円周率3.14の動揺と3.16の復活の謎

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At first Pi was valued as 3.16… in Japan, but it was corrected to 3.14… by S. Muramatsu in his Sanso published in 1663. And by the 1680s most books of mathematics published in Japan had adopted 3.14… as the value of Pi, and the 3.16… value had nearly disappeared. However, the 3.14 value, once seemingly established, started to fluctuate for some reason, and our study on texts of Japanese mathematics published in the Bunsei Era (1818-1830) makes it clear that the texts which adopted the 3.16 value were of greater number than those which adopted the 3.14 value. As far as the published works of mathematics are concerned, this was a retrogression. Consequently, the writers of this report, tried to solve "the puzzle of the changing value of Pi" and succeeded in clarifying it as follows: (1) S. Nozawa first adopted the Pi value of 3.14 in his Dokaisho published in 1664, but he showed two values of Pi in Sankyukai published thirteen years later in 1677, saying that it was 3.162 (≒√<10>) when calculated from "Ri (reason)", and 3.1404 when computed from "Katachi (form)" "Neither of them can be abandoned," says he, "and either alone brings a deadlock." (2) When S. Ogyu (1666-1728), a famous Confucianist, had a discussion with G. Nakane (1662-1733), an eminent scholar of native mathematics, on the value of Pi, he could never understand the mathematician's explanation on how to arrive at the value of Pi. And he refuted this, saying "There are both 'theory' and 'practice' in everything in this world. One can discuss 'theory' but one cannot always apply it to 'practice'. And 'theory', applied to 'practice', is not always fully developed." (3) This is because, one may say, native mathematicians only drew polygons inscribed within a circle, not adopting the method to put the circle between inscribed and circumscribed polygons. T. Kamata (1678-1747) had shown his method, in his Takuma-ryu Enri (manuscript in 1722), to obtain the maximum and minimum value by putting a circle between inscribed and circumscribed polygons. Unfortunately it was not handed down as a tradition. (4) Judging from this state of circumstances, it seems, N. Tachibana (1754-1806) had to conclude in his essay Hokuso Sadan that the value of Pi "has been discussed to be either 3.16 or 3.14 but saw no definite conclusion as yet. It may be due to the fact that a circle is not suitable for mathematics as it is positive and variable." (5) The correct value of 3.14… came to be adopted in almost all texts of native mathematics by the 1680s, and it seemed to have achieved a general consensus. K. Sawaguchi, however, adopted the outdated 3.16 value in his text Kokon Sanpoki published in 1671 although he was well informed of the 3.14 value. Though this was the exception, the fact that Sawaguchi, one of the finest scholars of native mathematics of the day, was in favor of 3.16 as the value of Pi might have caused the correct value of 3.14 to waver. (6) After the value of 3.14 appeared to have achieved a consensus in the 1680s, as it was adopted in most of all texts of native mathematics, both in S. Kambara's Sankan-ki published in 1685 and S. Murata's Sampo Meisui-ki published in 1693 did the once outdated value of 3.16 come to life again. That revival carried over to the eighteenth century, and the value of 3.16 appeared in most all popular books on mathematics which were published in large quantities in those days.
日本科学史学会の論文
1984-10-31

円周率3.14の動揺と3.16の復活の謎

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