# Talk:Deferent and epicycle

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## Epicycles on Epicycles

I ran across someone elsewhere (slashdot) mentioning that the notion of epicycles upon epicycles being added in the middle ages is bad history, and that this is one that Britannica had some very bad articles on at one point. So someone (unlike me) who does know this well may want to double-check this entry and maybe comment on other mis-information floating around...

Yeah, I have mentioned that on slashdot a couple of times. And indeed, I think this article is still inaccurate. In fact, I think we have some good ammunitation against Britannica here, because Britannica carries a lot of the responsibility for the exaggarations :-) . I haven't worked on this subject for the last few years, and some of my books are packed down somewhere, but let me take something from memory.

The most fundamental thing here is that Owen Gingerich recomputed the Alfonsine Tables, which was computed in the 13th century, and found that they were based on a purely Aristotelian model. Still, Alfonse lamented the complexity of the calculations, and that has often been interpreted as complexity had been added. I think we can safely say that if epicycles were added, they had no practical influence whatsoever.

The Alfonsine Tables is not the only piece of evidence. We have several textbooks from the 13th-16th centuries confirming that the astronomical models were almost the same as Ptolemy's. Not to get into the details of what was changed, but nobody was adding epicycles. Anyone care to guess what's the first western book to have epicycles-on-epicycles? I'll give you a hint: it was published in 1543. The story about Alfonso saying that he could have given God some advice, that is apocryphal. I think Gingerich wrote a short piece on it, which would be a good thing to cite here.

So, to the question of whether it is correct to say that epicycles were added upon epicycles. There is a short discussion in a book by J. L. E. Dreyer where there is also a figure of a world system with circles within circles. I guess you could point to that and say "see, they did", but there is very little record of this having any substantial influence on astronomy, and so, I still think that this article needs a good rewriting. I'd like to do it one day, I just need the time... :-) Kjetil Kjernsmo 00:10, 5 November 2005 (UTC)

I assume you mean his History of Astronomy/World Systems? I would like to know what this could be. Sounds like a cosmology section showing the order of the planets, not epicycles. Maestlin 21:59, 11 March 2006 (UTC)

You have added far too many (disputed - see talk page) comments - surely one would be enough?? I think that this should be changed. I will modify it a little.

Regarding the last part:

"The switch to the Sun-centered model removed epicycles for a while, but the original versions insisted on circular orbits for the planets. (disputed — see talk page) Better observational data from improved telescopes once again showed data counter to the model, and epicycles were brought back to plug the holes. (disputed — see talk page) It wasn't until Kepler developed the elliptical orbital model that epicycles were finally eliminated."

I have some disagreements here too. Who is disputing that Copernicus gave the planets circular orbits? Also, I have Never heard of epicycles being added to his system - where is the evidence for this? There is some discussion on the Copernicus site but is there a translation of his book availiable to firm up what he did or didnt say? Copernicus' model was disputed by the Church, but one of their objections was that his system gave predictions for planetary positions that was as bad as, or worse, than Ptolemy's system.

Kepler realised that the planets had elliptical orbits and this straight away removed the errors with the Copernican system. I'll modify some of the text and come back at another date to discuss some more. Adrian Baker 15:10, 10 December 2005 (UTC)

Kjetil is right about the myth of epicycles-upon-epicycles (Gingerich discuss it in The Book Nobody Read, pp 58-60). Also, Copernicus did in fact add epicycles to his system (Gingerich called them "epicyclets" as they were considerably smaller than the previous ones; p 265)--this is well-known among history of astronomy scholars. And to say Kepler "straight away" removed replaced circles with elliptical orbits is absurd; he published his first book in support of them, and only with Tycho Brahe's data did he realize circles couldn't be matched up to observations.--ragesoss 18:05, 14 January 2006 (UTC)

Thanks for the source by Gingeriich. I'll try and get a copy and read it. This would help clarify some of the points above. Regarding your comment about "Kepler 'straight away'.... ...is absurd", you have misread what I have written. I did NOT say that Kepler straight away hit the right solution, I said that "Kepler realised that the planets had elliptical orbits and this straight away removed the errors..." - two very different things. Adrian Baker 23:16, 15 January 2006 (UTC)

My mistake. Sorry.--ragesoss 01:08, 16 January 2006 (UTC)

With regard to "epicycles upon epicycles": it has been pointed out in Ptolemaic System (referred from Equant) that "The eccentric in the figure below (the figure of eccentric) is fixed; it could also be made movable. In this case the center of the large circle was a point that rotated around the Earth in a small circle centered on the Earth. In some constructions this little circle was not centered in the Earth."

It seems to me that the cited statement is mathematically equivalent to the following: we add one epicycle on another one. However, it's not usually called that way. Besides, due to the difference in size of these circles, the whole system looks quite different. --Fir-tree 17:47, 22 February 2006 (UTC)

Yes, you could convert the motion described into an epicycle, but AFAIK hardly anybody pointed this out or tried to do it in a serious way until Copernicus. And that small circle that's not centered on the Earth? ...Copernicus. Most modern readers do not realize that the full-blown system of De revolutionibus is beastly complicated, and that the headaches they think of as symptoms of Ptolemaic failure only become major players on the astronomical scene after the 1540s.
I think an earlier version of this is the wingnut article that discouraged me from joining Wikipedia some time back. It looks like the disputed tag has been up for months and nobody seems to be committed to the old text. If it gets fixed, does the fixer take the disputed tags out? Maestlin 21:59, 11 March 2006 (UTC)
Not necessarily epicycles in epicycles, but there had to be cycles inside epicycles, because they were aware of the moons of Jupitor which was what finally with Tycho Brahe's observation was observed to be elliptic. Before Tycho Brahe the moons was also known but as moving in cycles around epicycles. Maybe this where the idea of multiple layers of cycles comes from? 80.187.103.114 (talk) 13:31, 12 May 2012 (UTC)

## Ptolemy and "orbits"

Is it correct to speak of planetary "orbits" in Ptolemaic astronomy? As I understand it (someone correct me if I'm wrong), before astronomers began to seriously attack that idea of the heavens being eternal (e.g., Tycho Brahe's calculations that comets move through different planetary spheres and are not sub-lunar, widespread observation of novae, etc.), the basic cosmology involved planets embedded within rotating spheres and not actually orbiting anything.--ragesoss 18:52, 19 May 2006 (UTC)

I agree, but because there are so many presentist histories and translations that say "orbit," you might find some opposition unless you dig up a source that does terminology. Maestlin 20:29, 19 May 2006 (UTC)
One of the meanings of orbit is sphere. [1] -- Another Stickler (talk) 22:44, 4 December 2008 (UTC)

## Modern Epicycle approximation

I realise that it's important not to confuse the two issues, but it should be pointed out that the epicycle approximation is a useful approximation made by people studying stellar dynamics to this day. I don't know whether it should get a mention here, or in separate page which gets linked from here?

## 1898?

I find this sentence: "The popular total of about 80 circles for the Ptolemaic system seems to have appeared in 1898" rather strange. Could it be a mistake or a typo? Or maybe it suggests that this large number was added in posterior sources (modern encyclopedias)? Hugo Dufort 02:04, 16 November 2006 (UTC)

Yes, it's suggesting that there wasn't ever a system actually in use with 80.--ragesoss 03:10, 16 November 2006 (UTC)

## Fix errors

The last paragraph starts "The difficulty with this account" and corrects some errors in the previous several paragraphs. As it is, the "Epicycles on epicycles" section is confusing and contradictory. I hope someone fixes this. Roger 06:00, 28 December 2006 (UTC)

## Is it a matter of viewpoint

(1) Looking at the java applet (external link to University of Nantes) I'm getting confused by geometry. Epicycles are usually presented as a bad attempt to explain the solar system with the earth at the middle instead of the sun. But surely if this were the only problem, the simplest epicycle model would have been to give all planets exactly the same deferent, with the earth at its middle, and the sun at the point where all the planets have their epicycle. Then each planet would have an epicycle corresponding to its orbit, and everything would be fine. Heliocentricity or Earth-centricity is merely a matter of viewpoint! The Java applet makes this clear, as both diagrams trace out the same relative positions of Earth/Planet/Sun, but one diagram moves around on the screen, the other doesn't.

So surely epicycles should be presented rather as a bad solution to eliptical orbits.

(2) And this brings me to point 2. Is it worth a mention that even after heliocentricity caught on, epicycles were still in practical use. A glance at the page on the "Orrery" will show a mechanism where the moon is on a cranked arm, which presumably rotated once per orbit, in a clockwise direction, to simulate the eliptical part of the moon's orbit, and is obviously a mechanical epicycle in action. I'm no mathematician: is this truly an elipse?

Should there, in any case, be a link from some part of the solar system set of Wiki pages to the Orrery page?

195.92.194.11 (talk) 19:58, 9 February 2008 (UTC)

There is an excellent series on the equivalence of Ptolemy, Copernicus and Brahe's theories at http://science.larouchepac.com/kepler/astronomianova/. Fortunately, it doesn't seem to drag LaRouche's philosophy or politics in at all; part 1 is an exposition on the three theories befor Kepler, and Part 2 ff an explanation of Kepler's derivation of his final planetary theories with excellent graphics. I'd recommend it.
As for the mention of orreries, circular gears can only approximate the elliptical motion of planets in the same way that epicycles do; in fact while epicycles are required of Greeks philosophically, orreries required them mechanically. No, no such mechanism can exactly reproduce an elliptical orbit, but they can easily come close enough to make the difference on a clock face (even one a foot across) moot. See the article on the Antikythera Mechanism, an orrery of sorts which used gearing to not only position the moon on the ecliptic, but to represent the uneven motion (specifically, accelerations and decelerations) of the moon and even to show that the unevenness precesses around the Earth once every 8.8 years. All that (and much more) in hand-built bronze gearwork executed 300 years before Ptolemy.

## Equant merger

I propose that Equant be merged into this article and the article be renamed as to have a more general name (such as Elements of Ptolemaean astronomy or something) as if deferents and epicycles are covered in one article, so should equants be, presumably. I am basing the assumption that all the aforementioned items are elements of Ptolemaean astronomy on pp. 30-31 of ISBN 0 00 715252 3. It Is Me Here (talk) 18:41, 7 July 2008 (UTC)

I disagree. The equant is peculiar to Ptolemy's models. Copernicus was able to eliminate it from his models. Ptolemy was not the only one to use the deferent/epicycle method for modeling planetary orbits. Virgil H. Soule (talk) 05:09, 6 November 2008 (UTC)
I disagree too. Since the merge template has been on this page for over a year and a half, I will remove it. --SteveMcCluskey (talk) 14:27, 3 May 2010 (UTC)

Characterizing use of epicycles as "Bad Science" is nonsense. Observers of the time of Ptolemy were trying to find ways to derive mathematical descriptions of the planetary motion they saw against the (apparently) invariant star field. They didn't know orbits. They didn't know geocentric. They didn't know science. They may not have been aware that the Earth rotated about its own axis. They saw the star field moving cyclically on a daily basis so naturally they assumed that the Earth was fixed at the center of the universe. This fit in with religious dogma (particularly Christian) that asserted that the universe was miraculously created and the Earth was the center of that creation.

They used epicycles to trace out planetary motions because that's what worked. The problem was made more complicated by the retrograde motion that the planets displayed from time to time. The solution was to superimpose more epicycles and bring in the equant idea to make the math work. The cost was inordinate complexity in the model. They did the best they could with what they had at the time.

Copernicus' great contribution was in discovering that moving the center of the universe to the Sun greatly simplified Ptolemy's model. The motion of the planets as seen from the Sun is much simpler than that seen from the Earth. He also introduced the notion that the Earth rotated on its axis and presented the trigonometric transformations needed to resolve observations made on Earth to Heliocentric coordinates. Again, he used epicycles because that's what worked. They worked because the orbits of the then-known planets are nearly circular. Copernicus probably didn't know orbits, however. The idea of the Earth moving about the Sun was not well received particularly by the Catholic Church but also by many astronomers of the time.

The notion of orbital motion came with Galileo's discovery of the moons of Jupiter. Kepler refined the idea with his discovery that planetary orbits were elliptical rather than circular.

Characterizing use of epicycles as bad science implies that people like Ptolemy were cranks or quacks who didn't know what they were doing. They were honest people trying to make sense of the universe they saw around them. Virgil H. Soule (talk) 19:31, 19 October 2008 (UTC)

Yes it is nonsense, but what do you suggest? Roger (talk) 20:15, 19 October 2008 (UTC)
Bad science confuses theories of motion with descriptions of motion. The Earth based observations are what is known and these support a description whether it be epicycles or osculating Keplerian orbits. Egotism and predujice also taint interpretations as does the propaganda battle between the "old" world and the "new." --Jbergquist (talk) 18:57, 4 July 2011 (UTC)

Given that the current best models of the planetary motion (VSOP) use thousands of terms, surely it is enormous modernistic hypocrisy to denigrate epicycles that did work with "only" 80 terms? Old_Wombat (talk) 07:53, 30 July 2011 (UTC)

I just read the article and viewed the Java applet it links to, and it really doesn't adequately describe the subject. From the article I get the impression that if you approximate all the orbits in this system, then you'd get a model that is heavily centered on the angle between the earth and the sun. And that in turn should have triggered the question of what made the sun so special much sooner, especially for the inner planets. Then furthermore I can't square the Java applets with the text. The Ptolemy version clearly shows deferents and epicycles, but no hint of equants. Also for the outer planets the epicycles are somewhat unexpected, although I can kind of see the behaviour it tries to model, I can't say it's really understandable. What is the Ptolemaic location versus the real location of the planets in these cases? Then there's Brahe's version, which is what you'd expect to find, but with no equant again. I can't make sense of these mismatches and the article is not sufficiently referenced or illustrated to really tell what's going on... what was the motivation for example for the equant, and the motion of the epicycles with respect to these? Why this and not something else? What did it really look like? The illustration in the article doesn't show this at all, and the applets appear inconsistent with the article. What was the modelled location and the now known real location of the planets in various situation in the various models? Also, part of the article needs to be rewritten for accessibility to laypeople. 82.139.86.37 (talk) 01:48, 21 March 2009 (UTC)

## Dubious

The ancients worked from a geocentric perspective because the Earth was the platform on which they stood. Some Greek astronomers (e.g., Aristarchus of Samos) had speculated that the planets (Earth included) orbited the Sun but the mathematics needed to transform geocentric observations to a heliocentric perspective didn’t exist in Ptolemy’s time.

The latter claim is highly dubious. Indeed, considering this property of the Ptolemaic system:

Despite the fact that the Ptolemaic system is considered geocentric, the planets' motion was not thought to be actually centered on the Earth. Instead, the deferent was centered around a point halfway between the Earth and another point called the equant. The epicycle, meanwhile, rotated and revolved along the deferent with uniform motion. The rate at which the planet moved on the epicycle was fixed such that the angle between the center of the epicycle and the planet was the same as the angle between the earth and the sun.

the claim can be seen to be nonsense, since a heliocentric system requires less complication in the math than a deferent-centric system of the kind described. It would be a mistake to assume that anything as general as a modern understanding of coordinate transformations would have been required.

Also, according to Bartel Leendert van der Waerden's "The Heliocentric System in Greek, Persian and Hindu Astronomy", there is evidence that the math needed was available to and used by Seleucus of Seleucia, well before Ptolemy's time.

There is some discussion of more plausible reasons why heliocentric theories might not have been accepted earlier in the Heliocentrism article. --David-Sarah Hopwood ⚥ (talk) 00:14, 6 January 2010 (UTC)

## Theories that fit the data yet become abandoned due to other reasons

Both Deferent and epicycle and Phlogiston theory were theories that fit the data but were abandoned due to other reasons. 96.26.97.207 (talk) 00:54, 27 February 2012 (UTC)

In the case of Ptolemy's theory, the problem was that with a fixed number of epicycles per body you can only approximate the elliptical orbits that we now know are correct (ignoring perturbations), so eventually the data outed the theory. So, no, Ptolemy's theory doesn't match the data. The whole idea in experimental science is to design an experiment (produce data) which will differentiate between competing theories. SkoreKeep (talk) 17:22, 3 August 2014 (UTC)

## Is the mathematics in the "Mathematical Formalism" correct?

I'm no expert on complex valued Fourier series, but comparing the Mathematical Formalism subsection of this article with the Exponential Fourier Series subsection of the Fourier Series article suggests the ${\displaystyle k_{j}}$ terms are under specified (they are in fact left undefined). Comparing the mathematics in the Mathematical Formalism subsection with Hanson's paper, for the Fourier expansion in the latter there are two epicycles for each value of j, one for ${\displaystyle a_{j}e^{ik_{j}t}}$ and one for ${\displaystyle a_{-j}e^{-ik_{j}t}}$. Hanson sums from -N to N, whereas the series in the subsection sums from 0 to N. Furthermore, I think the ${\displaystyle k_{j}}$ must be defined using integer multiples of the fundamental frequency, e.g., ${\displaystyle k_{j}={\frac {2j\pi }{T}}}$.

Perhaps it is possible to fix things up so the sum goes from 0 to N, but I think there are advantages to summing from -N to N. Specifically, for j = 0, there would be only one term, which can be used to center the deferent on a point other than the origin of the coordinate system. This seems to fall out naturally, since ${\displaystyle a_{0}e^{i(0)t}=a_{0}}$, which is a constant.

One other point. It probably is prudent to indicate in this section that the Mathematical Formalism provided is an extension of the Deferent and Epicycle system used in the Ptolemaic model. I believe the latter only used one epicycle per deferent. Dnessett (talk) 16:50, 17 April 2013 (UTC)

I have investigated further and now believe that the formulation given in the article is correct. The expansion ${\displaystyle z_{N}=\sum _{j=0}^{N}a_{j}e^{ik_{j}t}}$ is a type of Fourier Series known as a Besicovitch almost periodic function. I have added a reference to the Wikipedia article that describes such functions. Dnessett (talk) 15:36, 7 May 2013 (UTC)
Actually it's a Bohr almost periodic function, the sum being finite. The Besicovitch almost periodic functions are one of several generalizations listed in the article you cite to infinite sums; in Besicovitch's case the sequence of coefficients aj is required to come from an ℓ2 space, i.e. the sequence must be square-summable, which holds vacuously for finite sums. Vaughan Pratt (talk) 17:47, 18 September 2014 (UTC)

## Too much

It seems to me that there is a lot more information in this article than there needs to be. It's about "deferents and epicycles", but it gets into eccentrics, equants, Ptolemy, Copernicus and Brahe, stuff related to but way beyond the named subjects, all of which should be moved to articles on geocentrism and the planetary theories of the afore-mentioned worthies. Compare, for example, to the article on the equant. I have myself lately been guilty of gilding the lily on this article, before I pulled back somewhat for a larger picture.

I propose to rewrite it to that standard, and while I'm waiting for comment, I'll wander on over to geocentrism and see what needs to happen there.SkoreKeep (talk) 20:01, 5 August 2014 (UTC)

I agree it does wander too much into the history of geocentrism (e.g. The 6 paragraphs starting 'Owen Gingerich') but I don't agree with excluding eccentrics and equants. The title may not be terribly appropriate but it's the only article covering this whole Ptolemaic system and I think it's useful for that. Where else would one look? Chris55 (talk) 21:57, 5 August 2014 (UTC)

All the topics you list as ‘too much’ are for me necessary context to make sense of the rest of the article. In fact, I think that Brahe, who currently is but a footnote in this article, should get a bit more coverage here. Wikipedia aims to be an encyclopaedia, not a dictionary. It isn't sufficient to provide a one-paragraph definition. Rather, a topic should be fully covered, including common details, history, competing theories or related topics, people involved and so on and so forth. — Preceding unsigned comment added by 80.114.146.117 (talk) 21:42, 9 May 2016 (UTC)

## Observations

Babylonian observations are mentioned in the article. I think earlier Sumerian observations were made, also.

At the moment, our knowledge of Sumerian astronomy is merely indirect.

## Any path—periodic or not, closed or open—can be represented with an infinite number of epicycles.

I'm not much of a mathematician, but I don't believe this statement is true. Certainly many complex paths can. But a straight line cannot, and does not have a Fourier transform valid for all time. Neither does a non-periodic trivial predator-prey model. If there is no analytical expression possible, then while an infinite set of points could be said to represent such curves, one cannot represent such points exactly without the original "open-form" expressions which are not based on epicycles nor a fourier basis. Dpleibovitz (talk) 16:46, 9 February 2016 (UTC)

The key here is an "infinite" number of epicycles. They resolve down, I believe, to sinusoids and an infinite number of them will reproduce any given function (even a constant). That it does not do so near the end points is not a problem for them, because the end points physically are embedded in the chaotic beginning of the solar system and the far future. For a time interval of a couple of thousand years on either side of 1 AD is not a problem. To say you need to have an analytical solution is to negate the usefulness of Fourier, LaPlace and Taylor series in general. And there, you've also heard the sum of my math knowledge on the matter. SkoreKeep (talk) 20:59, 11 February 2016 (UTC)

## About the quote in the section Mathematical formalism

I question whether Hanson is in this section a reliable source in general, and for the quote in particular.

Having read the reference I cannot help but feel that Hanson is out of his depth on this matter. The relevant section is outside of his regular area and he clearly lacks the required historical knowledge to accurately present what he discusses.

Further, the quote in this article is pulled out of context. As it is quoted, it is subtly wrong. It would be correct if it said ‘infinite’ instead of ‘finite’, but that would go against what the reference is trying to say, namely that a wide variety of curves can be approximated by stacking a few epicycles on top of each other. But due to the quote being pulled out of context it appears to say that any curve in use can be exactly reproduced and that isn't true.

Considering that the quote is wrong, needs a lot of context to become right, isn't necessary nor particularly relevant to this article, doesn't really add anything to this article and comes from a relatively shaky source, I propose to remove it altogether. — Preceding unsigned comment added by 80.114.146.117 (talk) 22:31, 9 May 2016 (UTC)