Editing Deferent and epicycle (section)

==Mathematical formalism==
According to the [[history of science|historian of science]] [[Norwood Russell Hanson]]:
{{Blockquote|There is no bilaterally-symmetrical, nor eccentrically-periodic curve used in any branch of astrophysics or observational astronomy which could not be smoothly plotted as the resultant motion of a point turning within a constellation of epicycles, finite in number, revolving around a fixed deferent.|[[Norwood Russell Hanson]]|"The Mathematical Power of Epicyclical Astronomy", 1960<ref name="Hanson">{{cite journal
| issn = 0021-1753
| volume = 51
| issue = 2
| pages = 150–158
| last = Hanson
| first = Norwood Russell
| title = The Mathematical Power of Epicyclical Astronomy
| journal = Isis
| date = 1960-06-01
| jstor = 226846
| access-date = 2011-10-21
| url = http://www.u.arizona.edu/%7Eaversa/scholastic/Mathematical%20Power%20of%20Epicyclical%20Astronomy%20%28Hanson%29.pdf
| doi = 10.1086/348869
| s2cid = 33083254
| archive-date = 1 November 2020
| archive-url = https://web.archive.org/web/20201101052130/http://www.u.arizona.edu/~aversa/scholastic/Mathematical%20Power%20of%20Epicyclical%20Astronomy%20%28Hanson%29.pdf
| url-status = dead
}}</ref>}}

Any path—periodic or not, closed or open—can be represented with an infinite number of epicycles. This is because epicycles can be represented as a [[complex number|complex]] [[Fourier series]]; therefore, with a large number of epicycles, very complex paths can be represented in the [[complex plane]].<ref>See, e.g., [https://www.youtube.com/watch?v=QVuU2YCwHjw this animation] made by Christián Carman and Ramiro Serra, which uses 1000 epicycles to retrace the cartoon character [[Homer Simpson]]; see also Christián Carman's "[https://santi75.files.wordpress.com/2008/07/deferentes-epiciclos-y-adaptaciones.pdf Deferentes, epiciclos y adaptaciones]." and [http://www.cfh.ufsc.br/~principi/p142-3.pdf "La refutabilidad del Sistema de Epiciclos y Deferentes de Ptolomeo"].</ref>

Let the complex number
{{block indent|<math>z_0=a_0 e^{i k_0 t}\,,</math>}}
where {{math|''a''<sub>0</sub>}} and {{math|''k''<sub>0</sub>}} are constants, <math>i=\sqrt{-1}</math> is the [[imaginary unit]], and {{math|''t''}} is time, correspond to a deferent centered on the origin of the [[complex plane]] and revolving with a radius {{math|''a''<sub>0</sub>}} and [[angular velocity]]
{{block indent|<math>k_0=\frac{2\pi}{T}\,,</math>}}
where {{math|''T''}} is the [[period (physics)|period]].

If {{math|''z''<sub>1</sub>}} is the path of an epicycle, then the deferent plus epicycle is represented as the sum
{{block indent|<math>z_2=z_0+z_1=a_0 e^{i k_0 t}+a_1 e^{i k_1 t}\,.</math>}}
This is an [[almost periodic function]], and is a [[periodic function]] just when the ratio of the constants {{math|''k<sub>j</sub>''}} is [[rational number|rational]]. Generalizing to {{math|''N''}} epicycles yields the almost periodic function
{{block indent|<math>z_N=\sum_{j=0}^N a_j e^{i k_j t}\,,</math>}}
which is periodic just when every pair of {{math|''k<sub>j</sub>''}} is rationally related. Finding the coefficients {{math|''a<sub>j</sub>''}} to represent a time-dependent path in the [[complex plane]], {{math|''z'' {{=}} ''f''(''t'')}}, is the goal of reproducing an orbit with deferent and epicycles, and this is a way of "[[saving the phenomena]]" (σώζειν τα φαινόμενα).<ref>{{cite book
| publisher = University of Chicago Press
| last = Duhem
| first = Pierre
| title = To save the phenomena, an essay on the idea of physical theory from Plato to Galileo
| location = Chicago
| year = 1969
|oclc=681213472
|author-link=Pierre Duhem
}} ([https://books.google.com/books?id=UofBybolmREC&pg=PA131 excerpt]).</ref>

This parallel was noted by [[Giovanni Schiaparelli]].<ref>Giovanni Gallavotti: [[arxiv:chao-dyn/9907004|"Quasi periodic motions from Hipparchus to Kolmogorov"]]. In: ''Rendiconti Lincei – Matematica e Applicazioni.'' Series 9, Band 12, No. 2, 2001, p. 125–152. ([https://web.archive.org/web/20051218204530/http://www.lincei.it/pubblicazioni/rendicontiFMN/rol/pdf/M2001-02-06.pdf PDF; 205 KB])</ref><ref>Lucio Russo: ''The forgotten revolution. How science was born in 300 BC and why it had to be reborn.'' Springer, Berlin. 2004, {{ISBN|3-540-20068-1}}, p. 91.</ref> Pertinent to the [[Copernican Revolution]]'s debate about "[[saving the phenomena]]" versus offering explanations, one can understand why [[St. Thomas Aquinas|Thomas Aquinas]], in the 13th century, wrote:

{{Blockquote|Reason may be employed in two ways to establish a point: firstly, for the purpose of furnishing sufficient proof of some principle [...]. Reason is employed in another way, not as furnishing a sufficient proof of a principle, but as confirming an already established principle, by showing the congruity of its results, as in astronomy the theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained; not, however, as if this proof were sufficient, forasmuch as some other theory might explain them.|[[Thomas Aquinas]]|''[[Summa Theologica]]''<ref>''[[Summa Theologica]]'', [http://www.newadvent.org/summa/1032.htm#article1 I q. 32 a. 1] ad 2.</ref>}}