A History of Japanese Mathematics/Chapter 10

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In the midst of the unseemly strife that waged between Fujita and Aida in the closing years of the eighteenth century there dwelt in peaceful seclusion in Yedo a mathematician who surpassed both of these contestants, and who did much to redeem the scientific reputation of the Japanese of his period. A man of rare modesty, content with little, taking delight in the simple life of a scholar rather than in the attractions of office or society, almost unknown in the midst of the turmoil of the scholastic strife of his day, Ajima Manzō Chokuyen^[1] was nevertheless a rare genius, doing more for mathematics than any of his contemporaries.

He was bom in Yedo in 1739, and as a samurai he served there under the Lord of Shinjō, whose estates were in the north-eastern districts. He was initiated into the secrets of mathematics by one Iriye Ōchū^[2], who had studied in the school of Nakanishi. He afterwards became a pupil of Yamaji Shujú, and at this time he came to know Fujita Sadasuke with whom he formed a close friendship but with whose controversy with Aida he never concerned himself. And so he received a training that enabled him to surpass all his fellows in solving the array of problems that had accumulated during the century, including all those which had long been looked upon as wholly insoluble. Such a type of mind rarely extends the boundaries of mathematical discovery, but occasionally an individual is ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/208 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/209 ​of the altitude by the chord, of the diameter of the circle by

the altitude, and of the side of the square by the diameter of the circle, it is required to find the various quantities mentioned."

The problem derives its name from the fact that it was, with its solution, first hung before the Gion Temple in Kyōto by Tsuda Yenkyū, a pupil of Nishimura Yenri's^[3], the solution depending upon an equation of the 1024^th degree in terms of the chord. The solution was afterward simplified by one Nakata so as to depend upon an equation of the 46^th degree. Ajima attacked the problem in the year 1774, and brought it down to the solution of an equation of the 10^th degree. This is not only a striking proof of Ajima's powers of simplification, but it is also evidence of the improvement constantly going on in the details of Japanese mathematics in the eighteenth century.

Ajima considers in his Fujin Isshū (Periods of decimal fractions) the problem of finding the number of figures contained in the repetend of a circulating decimal when unity is divided by a given prime number. Although he states that the problem is so difficult as to admit of no general formula, he shows great skill in the treatment of special cases. To assist him he had the work of at least two predecessors, for Nakane Genjun had studied the problem for special cases in his Kantō Sampō of 1738, and in the Nisei Hyōsen Ban Seiyei of Ōsaka had given the result for a special case, but without ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/211 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/212 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/213 ​cepted by them, that is, the area ABCD in the figure. Here we divide the chord c of the arc into a equal parts.^[4]

Then from the figure it is apparent that

where p_r is the r^th parallel from the diameter d.

Ajima now expands p_r, without explaining his process (evidently that of the tetsujutsu), and obtains

​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/215 ​Page:A history of Japanese mathematics (IA historyofjapanes00smitiala).pdf/216 ​lacked the simple symbolism of the West, but he had the spirit of the theory, and although his contemporaries failed to realize his genius in this respect, it is now possible to look back upon his work, and to evaluate it properly. As a result it is safe to say that Ajima brought mathematics to a higher plane than any other man in Japan in the eighteenth century, and that had he lived where he could easily have come into touch with contemporary mathematical thought in other parts of the world he might have made discoveries that would have been of far reaching importance in the science.