### A new interpretation of the ancient Chinese rule for spherical segment

By Professor R. C. Gupta, Ph. D. (Hist. of Math.)
R-20 Ras Bahar Colony, P. O. Sipri Bazar
JHANSI-284003, India

The Jiu Zhang Suan Shu (JZSS), or “The Nine Chapters of Mathematical Art”), is the most important work of ancient China on mathematics. Although based much on earlier material, its present form is placed in the first century of our Common Era (CE). Its earliest extant version is that of Liu Hui (263 CE), whose commentary exerted profound influence on Chinese mathematics for well over a millennium.

As the title indicates, the JZSS is divided into nine chapters and has a total of 246 problems. The first chapter is entitled Fang-tian (Mensuration of Fields) and contains 38 problems. It is devoted to calculation of areas of various geometrical figures. Problems 31 and 32 are on the area of a circle of circumference c and diameter d and contains the popular exact formula

$Area = (c \cdot d)/4 \hspace{6cm} (1)$

along with two approximate rules $(3/4)d^2$ and $c^2/12$, which imply the use of $\pi=3$. Problems 33 and 34 are about the area of the surface of a spherical segment. However, there is some disagreement on this issue and great divergence in interpreting the prescribed rule. The purpose of the present article is to briefly mention the various views and give some new interpretations.

Problems 33 and 34 ask us to find the area A of what it calls a wan tian or wan field whose lower zhou (circumference or perimeter p) and jing (diameter or width w) are given. The formula (prescribed and) used is

$A = (p \cdot w)/4. \hspace{6cm} (2)$

Modern scholars agree with Liu Hui that this is about spherical segment (the next two problems are on hu tian or circular segment.) However, a century ago, Mikami took wan tian as a sector of a circle. In this case, p will be the circular arc $E_{1}FE_{2}$ and w will be the diameter of the circle (see Fig. 1), and the rule (2) will be mathematically exact. Vogel (1968) and Ho (1973) followed Mikami’s interpretation.

Fig 1

Fig 2

In Crossley and Lun’s translation (1987) of Li and Du’s original Chinese book, we find that the JZSS rule for the area of a spherical segment (called “domed garden field” there, pp. 41-42) is given as

$Area = (p \cdot d)/4 \hspace{6cm} (3)$

where p is the perimeter and d the diameter of the base circle (see Fig. 2). This interpretation of relating p and w (which is taken here as d) to the same base circle is not supported by the numerical examples of JZSS (see Table 1).

Table 1: Data on Spherical Segment

 Example Number Source (Text) p w Area (in text) 1 JZSS problem 33 30 16 120 2 JZSS problem 34 99 51 1262.25 3 Wucao Sunjing (Lam, 1977, p. 95) 640 380 243200 4 GSS, VII. 26 56 27 378 5 GSS, VII. 27 36 15 135

We see that w cannot be the diameter of the corresponding base circle of perimeter p and the interpretation (3) does not clarify what w is! It is worth mentioning that the same misinterpretation has been also found in the Indian translations (up to 2000 CE) of the Sanskrit work Ganitasāra Sangraha (GSS) of Mahāvira (9th cent. CE), where a similar rule appears.

We must be careful about the old terminology used in the original rule. The Sanskrit word vyāsa (or wiskambha) ordinarily means ‘diameter’ but was also used to denote width (vistāra) of all sorts. The same is true about the Chinese jing which normally means ‘diameter’ but can also have other meanings and can denote any width in general. Also exact formulas for surface of a spherical segment were not known in China (and India) in those times. So JZSS rule (2) is not expected to be exact and credit for exactness can come only by some twisted interpretation. Mikami’s twist has been already mentioned above.

Fig 3

Earlier, the wise Liu Hui hinted at another case where (2) can be used to get an exact result. Consider a right circular cone of base radius r and slant height k (Fig. 3). The width of the plane base is measured by its diameter EF. For the elevated curved area, the double oblique distance EVF (2k) may be taken as its width w. Also its lower perimeter p will be equal to $2\pi r$. Thus the area of the curved surface of the cone will be given by (2) as

$A_0 = (2\pi r) \cdot (2k)/4 = \pi rk \hspace{4cm} (4)$

And indeed this is mathematically exact! Interestingly this curved surface can be developed into the plane sector $VE_{1}FE_{2}$ of Fig. 1 by cutting it along the slant edge VE.

Fig 4

A quite different interpretation of the JZSS rule is found in Lam’s writings (using either a translation or notes on the text) where w is taken as the diameter D = 2R (see Fig. 4) of the sphere of which the segment forms a part. Accordingly, the rule (2) takes the form

$A_1 = (p \cdot D)/4 = (p \cdot R)/2 \hspace{4cm} (5)$

Lam’s 1977 book (p. 95) presents this as Yang’s version of JZSS problem 33 but it is not clear as to on what basis jing or ‘diameter’ is taken as the “diameter of the sphere”. This enables us to find the height VM (h) of the segment (Fig. 4) by applying the simple formula

$h = R \pm \sqrt{R^2 - (p/\pi )^2} \hspace{5cm} (6)$

The correct surface $S_0$ is then obtained by using the exact formula (proved by Archimedes more than 2222 years ago)

$S_0 = 2\pi Rh \hspace{7cm} (7)$

Taking the lower sign in (6) and $\pi = 3$, Lam calculated $S_0$ to be 84.24, while the JZSS answer is 120. Using the alternative sign will yield nearly 684. Furthermore, there will be practical difficulty in finding or measuring R when a segment is given as an independent figure, such as a knoll or round hillock.

Fig 5

Following the exposition given by Liu Hui, our new interpretation takes the curvilinear distance PVQ (s) of the segment (see Fig. 5) as the jing or width w in (2) thereby obtaining the new rule

$S = (p \cdot s)/4 \hspace{6cm} (8)$

In this interpretation, a segment will be smaller than, equal to, or greater than a hemisphere as s is less than, equal to, or greater than p/2, respectively. Thus all three Chinese examples in Table 1 are cases of segments bigger than a hemisphere, i. e., the arc PVQ (s) is greater than the semicircle $EVF = \pi R$ (see Fig. 5). With angle $EOP = \phi$, we have the relations

$\phi = (s - \pi R)/2R \hspace{5cm} (9)$

$R cos \phi = PQ/2 = p/2 \pi \hspace{4cm} (10)$

Substituting R from (9) into (10) leads to an equation from which $\phi$ can be found, and subsequently R from (9).

In this way, the correct area will be, by (7),

$S_0 = 2 \pi R^2 (1 + sin \phi ) \hspace{5cm} (11)$

For the JZSS Example 1 (p = 30, s = 16), we found $\phi$ to be 5.5 degrees (.096 radians), R is 4.8 and $S_0$ is 158.7. Thus the text value 120 has error of + 43 % as per Lam’s interpretation (5) and about – 25 % by our method (8).

A separate article is needed to discuss the historical and theoretical aspects of the different interpretations in detail by various methods (e. g., making comparative tables and drawing graphs).

In closing, an interesting historical note should be mentioned. For a hemispherical surface (s = p/2), we obtain $p^2/8$ from (8). Thus, we have the expression $C^2/4$ for the surface of a sphere where C is the perimeter or circumference of the great circle. This expression appeared in India in the 13th century and in Japan as late as in the 17th century CE (Mikami, p. 206)!

BIBLIOGRAPHY
1. J. N. Crossley and A. W.-C. Lun (transl.): Chinese Mathematics: A Concise History (a transl. of Chinese book by Li and Du), Clarendon Press, Oxford, 1987.
2. R. C. Gupta: “Mahāvira’s Rule for Surface Area of a Spherical Segment”, Tulasi Prajnā 1(2) (1975), 63-66.
3. R. C. Gupta: “On Some Rules from Jaina Mathematics”, Ganita Bhārati 11 (1989), 18-26.
4. T. Hayashi: “Calculation of the Surface of a Sphere in India”, Sci. & Eng. Review of Doshisha Univ. 37(4) (1997), 194-238.
5. P.-Y. Ho: “Liu Hui”, Dictionary of Scientific Biography, Vol. VIII (1973), 418-425.
6. L.-Y. Lam: A Critical Study of the Yang Hui Suan Fa, Singapore Univ. Press, 1977.
7. L.-Y. Lam: “Chu Shih-chieh’s Suan-hsueh ch’i-meng”, Archive for History of Exact Sciences, 21(1) (1979), 1-31.
8. L.-Y. Lam: “Jiu Zhang Suan Shu”, Archive for History of Exact Sciences, 47(1) (1994), 1-51.
9. J.-C. Martzloff: A History of Chinese Mathematics (transl. by S. S. Wilson from the French), Springer, Berlin, 1987.
10. Y. Mikami: The Development of Mathematics in China and Japan, Chelsea, NY, 1961 (Reprint of 1913 ed.).
11. J. Needham: Science and Civilization in China, Vol. III, Cambridge Univ. Press, Cambridge, 1959.
12. Padmavathamma: The Ganita Sāro Sangraha of Mahaviracharya (editions and translations), Hombuja, 2000.
13. K. Vogel: Neun Bücher Arithmetischer Technik, Braunschweig, 1968.
14. D. Wagner: “Translation of the Discussions of the Circle-Mensuration in the Jiu Zhang Suan Shu and the Commentaries by Liu Hui etc.” Typed manuscript (1973) of 41 pages so kindly made available by him to RCG in 1989.

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