The Mystery of 355/113 (2010)

https://mathworld.wolfram.com/PiContinuedFraction.html > The simple continued fraction for pi is given by [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...] (OEIS A001203). > The very large term 292 means that the convergent > [3;7,15,1]=[3,7,16]=(355)/(113)=3.14159292... > is an extremely good approximation good to six decimal places that was first discovered by astronomer Tsu Ch'ung-Chih in the fifth century A.D. (Gardner 1966, pp. 91-102). You can see how the terms…

In Chinese the fraction 355/113 has been known as the 密率 ("detailed ratio") and 22/7 as the 约率 ("approximate ratio"). It is documented in the Book of Sui, volume 16: 宋末，南徐州從事史祖沖之，更開密法，以圓徑一億為一丈，圓周盈數三丈一尺四寸一分五厘九毫二秒七忽，朒數三丈一尺四寸一分五厘九毫二秒六忽，正數在盈朒二限之間。密率，圓徑一百一十三，圓周三百五十五。約率，圓徑七，週二十二。 [1] While it suggests that the value was calculated around the end of the Liu Song Dynasty (~479 CE) by Zu Chongzhi to have an upper bound of 3.1415927 and lower bound of 3.1415926, the 密法 method is not explained. It has been thought that Zu…