# 均輪和本輪

（地球半徑）

（現代/托勒密）

（現代/托勒密

## 歷史

[4]描述發生在1504年，很顯然是由哥白尼觀測的一次行星合。哥白尼在阿方索星表副本中的筆記評論說："火星的位置超出2度以上，土星則落後了1.5度。"使用現代的電腦程式計算，金格瑞契發現在合的時刻，土星確實落後表中數值1.5度，火星則超前約2度。次外，他發現托勒密在當時對木星的預測相當準確。因此，哥白尼和與他同一時代的人都使用托勒密的方法來尋找行星，並在托勒密的原創作品出版一千多年後，發現它們還是可以信賴。

## 周轉圓

——逐漸接受哥白尼的宇宙論"，1917年[18]

——大英百科全書，1968年 [20]

## 數學形式主義

——諾伍德·羅素·漢森，"數學在周轉圓天文學的功能"，1960[22]

${\displaystyle z_{0}=a_{0}e^{ik_{0}t}\,,}$

${\displaystyle k_{0}={\frac {2\pi }{T}}\,,}$

${\displaystyle z_{2}=z_{0}+z_{1}=a_{0}e^{ik_{0}t}+a_{1}e^{ik_{1}t}\,.}$

${\displaystyle z_{N}=\sum _{j=0}^{N}a_{j}e^{ik_{j}t}\,,}$

## 註解

1. ^
2. ^ Andrea, Murschel. The Structure and Function of Ptolemy's Physical Hypotheses of Planetary Motion. Journal for the History of Astronomy. 1995, (xxvii): 33–61 [2 August 2014]. Bibcode:1995JHA....26...33M.
3. ^ For an example of the complexity of the problem, see Owen Gingerich, The Book Nobody Read, Walker, 2004, p. 50
4. ^ Gingerich, Chapter 4
5. ^ One volume of De Revolutionibus was devoted to a description of the trigonometry used to make the transformation between geocentric and heliocentric coordinates.
6. ^ Gingerich, p. 267
7. ^ Gingerich, p. 54
8. ^ Palter, Robert. Approach to the History of Astronomy. Studies in the History and Philosophy of Science. 1970, 1: 94.
9. ^ Owen Gingerich, "Alfonso X as a Patron of Astronomy", in The Eye of Heaven: Ptolemy, Copernicus, Kepler (New York: American Institute of Physics, 1993), p. 125.
10. ^ Gingerich, "Crisis versus Aesthetic in the Copernican Revolution", in Eye of Heaven, pp. 193–204.
11. ^ "The popular belief that Copernicus's heliocentric system constitutes a significant simplification of the Ptolemaic system is obviously wrong ... [T]he Copernican models themselves require about twice as many circles as the Ptolemaic models and are far less elegant and adaptable." Neugebauer, Otto. The Exact Sciences in Antiquity 2. Dover Publications. 1969 [1957] [2020-09-18]. ISBN 978-0-486-22332-2. （原始内容存档于2020-08-14）., p. 204. This is an extreme estimate in favor of Ptolemy.
12. ^ Koestler, Arthur. The Sleepwalkers. Arkana, Penguin Books. 1989 [1959]., p. 195
13. ^ Palter, Approach to the History of Astronomy, pp. 113–114.
14. ^ Koestler, Arthur. The Sleepwalkers. Arkana, Penguin Books. 1989 [1959]., pp. 194–195
15. ^ A deferent/epicycle model is in fact used to compute Lunar positions needed to define modern Hindu calendars. See Nachum Dershovitz and Edward M. Reingold: Calendrical Calculations, Cambridge University Press, 1997, Chapter 14. (ISBN 0-521-56474-3)
16. ^ Goldstein, Bernard R. Theory and Observation in Medieval Astronomy. Isis. 1972, 63 (1): 39–47 [40–41]. doi:10.1086/350839.
17. ^ Kollerstrom, Nicholas. Newton's Forgotten Lunar Theory. Green Lion Press. 2000. ISBN 1-888009-08-X.
18. ^ Dorothy Stimson, The Gradual Acceptance of the Copernican Theory of the Universe (New York, 1917), p. 14. The quotation is from John Milton's Paradise Lost, Book 8, 11.82–85.
19. ^ Robert Palter, An Approach to the History of Early Astronomy
20. ^ Encyclopædia Britannica, 1968, vol. 2, p. 645. This is identified as the highest number in Owen Gingerich, Alfonso X. Gingerich also expressed doubt about the quotation attributed to Alfonso. In The Book Nobody Read (p. 56), however, Gingerich relates that he challenged Encyclopædia Britannica about the number of epicycles. Their response was that the original author of the entry had died and its source couldn't be verified.
21. ^ Gingerich, The Book Nobody Read, p. 57
22. ^ Hanson, Norwood Russell. The Mathematical Power of Epicyclical Astronomy (PDF). Isis. 1960-06-01, 51 (2): 150–158 [2011-10-21]. ISSN 0021-1753. JSTOR 226846. doi:10.1086/348869. （原始内容存档 (PDF)于2020-11-01）.
23. ^ See, e.g., this animation页面存档备份，存于互联网档案馆） made by Christián Carman and Ramiro Serra, which uses 1000 epicycles to retrace the cartoon character Homer Simpson; cf. also Christián Carman's "Deferentes, epiciclos y adaptaciones页面存档备份，存于互联网档案馆）." and "La refutabilidad del Sistema de Epiciclos y Deferentes de Ptolomeo"页面存档备份，存于互联网档案馆）.
24. ^ Cf. Duhem, Pierre. To save the phenomena, an essay on the idea of physical theory from Plato to Galileo. Chicago: University of Chicago Press. 1969. OCLC 681213472. (excerpt页面存档备份，存于互联网档案馆）).