【汉英对照】祖冲之圆周率 Zu Chongzhi's Value of π
【原文】
《隋書·律曆志》
古之九数，圆周率三，圆径率一，其术疏舛[1]。自刘歆[2]、张衡[3]、刘徽[4]、王蕃[5]、皮延宗[6] 之徒，各设新率，未臻折衷。
宋末[7]，南徐州从事史[8] 祖冲之，更开密法，以圆径一亿为一丈， 圆周盈数[9] 三丈一尺四寸一分五厘九毫二秒七忽，朒数[10] 三丈一尺四寸一分五厘九毫二秒六忽，正数在盈、朒二限之间。密率，圆径一百一十三，圆周三百五十五。约率，圆径七，周二十二。
【English Translation】
"Book of Sui'' (Treatise on the Calendar)
According to the nine computational methods of antiquity, the ratio of the circumference to the diameter is 3 to 1. This method is inaccurate. Liu Xin, Zhang Heng, Liu Hui, Wang Fan, Pi Yanzong each found a new ratio, yet could not reach a consensus.
Towards the end of Song, the assistant prefectural governor of southern Xu Zhou, Zu Chongzhi, further devised a precise method. Taking a circle of diameter 10000000 to be 1 zhang, the excess value of the circumference is 3 zhang 1 chi 4 cun 1 fen 5 li 9 hao 2 miao 7 hu, and the deficit value is 3 zhang 1 chi 4 cun 1 fen 5 li 9 hao 2 miao 6 hu. The actual number lies between the excess and deficit values. The precise ratio is a diameter of 113 to a circumference of 355. The approximate ratio is a diameter of 7 to a circumference of 22.
【注释】
[1] 疏舛shū chuǎn：粗疏，因为太小而不准确。
Crude, inaccurate for being too small.
[2] 刘歆（公元前50年 – 公元23年）汉代经学家、天文学家。
Liu Xin (50 BC – 23 AD) scholar of classics and astronomy of the Han dynasty.
[3] 张衡（公元78年 – 139年）汉代文学家、天文学家、发明家。
Zhang Heng (79 – 139 AD) literary scholar, astronomer, and inventor of the Han dynasty.
[4] 刘徽（约公元225年 – 295年）魏晋时期数学家。
Liu Hui (c. 225 – 295 AD) mathematician of the Three Kingdoms and Jin dynasty.
[5] 王蕃（公元228 – 266年）三国时代吴国天文学家、数学家。
Wang Fan (228 – 266 AD) astronomer and mathematicians of the Three Kingdoms period.
[6] 皮延宗（公元5世纪）南北朝数学家。
Pi Yanzong (fl. 5th century AD) mathematician of the Northern and Southern dynasties period.
[7] 宋sòng：南朝宋（公元420年 – 479 年）。
The Liu-Song dynasty (420 – 479 AD).
[8] 从事史cóng shì shǐ：负责主众曹文书的官员。
An official in charge of compiling and managing archives.
[9] 盈数yíng shù：过大数值。
Excess number or excess value.
[10] 朒数nǜ shù：不足数值。
Deficit number or deficit value.
【今译】
据古代的九数，圆周与直径的比率是3:1。此算法粗疏。自刘歆、张衡、刘徽、王蕃、皮延宗，各设新圆周率，但未达到折衷。
宋末，南徐州的从事史祖冲之，重新发现精密的圆周算法，以圆10000000的直径为1丈，圆周盈数是3丈1尺4寸1分5厘9毫2秒7忽，朒数是3丈1尺4寸1分5厘9毫2秒6忽。正数落在盈数和朒数二限之间。密率的直径113比圆周355。约率的直径7比圆周22。
