TRANSACTIONS OF SECTION A. 3290 
5. On Japanese Mathematics. By Professor Paut Harzer. 
The astonishing progress of Japanese mathematics in recent years has induced 
me to seek for signs of any independent mathematical work of the Japanese 
before their contact with occidental culture in our times, Our knowledge of 
these ancient times in general comes to us from very scanty sources, not easily 
accessible; a great part of them are in Japanese, and require very careful 
criticism ; the sources for Japanece mathematics in particular have been, moreover, 
kept secret by the Japanese mathematicians in the manner of the old Pythagorean 
schoo], and have become known to foreigners only in the last few years. 
Aided by a Japanese scholar, who has translated some Japanese texts for me, J 
have studied these sources, and am giving you an abstract of the results, which 
I have laid down in detail in a paper published in the ‘Jahresbericht der 
deutschen Mathematiker Vereinigung,’ vol. xiv. p. 312. I may add that in this 
study of Japanese sources I have never felt any doubt concerning their purity. 
The wide extension of the study of mathematics in Japan already in ancient 
times is to be inferred from the fact that the Imperial Library of Tokio contains 
more than 2,000 written and printed native mathematical works, going back to 
the year 1595, It is nut surprising that the higher of these works are con- 
secrated to the problem of the quadrature of the circle, which has been the 
centre of the mathematical efforts of all civilised nations, in such a degree that we 
may, with an English mathematician, assert that the history of this problem 
coincides nearly with the history of mathematics in general. 
Concerning the Japanese numerical value for 7—that is to say, of the ratio 
of the circumference of the circle to its diameter—we have knowledge of the 
fraction 32 about the year 1627. On expressing this value in the decimal system 
of numbers, in use in Japan since olden times, the first tavo places are right. 
To the second half of the seventeenth century belong decimal values of the 
precision of 9 and 10 places. About 1709 there is to Le found the famous 
fraction 333, right to 7 places, and about 1722 and 1739 the precision increases 
to 42 and 51 correct places. Moreover, about 1766 the two common fractions 
$419351 and 428224593349304 were known, representing 7 with astonishing pre- 
cision to 12 and 30 places; and already about 1760 there exists the value 
9848 for a, which is exact to 9 places. 
We are told about some of the methods by which these values were 
obtained. One form of solution of the problem depends on neatly the same 
principle as the famous method of Archimedes: an arbitrary are of the circle is 
divided into halves, each of the two halves once more into halves, and so on. The 
chords of these arcs get smaller and smaller, and their sum represents the 
length of the are nearer and nearer. Between the chord of the are and that 
of the half-are exists a quadratic equation, by whose solution, repeated for the 
subsequent chords, Archimedes arrived at a lower limit for 7. But whereas 
Archimedes made this calculation numerically only, one of the Japanese solutions 
makes use of analytical methods, and obtains its results in the form of infinite 
series quite in the way of modern higher mathematics. The other type of 
Japanese solution is connected with the consideration of the sphere, which is, in a 
manner already used by Archimedes, dissected by parallel planes, quite close 
to one another, into a large number, e.g. 400, of thin layers; and their contert is 
calculated and afterwards summed by elementary means. The connection between 
these two kinds of solution remained unknown to the Japanese mathematicians up 
to the year 1709. ; 
To explain Japanese methods of the first kind, which are of particular 
interest, 1 begin with that of the most famous Japanese mathematician, Kowa 
Seki (1642-1708). Let 2y, be the chord of the are wv of the circle of radius 1. 
The are being subsequently divided into halves, let 2y. be the chord of the arc 
3 Then we have 
4Ya%(l — ye’) = Yar”. : 
