The History of Mathematics: A Brief Course

256 9. MEASUREMENT

see the recursive operation: multiplication by (h/d)\2n^2 /(n + l)(2n + 1)]. Putting

these corrections together as an infinite series leads to the expression

when the full arc has length a.

In using this numerical approach. Takebe Kenko had reached his conclusion

inductively. This induction was based on a faith (which turns out to be justified)

that the coefficients of the power series are rational numbers that satisfy a fairly

simple recursive formula. As you know, the power series for the sine, cosine, expo-

nential, and logarithm have this happy property, but the series for the tangent, for

example, does not.

This series solves the problem of rectification of the circle and hence all problems

that depend on knowing the value of ð. In modern terms the series given by Takebe

Kenko represents the function

Takebe Kenko's discovery of this result in 1722 falls between the discovery of the

power scries for the arcsine function by Newton in 1676 and its publication by Euler

Was European calculus transmitted to Japan in the seventeenth century? The meth-
ods used by Isomura Kittoku to compute the volume and surface area of a sphere
and by Takebe Kenko to compute the square of a half-arc in terms of the versed sine
of the arc are at the heart of calculus. Smith and Mikami (1914, pp. 148-155) argue
that some transmission from Europe at this time is plausible in the case of Takebe
Kenko. They note that there was some contact, although very limited, between
Japanese and European scholars, even during the period of "closure," and that a
Jesuit missionary in China named Pierre Jartoux (1668-1721) communicated some
of the latest European discoveries to his Chinese hosts. After noting that "there is
no evidence that Seki or his school borrowed their methods from the West" (1914,
p. 142), they argue as follows (1914, p. 155):

Here then is a scholar, Jartoux, in correspondence with Leibnitz

[sic], giving a series not difficult of deduction by the calculus, which

series Takebe uses and which is the essence of the yenri, but which

Takebe has difficulty in explaining... [I]t seems a reasonable con-

jecture that Western learning was responsible for [Jartoux'] work,

that he was responsible for Takebe's series, and that Takebe ex-

plained the series as best he could.

Probably the question of Western influence on Japanese mathematics cannot
be decided. However, in allowing for the possibility of communication from West
to East, we must not neglect the possibility of some transmission in the opposite
direction, in addition to what was transmitted from the Muslims and Hindus earlier.
Leibniz, in particular, was fascinated with oriental cultures, and at least two of
his results, one of them a simple observation on determinants and the other a
more extensive development of combinatorics, were known earlier in India and
Japan. It should also be noted that in contrast to the Chinese mathematicians,

in 1737.