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

The angles
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.

(1)

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,

From (1),

Multiplying
by *d*^{2},

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
*

Let

*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 |
Figure 5.
Error in the calculation of |

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.