TRANSACTIONS OF SECTION A. 327 
Now the formula ¢ 
lim. (1a (2)° ee [wads 
Bs ‘ (¢) n=o B+1 2) B+ i aa 
was known to the Japanese mathematicians, and by making use of it they 
obtained 
(8) 
v oO : 
| 7m4- >8t2.8..@+"" 
In ancient Japanese mathematics only the two functions f(y)= /1—y? and 
al y were considered; that is to say, the series of the binomial theorem 
for the exponents + 4and —}. We are informed of three different series connected 
with Ajima’s name, and obtained by means of the method described from a 
quite elementary consideration of rectilinear figures in a circle. These series are 
the following ones :— 
; : <2) Lod a's oe (2B—L) sea 
JA La 4+ f- Fa0G 4) 7e+ 
(i.) Are sin y > 2-4-6 28 ee a 
: “te PR gh tee: NRE IS? CME RTE HOPE RIN 
(ii.) 3 (are sin y — y,/] —y?) = > B 3B 43 Sear er Re ‘ 
2 
Gil.) Are sin y = v1—; 
_ The third equation would by an integration lead back to the Seki series, but 
I did not find any indication as to whether this reduction had been the aim of 
Ajima’s remarkable transformation. 
From all three series, special values of 7 can be derived ; and we know that the 
value correct to 51 places had been calculated in 1739 by putting in the first 
seriesy=43. Concerning those very accurate common fractions for 7 and 7? which 
I have quoted, I have convinced myself, with astonishment, that they must have 
been obtained by the Japanese mathematicians by the means which we now apply 
for the approximate transformation of irrational numbers into common fractions— 
that is to say, the calculus of continued fractions. For these numbers are con- 
vergents, and even very high convergents of such fractions. 
After having heard of such unexpected Japanese work, you will not he 
astonished at learning that Ajima treated the ellipse in the same way; that at 
the beginning of the nineteenth century, there existed Japanese researches on the 
catenary and the cycloid; and we may trust the Japanese statement, not yet 
proved by documents accessible to me, that Seki and his school in the seventeenth 
century knew the general binomial theorem, some theorems. of the theory of 
numbers, of the theory of maximum and minimum, of determinants, of plane and 
spherical trigonometry, of analytical geometry, and of geodesy. 
Now the question is forced upon us whether we have in the flourishing 
condition of Japanese mathematics, from the seventeenth century onward, an 
independent development of native science, or only a transformation of imported 
acquisitions. 
We will first consider the calculus of continued fractions proved to exist in 
Japan about the middle of theeighteenth century. Now, this calculus had already 
been used by Euclid about 400 3.c. to free fractions of common divisors of the 
numerator and the denominator, and by applying this method to an irrational 
number, and by interrupting the algorithm at any step, we obtain the method 
of representing the fraction approximately by the convergents of a continued 
