1Recent studies on the models of solar orbit in the Qingdynasty, China, and natural philosophy(清王朝時代の太陽軌道モデルについての最近の研究と、自然哲学)
Yukio Ȏhashi （大橋由紀夫）

I.

Introduction
Western geometrical astronomy was introduced into China by Jesuit missionaries in

the late Ming (明) dynasty and the early Qing (清) dynasty.
In the case of the solar orbit, it seems that there was certain Chinese originality.
There are some recent studies on this topic.
(I.1) Western geometrical models
In ancient Greece, Apollonius (3rd century BC) used the epicyclic model for planetaryorbits, and showed that it is equivalent to the eccentric model. (See Figs.1 and 2.)

Then, Hipparcus (2nd century BC) used the simple eccentric model for the solar orbit.
(See Fig.5.) He also made a model of the lunar orbit, which is not discussed here.

2Ptolemaeus (2nd century AD) newly used the “equant” model for the planetary orbits,
and revised the model of the lunar orbit. However, the model of the solar orbit was thesame as Hipparchus, that is the simple eccentric model. (See Fig.2.)

In the simple eccentric model (Hipparchus for solar orbit), the object revolves at aconstant speed. (Fig.5.)
In the equant model (Ptolemaeus), the object revolves at a constant angular velocity
around the equant “E”. (Fig.6.)
The double epicycle model was created by ibn ash-Shātir (14th century AD). Theobject revolves at a constant speed on the small epicycle, and the small epicycle revolves
at a constant speed on the large epicycle. The large epicycle revolves at a constant
speed on the deferent. (Fig.7.) Its result is almost the same as the equant model.
(I.2) Introduction of Western sciences into China, ----- Jesuit missionaries
Matteo Ricci (1552 – 1610) arrived at Beijing in 1601, and introduced Westernsciences into China. Since then, several Jesuit missionaries visited China.
In China, some official astronomical works based on Western astronomy werecomposed as follows.

Chongzhen-lishu (Calendrical treatise in the name of Emperor Chongzhen) (崇禎暦書)(1631 – 34), where the simple eccentric model is used for the solar orbit.

Xiyang-xinfa-lishu (Calendrical treatise in Western new methodology) (西洋新法暦書)(1645), which is a re-edition of the Chongzhen-lishu. The Xiyang-xinfa-lishu (新法算書)
was later renamed as Xinfa-suanshu (Mathematical treatise of new methodology).

Lixiang-kaocheng (Treatise of calendrical phenomena) (暦象考成) (1722), where thedouble epicycle model is used for the solar orbit.

Lixiang-kaocheng-houbian (Second part of the treatise of calendrical phenomena) (暦象考成後編) (1742), where Kepler’s elliptic orbit is used.

3It should also be noted that there is one important non-official work composed in thisperiod, the Xiao’an-xinfa (New method of Xiao’an) (曉庵新法) (1663) of Wang Xichan (王錫闡) (alias Xiao’an) (1628 – 1682). There are also non-official but very importantworks of Mei Wending (梅文鼎) (1633 – 1721), which were edited by his successors, andwere very influential.
II.

Solar model in the Lixiang-kaocheng

In my Chinese paper [Ȏhashi (2007)], I wrote that the double epicycle model of thesolar orbit in the Lixiang-kaocheng may be a Chinese original device, because only thesimple eccentric model was used in the geocentric system in Europe at that time. In aseminar in 2012, I presented the similar view in English [Ȏhashi (2012)].
Recently, Chu Longfei and Shi Yunli also studied this topic, and expressed a similarview [Chu and Shi (2012) and Shi and Chu (2013)]. An anonymous referee of my paper[Ȏhashi (2012)] informed me about these works.
Chu and Shi (2012) pointed out that the equant model for the orbit of the sun wastried (with certain uncertainty) to apply in the Xiyang-xinfa-lishu (1645). It is alsomentioned in the Xinfa-suanshu (Vol.26). I think that this is certainly a Chinese newattempt to revise the theory of the orbit of the sun, and further research is needed.
Another important non-official work is the Xiao’an-xinfa (1663) of Wang Xichan (1628– 1682). According to the interpretation of Miyajima (1991), it used the combination ofan eccentric deferent and an epicycle for the sun, and according to the interpretation ofNing (2006), it used the “Double epicycle model” for the sun. The result of the both isthe same. (As this text Xiao’an-xinfa does not have any figure, it is difficult to saywhich interpretation is correct.) This may be a forerunner of the Lixiang-kaocheng.
Its relationship with the Lixiang-kaocheng should be investigated further.
And also, Shi and Chu (2013) pointed out that Mei Wending (1633 – 1721) suggested
to use two epicycles for the sun in his Lifa-yiwen (Queries about calendrical science) (暦学疑問) (Vol.3). This may also be a forerunner of the further development, and more
research is needed.
Then, the Lixiang-kaocheng clearly used the double epicycle model for the solar orbit
with clear figures.

(See Fig.8.)

The Lixiang-kaocheng (Part I, Vol.4) explains thereason to introduce double epicycle as follows. Firstly, the observed solar longitude
shortly after the solstices (i.e. around the apogee and perigee) is applied to the “Simpleeccentric model”, and the longitude of the apogee and perigee is determined. Then,from the difference between the mean sun and the observed longitude of the sun whenthe mean sun is at the distance of right angle from the apogee, determine the eccentric

4distance. And also, using the observed time of the equinoxes and “lixia” (立夏) (point oftime when the solar longitude is 45˚), determine the longitude of the apogee and theeccentric distance. The Lixiang-kaocheng says that these results are not harmonious
according to the “Simple eccentric model”, but are harmonious according to the “Doubleepicycle model”.
The “Double epicycle model” according to the Lixiang-kaocheng is as follows. Thereis a concentric deferent (“bentian”, 本天) whose centre is the earth. On the deferent isthe first (larger) epicycle (“benlun”, 本輪), whose radius is three fourth of the eccentric
distance of the “Simple eccentric model”.

Its centre revolves along the deferent

eastwards in a (tropical) year. On the first epicycle is the second (smaller) epicycle(“junlun”, 均輪), whose radius is one fourth of the eccentric distance of the “Simpleeccentric model”. Its centre revolves along the first epicycle westwards in a year, but
its speed is slightly slower than the speed of the centre of the first epicycle. (Thisdifference is due to the secular variation of the longitude of the apogee.) The sun
revolves eastwards on the second epicycle, and its speed is double of the speed of thecentre of the second epicycle. Now it is clear that this is basically the same as theordinary “Double epicycle model”. (See Fig.7.)
The “Double epicycle model” for the sun in the Lixiang-kaocheng (1722) is a special
feature of this work. There is a possibility that the “Double epicycle model” for theplanets was also applied to the sun when the Lixiang-kaocheng was edited in China.
Although

Keplarian

ellipticorbit

wasintroduced

intheSequel(Lixiang-kaocheng-houbian) to this work later, the “Double epicycle model” for the sunin the earlier Lixiang-kaocheng can also be considered to be a small improvement. The“Double epicycle model” for the sun in geocentric model is not found in Western
standard works, and might be a Chinese device. Though the geocentric system is used
here, this attempt to apply similar model to different heavenly bodies (the planets andthe sun) is a kind of a step to the recognition of the universal laws of nature.
Why this device was produced in China? Was there a kind of natural philosophy
behind this device? This point should be investigated further.
Chinese philosophy has a thought of a kind of “oneness”. This world was believed tohave been born from the ultimate one (taiji, 太極, literally, the “great pole”.).

AConfucianist classic Yijing (易經, Book of Changes) says that “Yi (易, principle ofchanges) has the taiji (ultimate one), it produces liangyi (two principles, that is yin(negative side of the nature) and yang(positive side of the nature)), ---”. (易有太極, 是生兩儀, ---) [Ekikyō (1969), vol.2, p.241 (繋辭上傳).] And also, a Taoist classic Laozi (老子, also called Daodejing 道徳經) says that “Dao (道, literally means “road”, that is the5fundamental principle of the world) produces “one” (existence), and the “one” produces“two” (heaven and earth) ---” [Rōshi (2008), p.203 (Chapter 42).]Fig.8, A page from the Lixiang-kaocheng6

I do not say that the Chinese solar model was directly influenced by the Conficianismor Taoism, but we should consider that China had a thought that this world wasoriginated in ultimate one. If so, there might have been a kind of thought of theuniformity of nature. This point should be investigated further.
III.

Conclusion

Eccentric models and epicycle models were used in Greek geometrical models for theorbits of the sun, moon and planets. These geometrical models were introduced into
China by Jesuit missionaries in the early 17th century. Since then, several works on
Western mathematical astronomy were composed in Chinese.
In the Greek model, a simple eccentric model was used for solar orbit, and it wasintroduced

to China firstly.

However, there isaChinese work entitled

Lixiang-kaocheng, composed in 1722, and a double epicycle model was used for the solar
orbit there. It is almost equivalent to the “equant model”.
At that time, Kepler’s model had not been introduced to China.

So, the double

epicycle model for the solar orbit in China may be Chinese original device.

References
Chu, Longfei and Shi, Yunli: “Chongzhen-lishu xilie lifa-zhong de taiyang yundonglilun” (The theory of Solar Motions in the Chongzhen Lishu Series) (in Chinese), Ziran

Kexue-shi Yanjiu (Studies in the History of Natural Sciences), Vol.31, No.4, (2012),pp.410 – 427. (褚龙飞 石云里:“《崇祯历书》系列历法中的太阳运动理论”,《自然科学史研究》,2012,31(4), 410-427)

Ekikyō (Yijing, Classical Chinese text with a Japanese translation), translated by
Takada Shinji and Gotō Motomi, 2 vols, (Iwanami-bunko), Iwanami-shoten, Tokyo, 1969.
（『易経』, 岩波文庫.）
Miyajima, Kazuhiko: “Ou Sekisen Gyou’an Shinpou no Taiyoukei moderu” (The modelof the solar system in the Xiao’an-xinfa of Wang Xichan) (in Japanese), Chūgoku Kodai

Kagakushi-ron, Zokuhen, Kyōto Daigaku Jinbunkagaku Kenkyūjo, Kyoto, 1991, pp.243－266. （宮島一彦：王錫闡『曉庵新法』の太陽モデル, 『中国古代科学史論』, 京都大学人文科学研究所, 京都, 1991, pp.243–266）
Ning, Xiaoyu: “Xiao’an-xinfa yu Xinfa-suanshu jisuan taiyang dixin huangjing fangfa
de bijiao” (Comparison of the method to calculate geocentric solar longitude of theXiao’an-xinfa and the Xinfa-suanshu) (in Chinese), Zhongguo Keji Dianji Yanjiu –Di-san-jie Zhongguo keji dianjie guoji huiyi lunwenji (Studies on Ancient Chinese7

Scientific and Technical Texts – Proceedings of the 3rd ISACBRST), Daxiang Chubanshe,Zhengzhou, 2006, pp.201－209. (宁晓玉: 《晓庵新法》太阳地心黄经的计算及其与《新法算书》的比较，《中国科技典籍研究 - 第三届中国科技典籍国际会议论文集》，大象出版社,
郑州,2006,pp.201-209)Ȏhashi, Yukio: “Lixiang-kaocheng zhong de taiyang yundong-lun” (On the Solar
Motion in the Lixiang-kaocheng) (in Chinese), Neimenggu Shifan Daxue Xuebao (ZiranKexue Hanwen-ban) (Journal of Inner Mongolia Normal University (Natural ScienceEdition), Vol.36, No.6, 2007, pp.662 – 665. (大桥由纪夫: 《历象考成》中的太阳运动, 内蒙古师范大学学报(自然科学汉文版),2007, 36(6), pp.662-665.)Ȏhashi, Yukio: Eccentric Model of the Solar Orbit in China (International Seminaron History of Mathematics, 2012, Ramjas College, University of Delhi, Delhi, India)(Its proceedings are forthcoming.)

Rōshi (Laozi, Classical Chinese text with a Japanese translation and commentary),translated and commented by Hachiya Kunio, (Iwanami-bunko), Iwanami-shoten,Tokyo, 2008. （『老子』, 岩波文庫.）
Shi, Yunli and Chu, Longfei: “Guanyu Chongzhen-lishu bianzuan guocheng-dexin-sikao” (On the new considerations in the process of editing the Chongzhen-lishu) (inChinese), Inter‐cultural and Intra‐cultural Perspectives on Scientific Exchanges inSeventeenth ‐ and Eighteenth ‐ Century East Asia, A Templeton International
Workshop , Seoul National University, 24‐25 May 2013. (石云里 褚龙飞“关于《崇祯历书》编纂过程的新思考 ——以日躔历和月离历的分析结果为例”.)

Yingyin Wenyuange Siku-quanshu (Photographic reproduction of the Complete
library in four treasuries in the Wenyuange library) (in Classical Chinese),
multivolumed reproduction, Taiwan Shangwu Yinshuguan, Taibei, 1983－86. （《景印・文淵閣四庫全書》, 臺灣商務印書館, 臺北.） The main text of the Lixiang-gaocheng is
included in its vol.790.