328 REPORT — 1905. 
fraction. Moreover, we linow that already about 500 a.p. this method had been 
transplanted to India, and had ripened there into a beautiful method of resolving 
indeterminate equations. I cannot doubt that this method had already in olden 
times arrived in China, within the reach of a regular connection with Japan. 
I suppose this part of Japanese mathematics to be of Greek origin, the more 
because I have, to my great surprise, found in a Japanese source the ancient 
Greek solutions of the famous Delian problem ascribed to Plate. 
By denying to the Japanese the independent discovery of the theory of 
continued fractions we withdraw from them only things belonging to the lower 
mathematics. Much more weight is to be attributed to the quoted Japanese 
infinite series. Now, the basis of all these Japanese researches is formed by the 
binomial theorem for particular, though not integral, values of the exponent, Seki, 
who possessed this theorem for the value + $ at least, but very probably for any 
rational value of the exponent, died in 1708 at the age of sixty-six. A more 
precise dating of his work is not possible, but we can surely suppose that he knew 
the basis of his researches, which he taught to his scholars, already a considerable 
time before the year 1700. Inthe Occident this theorem, one of the most important 
parts of mathematics in general, was actually discovered by Isaac Newton in 1666, 
The theorem was not published till 1685, by John Wallis, and only in 1704 by 
Newton himself. The near coincidence of the times when the binomial theorem was 
known in the Occident and the Far-East excludes, with a high probability, any con- 
nection between the two discoveries, if we take into consideration the slow rate 
of intercourse even between European scholars, the great distance between the two 
opposite parts of the earth, not to be overcome in much less than three years, the 
difficulty of the languages and of the accessibility of a country maintained in tho e 
times in the uttermost seclusion by laws of the highest severity. But we can 
go far beyond mere probability by comparing Seki's and Newton’s methods for 
the development of the root of a binomial. These methods both represent the 
analytical investigation of the numerical operations known formerly in both 
regions of the world; but the Eastern is as far different from the Western as things 
Japanese generally are from ours. Moreover, we find in Seki's other sagacious 
and audacious performances, for which any prototype in the Occidental sciences 
is entirely lacking, the revelation of a high mathematical genius, to whom the 
independent detection of the binomial theorem must without any hesitation be 
attributed. 
Whereas it is merely in the highest degree probable that Seki found the 
binomial theorem independently, the independence of his series for (arc sin y)? 
is raised above any doubt by the mere fact that this series was first published 
in the Occident in 1815, though we are now acquainted with the fact that a 
letter of Leonard Euler, of the year 1737, unpublished up to the year 1904, 
contains this series, but here, too, only about thirty years after Seki's death. 
So clear a result cannot be arrived at with respect to the question whether the 
three series which come down to us under the name of Ajima, living about 1760, 
are to be ascribed to native skill. This question is important, far beyond the 
special results, from the fact that the method applied —that is to say, the circle 
principle—is a method of the infinitesimal calculus ; the particular meaning of the 
question being, therefore: Had the Japanese independently discovered the elements 
of this calculus ? 
At first sight the appearance is entirely against the independence of the 
Japanese work, since two of the three series under consideration had been pub- 
lished in Kurope in 1685, by John Wallis, and, what is of high importance, bad 
been derived by him in very much the same manner as in Japan. But there 
are several indications which point the opposite way. 
The first of the three series was certainly known in Japan about 1739, 
and very probably much before 1722; by this fact the authorship of this series 
is withdrawn from Ajima, who was born about 1737. Owing to the remark of a 
Japanese source, that Ajima has been a disciple of a mathematician of Seki’s 
school, I had been induced to suppose Seki himself to be the author of this series ; 
and I am disposed to do so the more because the traces of the circle principle, 
