SS16.06 隋書·祖沖之圓周率 Book of Sui: Zu Chongzhi's Circumference Ratio

The following excerpt is from Volume 16 of the ***Sui Shu*** (隋書 suí shū; *Book of Sui*), the official history of the Sui dynasty (581 - 618 AD). The *Sui Shu* was authored by several authors, most notably Wei Zheng (魏徵 wèi zhēng; 580 - 643), Zhangsun Wuji (長孫無忌 zhǎng sūn wú jì; ? - 659), Kong Yingda (孔穎達 kǒng yǐng dá; 574 - 648), and Yan Shigu (顏師古 yán shī gǔ; 581 - 645). This passage gives the earliest surviving account of the mathematical accomplishments of Zu Chongzhi (祖沖之 zǔ chōng zhī; 429 - 500); most notably his accurate calculation of

π

{\displaystyle \pi }

, better known as the circumference ratio (圓周率 yuán zhōu lǜ) in Chinese. The *Sui Shu* also mentions the names of several Chinese mathematicians involved with the compulation of an accurate value of the circumference ratio, as well as Zu Chongzhi's lost mathematical work, the Zhui Shu (綴術 zhuì shù; *Method of Interpolation*). In this translation, I present an interpretation of the terms 開差冪 (kāi chà mì; opening the difference of areas) and 開差立 (kāi chà lì; opening the difference of areas) to mean the general algorithm for solving quadratic and cubic equations related to areas and volumes.

**《隋書·卷十六律曆志》**

古之九數，圓周率三，圓徑率一，其術疏舛[1]。 自劉歆[2]、張衡[3]、劉徽[4]、王蕃[5]、皮延宗[6]之徒，各設新率，未臻折衷。
宋[7]末，南徐州從事史[8]祖沖之，更開密法，以圓徑一億為一丈， 圓週盈數[9]三丈一尺四寸一分五厘九毫二秒七 忽，朒數[10]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，圓週三百五十五。 約率，圓徑七，週二十二。又設開差冪，開差立，兼[11]以正員[12]參之。 指要精密，算氏之最者也。 所著之書，名為《綴術》，學官[13]莫能究其深奧，是故廢而不理。

**Book of Sui (Treatise on Harmonics and Calendrics)**

According to the nine computational methods of antiquity, the ratio of the circumference to the diameter is 3 to 1. This method is inaccurate for being too small. 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 South Xuzhou, 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. He also established the method of opening the difference of areas, as well as the difference of volumes, with the incorporation of positive and negative edge coefficients. The methods are precise, and he is a mathematician of the highest caliber. The book he wrote, entitled the *Zhui Shu*, was so abstruse that students [at the Imperial Academy] could not study its profundity. For this reason, it was abandoned and ignored.

**Annotations**

[1] 疏舛 shū chuǎn：粗疏，因為太小而不準確。Crude, inaccurate for being too small.

[2] 劉歆 liú xīn：劉歆（公元前 50 年 – 公元 23 年）漢代經學家、天文學家。
Liu Xin (50 BC – 23 AD) scholar of classics and astronomy of the Han dynasty.

[3] 張衡 zhāng héng：張衡（78 年 – 139 年）漢代作家、天文學家、發明家。
Zhang Heng (79 – 139 AD) literary scholar, astronomer, and inventor of the Han dynasty.

[4] 劉徽 liú huī：劉徽（約 225 年 – 295年）魏晉時期數學家。Liu Hui (c. 225 – 295 AD) mathematician of the Three Kingdoms and Jin dynasty.

[5] 王蕃 wáng fān：王蕃（228 年– 266年）三國時代吳國天文學家、數學家。Wang Fan (228 – 266 AD) astronomer and mathematicians of the Three Kingdoms period.

[6] 皮延宗 pí yán zōng：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] 從事史 cóng shì shǐ：負責主眾曹文書的官員。An official in charge of the archives.

[9] 盈數 yíng shù：過大數值。Excess number or excess value.

[10] 朒數 nǜ shù：不足數值。Deficit number or deficit value.

[11] 兼 jiān：應為"廉"，指系數。A typo that could read 廉 (lián), which means the coefficient.

[12] 員 yuán：應為"負"，指負數。A typo that could read 負 (fù), referring to negative numbers.

[13] 學官 xué guān：國子監的學生。Students who attend the Imperial Academy.

**Mathematical Notes**

(1) Zu Chongzhi calculated

3.1415926
<
π
<
3.1415927

{\displaystyle 3.1415926<\pi <3.1415927}

 and gave the rational expression

# π

355
113

{\displaystyle \pi ={\frac {355}{113}}}

.

(2) It is uncertain what the terms 開差冪 (kāi chà mì; opening the difference of areas) and 開差立 (kāi chà lì; opening the difference of areas) truly mean. The name suggests a connection to the square root algorithm 開方術 and cube root algorithm 開立方術 in the Jiuzhang Suanshu, as well as Liu Hui's terminology for the area 冪. There is a possible corruption of the characters 廉 and 負 with 兼 and 員 (圓 in some editions) respectively. If the above two characters are indeed typographical errors, then the terms 開差冪 and 開差立 could be referring to a general algorithm for solving quadratic equations

x

2

a
# x
A

{\displaystyle x^{2}+ax=A}

 and the cubic equations of the form

x

3

a

x

2

b
# x
V

{\displaystyle x^{3}+ax^{2}+bx=V}

, where

A
,
V

{\displaystyle A,V}

 are the areas and volumes of certain geometric shapes.