The Ellipse Seen from Nineteenth-Century China

Bréard, Andrea

doi:10.1007/978-3-319-93695-6_2

Andrea Bréard⁷

Part of the book series: Transcultural Research – Heidelberg Studies on Asia and Europe in a Global Context ((TRANSCULT))

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Abstract

This chapter uses the case of the ellipse as a new geometrical object introduced to China in the context of Aristotelian mechanics and Keplerian astronomy in order to analyse how authors in late imperial China, in particular before the introduction of differential and integral calculus, approached the rectification and quadrature of curves. It not only reveals the reliance on the classics and epistemic values when, for example, Dong Youcheng 董祐誠 (1791–1823) in 1821 proposed a solution to the problem of calculating the circumference of the ellipse by applying a problem from the Nine Chapters on Mathematical Procedures, but also argues that a shift of analysis was triggered by a change in scales of observation. It is shown in particular for the work of Xia Luanxiang 夏鸞翔 (1823–1864) that a panoptic point of view on conics, as compared to observing the ellipse as an isolated object, changes the representation, use and organization of Western concepts. The mathematical details of the calculation procedures discussed in this chapter are provided in Appendix C.

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Notes

1.
This passage is reminiscent of Isaac Newton’s classification of curves that is related in more detail by Elias Loomis (1811–1889) in his Elements of Analytical Geometry, and of Differential and Integral Calculus in 1859 (Loomis 1851) and translated into Chinese by Li Shanlan and Alexander Wylie (chin. 偉烈亞力, 1815–1887). See Li and Wylie (1859) vol. 8, p. 6B–7B.
2.
I.e. the cubical parabola, the biquadratic parabola, etc.
3.
Cf. Loomis (1851) p. 103: “the only curves whose equations are of the second degree, are the circle, parabola, ellipse, and hyperbola.”
4.
The expression Zhiqi shangxiang 制器尚象 is borrowed from The Great Appendix to the Classic of Changes《周易》易傳繫辭下. For a translation, see Legge (1963) app. III, chap. X.59, p. 367–369.
5.
The common phrase yang guan fu cha 仰观俯察 has its origins in The Great Appendix to the Classic of Changes《周易》易傳繫辭上: “仰以觀於天文, 俯以察於地理.” Translated in Legge (1963) app. III, chap. IV.21, p. 353:

(The sage), in accordance with (the Yî), looking up, contemplates the brilliant phenomena of the heavens, and, looking down, examines the definite arrangements of the earth.
6.
Translated from the Diagrammatic Explanations [of Procedures] for Curves (Zhiqu tujie 致曲圖解, 1861) in Xia (2006b) p. 438.
7.
See Chap. 6 p. 150 for a short explanation of the Chinese hexagrams.
8.
On Xia’s ideas concerning a comprehensive treatment of conics, see Liu (1990) p. 13.
9.
Projective transformations map lines to lines but do not necessarily preserve parallelism, whereas affine transformations respect parallel lines. The latter allow, for example, mapping a circle to an ellipse, but not an ellipse to a parabola or hyperbola.
10.
Swinden (1954) p. 45. It is uncertain if Xia Luanxiang had access to Kepler’s text (Kepler 1604), which had been part of the Beitang library according to Verhaeren (1949) p. 559 item no. 1901, and bound together with Kepler’s essay Strena seu de nive sexangula (1611) on the six-cornered snowflake.
11.
See the History of the Qing Dynasty (Qingshi 清史), scroll 506 卷五百六, Biographies 292 列傳二百九十二, Mathematician-Astronomers 2 疇人二 in Qingshi (1961) vol. 7, p. 5501–5502 and the entry on Xia Luanxiang in the anonymous Qingdai xueren liezhuan 清代學人列傳 (Biographies of Learned Men from Qing Dynasty), online http://ctext.org/wiki.pl?if=en&chapter=127281, §85:

All the principles of the ellipse and curves stem from the circle. If indeed, we unite what they have in common, then we “build tools to imitate the cosmos.” By “bowing [to the earth] and looking upwards [to the heaven], by observing [the sky] and investigating [the ground],” their applications are without limitations.

楕圓及諸曲線其理皆出於平圓, 苟會其通, 則制器尚象, 俯仰觀察, 為用無窮。
12.
See the entry on Xia Luanxiang in the anonymous Biographies of learned men from Qing dynasty (Qingdai xueren liezhuan 清代學人列傳) (Biographies of learned men from Qing dynasty), online http://ctext.org/wiki.pl?if=en&chapter=127281, §83–85.
13.
The term fangji used here is ambiguous, since it refers on the one hand to rectangular surfaces, but on the other to the binomial coefficients used in solving numerically polynomial equations for which Xia has developed a unique method. It is likely that the authors played on this ambiguity here in order to stress the relation between geometric objects of curved lines (here a chord of a circle) and the square (here more generally rectangular figures).
14.
Text quoted and translated from the History of the Qing Dynasty (Qingshi 清史), scroll 506 卷五百六, Biographies 292 列傳二百九十二, Mathematician-Astronomers 2 疇人二 in Qingshi (1961) vol. 7, p. 5501–5502.
15.
A list of Chinese writings related to conics can be found in Tables 2.2 and 2.3 p. 27–28.
16.
The Dai wei ji shi ji 代微積拾級 (Li and Wylie 1859), translated by Li Shanlan in collaboration with the British missionary Alexander Wylie (1815–1887) was based on Loomis (1851).
17.
Ming (1774). As Chen (2015) p. 493n7 points out that “no currently known documentation, Chinese or Mongolian, indicates Ming’s birth and death years.” On Jartoux and Ming’s work on infinite series see Martzloff (1997) p. 353–359.
18.
The “arrow” (chin. shi 矢) here is not what in the history of mathematics was also called the “sagitta,” a Latin word meaning “arrow,” and which came to be used interchangeably with “versed sine.” See Newton (1999) p. 306–307. In the Chinese context, the “arrow” is rather the versed sine of half the arc under consideration. See Fig. 2.2.
19.
For a complete list of the nine formulas, see Jami (1990) p. 58–60. For the procedure to find the arc with the versed sine at the bottom of page 59 there is a mistake. The series yields the square of the arc, not the arc itself. See Ming (1774) scroll 1 卷上, p. 5A.
20.
Nowadays an outdated trigonometric function, it is defined as \( \mathop {\mathrm {versin}} \nolimits z=1-\cos z\).
21.
In this text, Xia uses an unusual term for expressing the square of a number: suan , a variant of the character suan 筭, literally meaning “to calculate”; in its nominal form it designates a “counting rod”. In the present text, for example, the expression zhengxian suan 正弦 stands for the square of the sine, the expression banjing suan 半徑 is the square of the radius.
22.
Translated from Xia (1908) 2B, textually identical to the manuscript edition reprinted in Xia (2006a) p. 426.
23.
For the mathematical details, see Appendix C.
24.
As for Chap. 4, there is no correspondence in the Western tradition to the concept of Xia’s gui-lines. For a short discussion of these, see Liu (1990) p. 15–16.
25.
For some examples of Xia’s procedures and a discussion of errors, see Song and Bai (2008).
26.
The title is probably a pun on a book written by his teacher, Xiang Mingda 項名達 (1789–1850), entitled Images and Numbers—a Single Origin (Xiang shu yi yuan 象數一原), prefaces from 1843 and 1849. Reprint in Guo et al. (1993) vol. 5, p. 473–602.
27.
Quoted from The Gnomon of Zhou (Zhoubi suanjing 周髀算經), reprinted in Anonymous (1931), juan 上, p. 1B. English translation and commentary in Cullen (1996) p. 174.
28.
Style name of Xiang Mingda 項名達 (1789–1850).
29.
Style name of Dai Xu 戴煦 (1805–1860).
30.
Style name of Xu Youren 徐有壬 (1800–1860).
31.
Translated from Xia (1898a) Preface p. 1A.
32.
Idem scrolls 4–7.
33.
For mathematical details, see the Appendix p. 257.
34.
Xia (1898a) juan 3 p. 4A.
35.
For the derivation of this formula, see Loomis (1851) p. 225.
36.
See Xia (1898a) juan 4 p. 3B.
37.
For later reactions to Li and Wylie (1859), see for example Horng (1991) p. 342–353.
38.
See for example Xu Youren’s 徐有壬 (1800–1860) Correct Procedures for the Ellipse (Tuoyuan zhengshu 橢圜正術), posthumously published in 1872, in Xu (1985) p. 33–42, here p. 33:

The new [astronomical] methods all calculate with the ellipse for the expansion and contraction, acceleration and deceleration [of the planets], but calculation of their revolving trajectories are complicated. They all use approximations but no correct procedures.
39.
Translated together with John Edkins (chin. 艾約瑟, 1823–1905), see Li and Edkins (1898) and Whewell (1859). The original source of the three chapters on conics, the Theory of Conic Sections (Yuanzhui quxianshuo 圜錐曲線說) has not yet been identified. See Wang (1996) p. 353 and Han (2009) p. 104.
40.
See Li (1867c), Li (1867b) and Li (1867a), of which the first is a sequel to Xu Youren’s Correct Procedures for the Ellipse in Xu (1985) p. 33–42. For a discussion of Li’s work on conics, see Horng (1991) p. 353–362, Gao (2009).
41.
Li (1867a) p. 1a.
42.
The Nine Chapters give procedures for calculating a ring and segments of the circle and the sphere. See Chemla and Guo (2004) p. 141–143.
43.
1 mu = 240 bu [square], 1 jiao = 60 square bu [square].
44.
The value 10 is probably an approximation for π ².
45.
Translated from Qin (1842) p. 499. The three expressions in italic represent technical terms for the coefficients of a polynomial equation in the Chinese mathematical tradition. Here, the positive root x of the quadratic equation gives the double of the surface area of the banana-leaf.
46.
Libbrecht (1973) p. 109 remarks that Qin’s formula was probably meant as a better approximation for the case of a circular segment where the arrow is relatively small compared to the chord sustaining the arc. For the corresponding problems, algorithm and proof in the Nine Chapters, see problems I.35 and I.36 in Chemla and Guo (2004) p. 191–193.
47.
For Japan, where it was commonplace to pose highly original problems about the ellipse in early nineteenth-century sangaku temple geometry riddles, foreign influence is uncertain. See for example Fukagawa and Rothman (2008) problems 19 and 21.
48.
Quoted from Jami (2012) p. 289.
49.
Deng and Wang (1830) juan 1 p. 24B. As pointed out in Zhang et al. (2008) vol. 1, p. 98, this theorem was copied from Federico Commandino’s Liber de Centro Gravitatis Solidorum. See Theorema IIII in Commandino (1565) p. 6, where also a detailed proof is given:

In circulo & ellipsi idem est figuræ & gravitatis centrum.
50.
According to Yang (2004) p. 47n3 in particular in problems 17 and 18 of Ricci and Li (1614).
51.
The circle with a diameter from the point 卯 to the point 辰, shown in the upper left figure in Fig. 2.7.
52.
Ricci and Li (1614) p. 20B in Fig. 2.7.
53.
According to the catalogue (Verhaeren 1949) item 2612 the Jesuits had, for example, a 1597 edition by Adriaen van Roomen (1561–1615) of a book entitled In Archimedis circuli dimensionem Expositio & Analysis. Apologia pro Archimede ad Clariss. virum Iosephum Scaligerum … (Roomen 1597). Verhaeren (1949) item 4000 is the editio principes of the collection of Archimedean texts, Archimedes and Eutocius (1544).
54.
Verhaeren (1949) item 1288, for example, lists Clavius’s collected mathematical writings, the Opera mathematica (Clavius 1611).
55.
A comparison of reference in the commentary of proposition 18 to “Yuanshu 圓書 Book I proposition 31” for example, shows clearly that De Sphaera et Cylindro is what Ricci and Li consulted when they made a statement about the fourfold surface area of the sphere with respect to a circle with the same radius. See Archimedes and Eutocius (1544) De Sphaera et Cylindro Liber Primus vol. I, p. 31–32 and Ricci and Li (1614) p. 20B.
56.
Contained in Archimedes and Eutocius (1544).
57.
In the proof of Prop. 4 it is assumed that this auxiliary circle is the one having the major axis as radius.
58.
Paraphrases of the enunciations (which are longer and more involved) according to Heath (1897) p. 113–115, these propositions correspond to Propositions 5 to 7 in Archimedes and Eutocius (1544) p. 52–53.
59.
On the problem of dating Heron, see Héron d’Alexandrie (2014) p. 15–26.
60.
See Book I, prop. XXXIV translated in Héron d’Alexandrie (2014) p. 235. As pointed out in idem p. 235n310, this is not exactly the Archimedean formulation in De Conoidibus et Sphaeroidibus, where Archimedes gives the proportion:

Ellipse : circle = minor axis : diameter.
61.
Clavius (1604) Geometria Practica bk. IV, prop. V, p. 224.
62.
On the wider context of the book within the whole history of Jesuit astronomy in China, see Shi (2008a).
63.
Shi (2008b) p. 225. See also Hashimoto (1971).
64.
Translated from Qianlong 乾隆 (1773) p. 45A. According to Yang (2004) p. 49 this statement is to be found in juan 1 in the second part entitled “Finding the distance between two centers” (qiu liang xin cha 求兩心差), but it is in fact quoted from the section on “Finding the mean proportional between the major and minor axis of an ellipse” (qiu tuoyuan da xiao jing zhi zhonglü 求橢圓大小徑之中率).
65.
See Qianlong 乾隆 (1773) p. 41B–42A.
66.
Jami (2012) p. 320. For possible foreign sources, see also the introduction by Han Qi 韩琦 to the reprint edition of the Essence in Guo et al. (1993) vol. 3, p. 1–10.
67.
See Yunzhi 允祉 (1723) scroll 20, p. 17B–18B (Fig. 2.9). The preceding numerical example asks for the surface area of an ellipse with a major axis of 9 chi and a minor axis of 6 chi. In the solution, the argument by proportion is applied to a circle with a diameter equal to 10⁴ and a surface area of 78 539 816, thus assuming a value of π = 4 ⋅0.785 398 16 = 3.141 592 64.
68.
Translated from Yunzhi 允祉 (1723) scroll 20, p. 18A.
69.
See Yunzhi 允祉 (1723) scroll 3, p. 53A–55A:

In general, when the diameter of a circle is equal to the major axis of the ellipse another name is “duck-egg shape,” then the ratio between the surface area of the circle and the ellipse is also equal to the ratio between the rectangles circumscribed to the two figures. The ratio between the surface area of the circle and the ellipse is also equal to the ratio between the diameter of the circle and the minor axis of the ellipse.
70.
Dates unknown, Jinshi in 1801. See Yan (1990) entry 0174, p. 48.
71.
A county in Zhejiang province, now Jiaxing 嘉興.
72.
Dong (1821) p. 1A.
73.
Chapter 9 of the Nine Chapters concerned with problems related to right triangles.
74.
Quoted from Ruan and Luo (1882) scroll 51 卷五十一, entry on Dong Youcheng 董祐誠.
75.
Equivalent, but historically unrelated, procedures were found by Colin MacLaurin (1698–1746) in his Treatise on Fluxions (1742) and by the Japanese mathematician Sakabe Kōhan 坂部広胖 (1759–1824) in his Guide to Tenzan [Mathematical Methods] ([Sampō] Tenzan shinan roku [算法] 点竄指南錄 ) in 1815. See MacLaurin (1742) § 806, p. 658–659 and Mikami (1912).
76.
Lü is a technical mathematical term conceptually introduced by Liu Hui in his 263 commentary to the Nine Chapters where it designates a number that is in a ratio with another one. See Chemla and Guo (2004) p. 956–958.
77.
For convenience only, the expression \(\sqrt {\frac {a^2-b^2}{a^2}}\) is replaced in my transcription by e, the eccentricity of an ellipse, a concept measuring its deviation from circularity not present in Xiang Mingda’s text.
78.
Translated from Xiang (1888) vol. 6, p. 27B–28A.
79.
Translated from Xiang (1888) vol. 6, p. 28A. A different version of Xiang’s procedure is given by Dai Xu in his Diagrammatic Explanations [to the Procedure] to Find the Circumference of an Ellipse (Tuoyuan qiu zhou tujie 橢圓求周圖解, 1857), which was published as an appendix to the posthumous publication of Xiang Mingda’s Images and Numbers—a Single Origin (Xiang shu yi yuan 象數一原). See Dai (1888) vol. 7, here p. 53A–53B.
80.
For a formulaic transcription of Dai Xu’s work, see Li (1955).
81.
See MacLaurin (1742) § 806, p. 658–659. A more detailed comparison between Xiang Mingda’s, Dai Xu’s and MacLaurin’s approach is in preparation for a future publication.

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Bréard, A. (2019). The Ellipse Seen from Nineteenth-Century China. In: Nine Chapters on Mathematical Modernity. Transcultural Research – Heidelberg Studies on Asia and Europe in a Global Context. Springer, Cham. https://doi.org/10.1007/978-3-319-93695-6_2

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DOI: https://doi.org/10.1007/978-3-319-93695-6_2
Published: 26 June 2019
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