholars could understand Chinese.  The republished book of Suan Xue Qi
      Meng was published by Hisada Gentetsu and Haji Doun  “yŽt“¹‰_ (?-?).
(10): Endo Toshisada, 1896; 1981: 73-4, Mikami's comment.
(11): Jochi Shigeru, 1991.
(12): Kano Ryokichi Žë–ì‹œ‹g. 1902. "Seki Kowa 200-Nensai Kinen Honcho Sugaku
      Tsuzoku Koen-Shu" 萍F˜a“ñ•S”NÕ‹L”O–{’©É›{’Ê‘­u‰‰W (Transcript of
      Lectures for Popularising Japanese Mathematics, in Remembrance of Seki
      Kowa on 200th anniversary of his death) (Nihon Gakushi-in, 1954, vol.2:
      143).
(13): Oba Osamu, 1967: 689.  Moreover Ce Yuan Hai Jing Xi Cao ‘ªš¢ŠC‹¾×‘
      (Comments of Ce Yuan Hai Jing) was also introduced in 1726 (Oba Osamu,
      1967: 689)    .
(14): It is not Ce Yuan Hai Jing (Sea Mirror of Circle Measurement) itself, but
      it quotes the whole sentence of Ce Yuan Hai Jing, so Japanese
      mathematicians had access to the contents of Ce Yuan Hai Jing.
(15): Nihon Gakushiin, 1954, vol.5: 428.
(16): Oba Osamu, 1967: 687.
(17): Nihon Gakushiin, 1954, vol.2: 143.
(18): Nihon Gakushiin, 1954, vol.2: 7 and 17.  Shimodaira Kazuo, 1965, vol.1:
      188.

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