Donald B. Wagner
11 October 2021
Some 50 years ago Nathan Sivin challenged me to explain why a number of ancient Chinese mathematicians continued to use π ≈ 3 long after Zu Chongzhi 祖冲之(429–500) had calculated π to seven places (3.1415926 < π < 3.1415927). This was especially odd in the case of the Season Granting Astronomical System (Shoushi li 授時曆), by Guo Shoujing 郭守敬 (1231–1316) and others, presented to the Throne in 1280 (see now Sivin 2009). I speculated at the time that the error in the value of π might have compensated for other errors in the full calculation. In 1982 I was able to do the calculations, determine that my speculation was correct, and write a short paper about it. But before I got around to submitting it, it turned out that Nathan had put the same challenge to Jean-Claude Martzloff, and that he had come to the same conclusion (Martzloff 1987/1997: 329–335). There is therefore no reason to publish the present revised version of that paper except that I like the way I present the problem and the calculations.
This article concerns a narrow computational point, and I do not consider wider historical issues. I have not read the text of the computation in the Ming shi 明史, but follow Qian Baocong’s description of it (1932: 148–151; 1956/1983: 370–373; 1964: 210–214; see also Yan Dunjie 1966: 221–224).
Figure 1. Octant of the celestial sphere, related to the conversion of ecliptic to equatorial coordinates.
The calculation to be considered here is equivalent to calculating, for a given angular position on the ecliptic, its declination and right ascension. However, in ancient Chinese astronomy angles were measured from the solstices rather than from the vernal equinox, so in modern terms the problem is: given the complement of the angular position on the ecliptic, to calculate the declination and the complement of the right ascension. See Figure 1.
A is the vernal equinox
D is the summer solstice
arc AD is a quadrant of the ecliptic
arc AE is a quadrant of the equato
arc DE is the obliquity of the ecliptic (modern value 23.45°)
B is an arbitrary point on the ecliptic
of arcs DE and BD are known, and the angles of arcs BC and CE are to be
calculated. The correct calculation of these angles uses
These formulas follow directly from the Fundamental Formula of spherical trigonometry (Smart 1962: 7).
Figure 2. Related to Shen Gua’s and Guo Shoujing’s approximation formulas.
Guo Shoujing does not explicitly use trigonometric functions, but his calculations are essentially the same as these. In the process he needs a means of approximating one trigonometric function and the inverse of one trigonometric function. He uses a version of an approximation attributed to Shen Gua 沈括 (1031–1095) for the length of an arc (Wagner 2012). See Figure 2.
This approximation is used in the Shoushi li to approximate an angle whose versine and sine are known.
(The versine is defined by ).
Thus it serves in situations in which we would use the inverse of a trigonometric function.
The other approximation used in the calculation finds the sagitta when an arc and its chord are known. Again in Figure 2,
This can be derived from (1) as follows. By the Pythagorean Theorem,
Multiplying by d2,
The value of s is found using the Chinese version of Horner’s Method for the roots of polynomials (see Wagner 2017). This is functionally equivalent to approximating the versine of an angle.
In the Shoushi li angles are measured in du 度. A circle is divided into 365.25 du, so that one du is equal to about 0.986 modern degrees. Many Western writers on ancient Chinese astronomy translate du as ‘degree’, but I prefer to leave it untranslated.
A curiosity in the Shoushi li is that the du is treated as a kind of linear unit. The computation of a ‘diameter’ of the celestial sphere in du, using π ≈ 3, gives a diameter of du. This is the only use of any value of π in the computation.
For ease of exposition define
This is (1), the version of Shen Gua’s arc-length formula used by Guo Shoujing, an approximation of .
guo(a, d) = the lesser of the two roots of
This is Guo Shoujing’s approximation of .
Figure 3. Figure 1, with additional constructions.
See Figure 3, which is the same as Figure 1, with several constructions added. Note that BLMN is a rectangle, so that
BL = MN
BN = LM
d = 121.75 du; the ‘diameter’ of the celestial sphere
= the ‘radius’
The two known quantities are the arcs DE and BD, and the quantities to be determined are BC and CE. As noted above, the correct computation is
We shall see that the Shoushi li calculation corresponds step-by-step to this modern calculation.
The first step in the computation corresponds to the calculation of sin DE. The value used for the obliquity of the ecliptic, DE, is 24 du, which is equivalent to 23.66 degrees. This may be compared with the modern value, 23.45 degrees.
The next step corresponds to the calculation of the cosine and sine of BD:
Next the calculation of sin BC: because the triangles OLM and ODK are similar, and BN = LM,
Then the calculation of the arcsine of sin BC:
This completes the calculation of BC. For the calculation of CE most of the work has been done. First sin CE:
And then the arcsine of sin CE: because triangles OCP and ONM are similar,
And this completes the computation.
Figures 4 and 5 show the error in the approximations for BC and CE for the full range of interest, 0 ≤ BD ≤ 365.25/4 du. Using π ≈ 3, the error in the calculation of BC is always less that 0.22 du, while the error using the correct value of π is much larger for large values of BD. The comparison is less impressive in the case of CE, but the maximum error using π ≈ 3 is less than 0.25 du, while the maximum error using the correct π is about 0.34 du.
It is a matter for speculation, without much hope of a resolution, whether the Shoushi li creators were aware of the difference or were just ‘lucky’ in their choice of a value for π.
Figure 4. Error in the calculation of BC in the full range of BD, using two different values for π.
Figure 5. Error in the calculation of CE in the full range of BD, using two different values for π.
Martzloff, Jean-Claude. 1987/1997. A history of Chinese mathematics. Berlin: Springer. Orig. Histoire des mathématiques chinoises, Paris: Masson, 1987.
Qian Baocong 錢寶琮. 1932. Zhongguo suanxue shi 中國算學史 (‘A history of Chinese mathematics’). Beiping: Academia Sinica 國立中央研究院. www.scribd.com/doc/285715053/
———. 1956/1983. ‘Shoushi lifa lüelun 授時曆法略論’ (Notes on the methods of the Season Granting Astronomical System). In Qian Baocong kexueshi lunwen xuanji 錢寶琮科學史論文選集, edited by Chen Yongjiang 陳永鏘 and Bi Ying 畢穎. Beijing: Kexue Chubanshe, pp. 352–376. Orig. Tianwen xuebao, 天文學報 1956, vol. 4, no. 2.
———. 1964/1981. Zhongguo shuxueshi 中國數學史 (The history of mathematics in China). Beijing: Kexue Chubanshe Repr. 1981 with minor corrections.
Sivin, Nathan. 2009. Granting the Seasons: The Chinese astronomical reform of 1280, with a study of its many dimensions and a translation of its records. (Sources and studies in the history of mathematics and physical sciences). New York: Springer.
Smart, W. M. 1962. Text-book on spherical astronomy. 5th edn. Cambridge: Cambridge University Press. Repr. 1965.
Wagner, Donald B. 2012. ‘Shen Gua and an ignorant editor on the length of an arc’. donwagner.dk/Shen-Gua-arc.htm
———. 2017. ‘The classical Chinese version of Horner’s method: Technical considerations’. donwagner.dk/horner/horner.html
Yan Dunjie 顏敦傑. 1966. ‘Song Jin Yuan lifazhong de shuxue zhishi 宋金元曆法中的數學知識’ (Mathematical knowledge in the astronomical methods of the Song, Jin, and Yuan periods). In Song Yuan shuxueshi lunwenji 宋元數學史論文集, edited by Qian Baocong 錢寶琮. Beijing: Kexue Chubanshe, pp. 210–224.