326 REPORT—1905. 
Seki possessed a method of developing the root 
ya =3(1- V1 —y,1’) 
of this equation in series of powers of y,_,*. He got, by a very curious recurring 
algorithm, quite after the Far-Eastern manner of reckoning, which I have not the 
time to explain in this place, one member of the series after another up to the 
sixth. It is a highly astonishing fact that Seki obtained by this method, from 
the equation 
yy? =3(l- V1-y,"); 
the infinite binomial series for the exponent 4. Then, making use of the series for 
y,’, he derived in the same manner y,”, by means of the equation 
y," =3(1- V1 —y,"), 
in a series of powers, not of y,?, but of y,?. Continuing in this way, he obtained 
for y37, ¥,7, ... up to ¥,°, series of powers of y,. By passing to the limit 
a= oo, according to the formulas 
. ts . 
Peri? = (2 sin ati)? =27=(2 are sin y,)’, 
Seki finds the series 
(are sin y)? = y+2, 
ce 8 ee pees: 
> B+11.3.6...(28+1) 
The laws of the coefficients of all series are found by incomplete induction, 
which played a great part in Japanese mathematics; but it must be stated, with 
great astonishment ane admiration for Seki's skill and sagacity, that he, though 
proceeding on lines not quite legitimate, always arrives at right results. But we 
must allow a good deal for the non-existence of any researches concerning the 
validity of series in the ancient Japanese mathematics. 
vr wT 
St A, 
one of these values rests the formerly quoted common fraction for this quantity. 
Afterwards some other series were deduced by which not 7%, but m itself, 
could be calculated. All these series are connected with the name of Naomaru 
Ajima (1737-1797 nearly). The vehicle of these researches is a very singular 
method, called the ‘circle principle.’ In our mode of expression this principle is 
the following. If the integral 
Putting in Seki’s series v = special values of m? are obtained; on 
[ys (y)dy 
0 
is to be calculated, f(y) is developed in powers of y, thus :— 
[e@) (A) 
ee i pa na oe idle 
and the integral is replaced by a sum of a great number of terms, determined 
by the upper limit y =, and the equidistant values 
1 
y=", a=0,1, 2,3,..p, = 
—n being a very great number—are used to give the equation 
y 
P [ev ua ft a 
dy=1 ee 
\fw : no" &°19.8..8 n 
