254 9. MEASUREMENTIn Japan these techniques were first applied in the area called yenri (circle
theory),^11 a topic that had been studied extensively in China. The idea of ap-
proximating by shells or disks can be seen in the 1684 edition of the Ketsugi-shd
(Combination Book), first published in 1660 by Isomura Kittoku.
Isomura Kittoku explained the method as follows (Mikami, 1913, p. 204):If we cut a sphere of diameter 1 foot into 10,000 slices, the thickness
of each slice is 0.001 feet, which will be something like that of a very
thin paper. Finding in this way the volume of each of them, we
sum up the results, 10,000 in number, when we get 532.6 measures
[that is, a volume of 0.5326 cubic foot]. Besides, it is true, there
are small incommensurable parts, which are neglected.
The technique of obtaining extraordinary precision and using it to perform
numerical experiments which provide the basis for general assertions also appears
in some remarkable infinite series attributed to Takebe Kenko, as we shall see below.
Takebe Kenko's method of squaring the circle was based on a relation, which he
apparently discovered in 1722, between the square of half of an arc, the height h
of the arc,^12 and the diameter d of the circle. Here is his own description of this
discovery, as explained by Smith and Mikami (1914, pp. 147-149). He began with
height h = 0.000001 = 10~^6 and d = 10, finding the square of the arc geometrically
with accuracy to 53 decimal places.
The value of the square of this arc is
0.00001 00000 00333 33335 11111 12253 96833 52381 01394 90188 203+.Isomura Kittoku's method of computing the volume of a sphere.
© Stock Montage, Inc.
According to Smith and Mikami (1914, P- 148), the value given by Takebe Kenko
was
(^11) The symbol for circle here (yen) is also the symbol for the Japanese unit of currency; it is
actually pronounced "en."
(^12) This height is called the sagitta (arrow) by lens grinders, a name first bestowed on it in India.
It is now called the versed sine in mathematics.