In our last paper titled "The Value of Pi in the Edo Period" (Kagakusi Kenkyu, No.143, 1982. pp. 142-152), we did an exhaustive review of the books of native mathematics of Japan published in the Bunsei Era (1818-1830), and showed that these books could be divided into two types according to the two pi values (i.e. 3.14... and 3.16...) they respectively adopted. Specifically, the relatively advanced books of mathematics adopted 3.14..., while the value of 3.16... was generally used in popular booklets of the Jinkoki type and the like. It had been more than 150 years since Muramatsu correctly demonstrated the pi value of 3.14... in his Sanso published in 1663, but a considerable number of books still adopted 3.16... as the pi value in our period of study.
Then, the next question would be how the correct value of 3.14... demonstrated by Muramatsu was handed down to the mathematicians of the Edo Period and disseminated. We carried our study a step further in this direction and tried to clarify the adoption process.
As a result of our extensive research and analysis, we believe that we have successfully traced the adoption process of 3.14... instead of 3.16... as the value of pi.
Among the various issues treated in this paper, the following points would be of particular interest.
1. After Muramatsu's Sanso (published in March,1663), the first book with 3.14... as the value of pi was Nozawa's Dȏkaishȏ (dated August, 1663 in the preface and published in November,1664), the interval between these books is less than two years.
2. Among the books of mathematics published during the ten years between 1663 and 1673, every one of those with 3.14... as the value of pi made an intentionnal alteration to the value adopted by its predecessor, such as 3.14 (→3.1404)→3.142→3.1416. This phenomenon had some connection with thebmovement to take over the traditional unsolved problems and it continued up to Miyake's Guȏ-sampȏ (published in 1699), in which the value of pi was further changed from 3.1416 to 3.141593. There were even a few cases of alteration from 3.142 to 3.14.
3. With the publication of Zȏho-sampȏ-ketsugishȏ (1684), Zohȏ-shimpenjinkȏki (1686) and Kaizanki-Kȏmoku (1687), the value of pi in the three most widely-read books of native mathematics in the Edo Era, Jinkȏki, Kaizanki and Sampȏ-Ketsugishȏ was altered from 3.16... to 3.14...
4. Upon examining all the books of native mathematics published between 1681 and 1690, we found that there was only one book (i.e. Kambara's Sankanki published in 1685) that had not altered the value of pi to 3.14.... and still used 3.16... All the remaining ten books adopted the value of 3.14...
Once having attained this stage, how did it come about that the popular books of native mathematics fell back to the value of pi of 3.16... without any apparent hesitation ? A report, on this issue is now in preparation.
In the Sanso of 1663, Shigekiyo Muramatu (1608〜1695) made calculations of the circumferences of a regular square, a regular octagon, a regular sixteen-sided polygon, …… inscribed in the circle with the diameter 1,and ended in the circumferences of regular 2¹⁵ -sided polygon. And he got π=3.141592648777698869248 from this conclusion.
Similarly Takakazu Seki (1640?〜1708) wrote the Katsuyȏ-Sanpȏ in 1712, in which he showed his measurement of the circle. From the perimeter of an inscribed 2¹⁷-sided polygon, he let the circumferences of a regular 2¹⁵ sided polygon be a, that of a regular 2¹⁶-sided polygon be b, and that of a regular 2¹⁷-sided polygon be c, and he made calculations as follows:
b+(b-a)(c-b)/(b-a)-(c-b)=3.14159265359 "extremely weak".
Shu-Li-Jing-Yun was introduced into Japan in about 1723. In this book Ludolf (1540〜1610) also showed his measurement of the circle, and he got π=3.141592653589793238431541553377501511680 from the perimeter of an inscribed 2³⁵-sided polygon.
In this paper we made a comparative study with the process of their measurements, which is an oblique side of rectangular triangles produced in each process.