TRANSACTIONS OF SECTION A. 329 
too, lead back to this mathematician. We can safely rely upon this fact, the more 
because we know that a generalisation of this principle from two to three dimen- 
sions, on which reposes the already mentioned second kind of Japanese com- 
putations of the number 7, had been developed in Japan from 1627 onward, 
whereas the distinct traces of this method in the Occident cannot be found before 
1658. It may be that the circle principle rests, both in the Occident and in 
the Far-East, on the researches of Archimedes; for there the geometrical sources 
may be found from which, after the lapse of centuries, the analytical method of 
the circle principle may have flowed independently in both regions. 
If, indeed, Seki was in possession of the circle principle, considering his 
mathematical genius, the omission (more than the existence of the series) would 
require explanation. Another fact which spealis in favour of the independence of 
the Japanese results is that the third and most interesting of Ajima’s series, though 
found in the Occident by Euler already .in Ajima’s lifetime, seems to be an 
undoubted property of the Japanese, since the mode of derivation is quite 
different from that applied to the same purpose by Kuler in the years 1755 and 
1768. As one more mark of independence may be mentioned the fact that, of the 
various series for are tany in powers of y, by which the Occident from the year 
1671 onward has supplied a much more pewerful tool for the determination of 
the numerical value of 7, no trace can be found in Japan, though the work by 
John Wallis of 1685, which formerly might be suspected of being the source of 
the first two Japanese series, contains the chief formula of this kind too. 
We may add, finally, that the startling similarity, notwithstanding the 
independence of the advances made by different persons in different countrivs 
nearly at the same time, has been often pointed out in the history of science. . 
I call to your mind the invention of the infinitesimal calculus and of the non- 
Kuclidean geometry. 
As a result of these considerations, I conclude that the invention and 
application of the elements of the infinitesimal calculus may be ascribed to the 
Japanese mathematicians, though not with certainty, yet witha sensible degree of 
probability. Only from the beginning of the nineteenth century I consider can 
distinct traces of direct influence of Occidental on Far-Eastern mathematics Le 
traced. 
In following the development of Japanese mathematics we meet with the sur- 
plising fact that to the tide-like advance of mathematics in Japan from the second 
half of the seventeenth century onwards, nearly equivalent in the importance of its 
results to contemporary Occidental mathematics, there corresponds no activity to 
be compared to that which existed in the Occident in the eighteenth century, led 
by the incomparable Leonard Euler. 
The concession of independent work to the old Japanese mathematicians does 
not exclude every Occidental—though indirect—influence. On the contrary, since 
the time of blossoming of the Japanese mathematics coincided with an active state 
of Dutch trade, which in 1609 introduced the fruits of Occidental civilisation into 
Japan, it must be regarded as highly probable that this connection called to life 
the slumbering inclination to the mathematics and paved the way for its 
development. 
I conclude these remarks with the affirmation that the Japanese have revealed 
by their own efforts in the arts, as has been long known, by their own productivity 
in the mathematics, as we have just now seen, and probably by performances in 
other domains, not yet accessible to foreign investigation, capabilities quite com- 
parable to our own. And we are entitled to conclude that Japan will in future, 
with the highest success, play her part in the eternal work of civilisation common 
to all nations, independently of casual political conjunctures which the day creates 
and the day destroys, 
