A History of Japanese Mathematics/Chapter 7

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Whether or not Seki can be called a great genius in mathematics, certain it is that his contemporaries looked upon him as such, and that he reacted upon them in such way as to arouse among the scholars of his day the highest degree of enthusiasm. Although he followed in the footsteps of Pythagoras in his relations with his pupils, admitting only a few select initiates to a knowledge of his discoveries,^[1] and although he kept his discoveries from the masses and gave no heed to the researches of his contemporaries, nevertheless the fact that he could accomplish results, that he could solve the puzzling problems of the day, and that he had such a large following of disciples, made him a stimulating example to others who. were not at all in touch with him. In view of this fact it is now proposed to speak of some of Seki's contemporaries before considering his own relation to the yenri, and at the same time to consider the question of possible Western influence at this period.

Two years before Seki published (1674) his Hatsubi Sampō; namely in 1672, Hoshino Sanenobu published his Kokōgen-shō, and in 1674 Murase, a pupil of Isomura, wrote the Sampō Futsudan Kai. A year later (1675), Yuasa Tokushi, a pupil of Muramatsu, published in Japan the Chinese Suan-fa Tung-tsong. In 1681 Okuda Yūyeki, a Nara physician, wrote the Shimpen Sansū-ki. Two years later, Takebe Kenkō published ​the Kenki Sampō, in which he solved the problems proposed in Ikeda Shōi's Sūgaku Fojo Ōrai of 1672, without making use of the tensan algebra of Seki, saying that "this touches upon what my mathematical master wishes kept secret," thus leaving unsolved those problems that required the senkan-jutsu and similar devices. It was in the work of Ikeda that the old Chinese value of ${\displaystyle \pi ,{\frac {355}{113}}}$ was first made known in Japan.

In the same year (1683) Kozaka Sadanao published his Knichi Sangaku-sho.^[2] He had been the pupil of a certain Tokuhisa Kōmatsu, founder of the Kūichi school of mathematics, a school that was much given to astrology and mysticism.^[3] Also in this year Nakanishi Seikō published his Kōkogen Tekitō-shū, a book that was followed in 1684 by the Sampō Zoku Tekitō-shū written by his brother, Nakanishi Seiri. These brothers had been pupils of Ikeda Shōi, and one of them^[4] opened a school called after his name.

In 1684 the second edition of Isomura's Ketsugi-shō appeared,^[5] and in the following year Takebe's commentary on Seki's Hatsubi Sampō was published. This latter made generally known the yendan method as taught by Seki.

In 1687 Mochinaga and Ōhashi published the Kaisan-ki Komoku,^[6] and in 1688 the Tōsho Kaisanki.^[7] In the first of these, works we already find approaches to the crude methods of integration (see Fig. 30) that characterized the labors of the early Seki school. In the year 1688 Miyagi Seikō, the teacher of Ōhashi, published the Meigen Sampō, to be followed in 1695 by his Wakan Samp^[8] in which he considers in detail the numerical equation of the 1458th degree already mentioned by Seki, and attempts to solve the hundred fifty problems ​in Satō's Kongenki and the fifteen in Sawaguchi's Kokon Sampō-ki (1670), all by the yendan process.

Miyagi founded a school in Kyōto that bore his name, and to him is sometimes referred a manuscript^[9] on the quadrature of the circle. He was highly esteemed as a scholar by his contemporaries.^[10]

In 1689 Ando Kichiji of Kyōto published a work entitled Ikkyoku Sampō in which the yendan algebra is set forth, and

Fig. 30. Early integration, from Mochinaga and Ōhashi's Kaisan-ki Kōmoku (1687).

in 1691 Nakane Genkei published a sequel to it under the title Shichijō Beki Yenshiki.

In 1696, Ikeda Shōi published a pamphlet on the mensuration of the circle and sphere,^[11] and in 1698 Satō Moshun ​
Fig. 31. Mensuration of the circle, from Satō Moshun's Tengen Shinan (1698). ​published his Tengen Shinan or Treatise on the Celestial Element Method. In this his method of finding the area of a circle is distinctly Western (Fig. 31), although it is so simple as to claim no particular habitat.

This list is rather meaningless in itself, without further description of the works and a statement of their influence upon Japanese mathematics, and hence it may be thought to be of no value. It is inserted, however, for two purposes: first, that it might be seen that the Seki period, whether through Seki's influence or not, whether through the incipient influx of Western ideas or because of a spontaneous national awakening, was a period of special activity; and second, that it might be shown that out of a considerable list of contemporary writers, only those who in some way came under Seki's influence attained to any great prominence.

We now turn to the second and more important question, did Seki and his contemporaries receive an impetus from the West? Did the Dutch traders, who had a monopoly of the legitimate intercourse with mercantile Japan, carry to the scholars of Nagasaki and vicinity, where the Dutch were permitted to trade, some knowledge of the great advance in mathematics then taking place in the countries of Europe? Did the Jesuit missionaries in China, who had followed Matteo Ricci in fostering the study of mathematics in Peking, succeed in transmitting some inkling of their knowledge across the China Sea? Or did some adventurous scholar from Japan risk death at the order of the Shogun,^[12] and venture westward in some trading ship bound homewards to the Netherlands? These are some of the questions that arise, and which there are legitimate reasons for asking, but they are questions that future research will have more definitely to answer. Some material for a reply exists, however, and the little knowledge that we have may properly be mentioned as a basis for future investigation.

It has for some time been known, for instance, that there ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/145 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/146 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/147 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/148 ​samurai, and to trace his wanderings, especially as he returned and could, at least in the secrecy of his family, have told his story. We are, however, quite uncertain as to any of these matters. His descendants have kept the tradition that his visit abroad was in the Manji era, and since this extended from 1658 to 1661, it included the time that Hartsingius was in Leyden. Tradition also says that he visited the capital of Namban, which at that time meant not only the Spanish peninsula, but the present and former colonies of Spain and. Portugal, and which included Holland. While in this city he learned medicine from someone whose name resembled Postow or Bostow,^[13] and after some years he again returned to Japan.

Arrived in his own country Nakashima was in danger of being beheaded for his violation of the law against emigration, and this may have caused the journeying from place to place which tradition relates of him. It is more probable, however, that his skill as a physician rendered him immune, the officials closing their eyes to a violation of the law which might be most helpful to themselves or their families in case of sickness. The danger seems to have passed through the permission granted by the Shogun that two European physicians, Almans and Caspar Schambergen should be permitted to practise at Nagasaki. Thereupon Nakashima became one of their pupils, began to practise in the same city, and assumed the name Nakashima Sōha.

It happened that there lived at that time in the province of Hizen, in Kyūshū, a certain daimyo who was very fond of a brood of pigeons that he owned. One of the pigeons having injured its leg, the daimyo sent for the young physician, and such was the skill shown by him, and so rapid was the recovery ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/150 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/151 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/152 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/153 ​way into Japan in the seventeenth century; that we have no definite information as to the nature of this work beyond the fact that mathematical astronomy was part of it; that there is no evidence that Seki or his school borrowed their methods from the West; but that Japanese mathematicians of that time might very well have known the general trend of the science and the general nature of the results attained in European countries.