Ptolemy reduces to Kepler as Epicycles → ∞?

[Geremia]

Does Ptolemy's theory of epicycles mathematically reduce to Kepler's 3 Laws in the limit of the number of epicycles tending to infinity? Thanks

 

[russ_watters]

Sure, but an infinitely long, ad hoc equation would be kinda difficult to create and solve, no?

 

[Ken G]

I agree, it would be both possible and useless. Presumably any periodic motion can be so reduced using epicycles that are commensurate with that period, but would be no more useful than simply tracing out the path the planet took, and saying, "now do that again orbit after orbit." What Kepler did was to simplify the mathematical content of the orbital path-- as usual, by approximating it. He probably didn't realize he was approximating it though-- that is a more modern recognition about how physics works.

 

[Geremia]

Does Ptolemy's theory of epicycles mathematically reduce to Kepler's 3 Laws in the limit of the number of epicycles tending to infinity? Thanks

There is a very good article on JSTOR:

The Mathematical Power of Epicyclical Astronomy
Norwood Russell Hanson
Isis
Vol. 51, No. 2 (Jun., 1960), pp. 150-158
(article consists of 9 pages)
Published by: The University of Chicago Press on behalf of The History of Science Society
Stable URL: http://www.jstor.org/stable/226846

 

[Chronos]

Ptolemy's epicycles was a very good description, but, wrong. You can't make wrong better with math.

 

[DrStupid]

Does Ptolemy's theory of epicycles mathematically reduce to Kepler's 3 Laws in the limit of the number of epicycles tending to infinity?

Yes, but today we would call it Fourier series.

 

[D H]

 

[Geremia]

Yes, but today we would call it Fourier series.

Actually, "[URL Power of Epicyclical Astronomy (Hanson).pdf"]Hanson's Isis article[/URL] shows that you don't need an infinite number of epicycles to reproduce an elliptical orbit.

He also shows how any continuous, periodic orbit can be represented by a finite number of epicycles down to an arbitrarily small accuracy. And any path—periodic or not, open or closed—can be represented by an infinite complex Fourier series.

 

[Geremia]

Ptolemy's epicycles was a very good description, but, wrong. You can't make wrong better with math.

Kepler's laws were an approximation to Ptolemy's epicycles. How are Ptolemy's epicycles "wrong"?

 

[D H]

Does Ptolemy's theory of epicycles mathematically reduce to Kepler's 3 Laws in the limit of the number of epicycles tending to infinity? Thanks

No, it doesn't.

In focusing solely upon epicycles, you are throwing out all of the incorrect baggage of Ptolemy's astronomy. The concept of celestial spheres is falsifiable and is incorrect. Those celestial spheres were central to Ptolemy's astronomy. Throw them out but maintain the concept of epicycles and you have something very different than Ptolemy's astronomy. What you have left can describe any cyclical path, and with generalizations, any closed path whatsoever. This is not a falsifiable description; it is not scientific.

This Ptolemaic theory stripped of the concept of celestial spheres theory can describe a path that traces out a likeness of Homer. There is nothing in this theory that says planets must follow ellipses with the sun at one of the foci(Kepler's first law), that a line joining a planet and the sun sweeps out equal areas during equal intervals of time (Kepler's second law), or that the square of the orbital period is proportional to the cube of the semi major axis (Kepler's third law).

Kepler's laws are much more restrictive than are this Ptolemaic theory stripped of the concept of celestial spheres. Mathematically this is a bad thing. This stripped-down Ptolemaic theory is much more expressive than are Kepler's laws. Scientifically, this is a good thing. It means that Kepler's laws are falsifiable (and they are indeed false; they are instead approximately true).

 

[DrStupid]

What you have left can describe any cyclical path, and with generalizations, any closed path whatsoever. This is not a falsifiable description; it is not scientific.

Empirical approximations (e.g. Shomate equations for thermodynamic properties or virial expansions for real gases) belong to the standard methods of science.

 

[BobG]

The Ptolemaic model, the Copernican model (circular orbits around the Sun with epicycles and deferents), and the Keplerian model are all descriptive models that have a high probability of predicting the future actions of planets based on the past. None of them provide any explanation of why the models work - the models just do. And accurate models are valuable to science even if the reason the models work isn't known (yet).

Newton's laws of motion provide basic physical laws of motion and explain why the Keplerian model has to be the most accurate model of the three. Without Newton's laws, we'd still be debating whether the Copernican or the Keplerian model were the 'true' model, and maybe Brahe's model as well (his model never lasted long enough to start adding epicycles and deferents). Simply saying Kepler's model must be correct because it was 'simpler' and therefore 'more aesthetic' isn't enough - there has to be some physical reason to explain why Kepler's laws are true or else they're no better than the Copernican model. Newton's Laws provide the reasons for the Keplerian model (in fact, if Kepler had known Newton's Laws, it surely wouldn't have taken him a decade to progress from his Second Law to his Third Law).

Galileo's obsevations with a telescope disproved the Ptolemaic model, which is a falsifiable model, but did not prove the Copernican model, per se - just that it was a better model than the Ptolemaic model. In some ways, Brahe's model was just a response to point out that Galileo hadn't actually proved anything - Galileo just disproved something (which actually is a very valuable contribution to science).

If you wanted, you could go a step beyond that and ask why momentum has to be conserved. In fact, there was quite a debate about whether conservation of momentum (Newton, DesCartes, et al) was the explanation for all motion or conservation of energy (Leibniz, et al) was the explanation for all motion, only to finally realize both were true.

 

[D H]

Empirical approximations (e.g. Shomate equations for thermodynamic properties or virial expansions for real gases) belong to the standard methods of science.

Empirical laws can come in very handy at times. Kepler's laws are an empirical model, for example. Another example is the JPL Development Ephemerides, a set of piecewise continuous Chebyshev polynomial coefficients that one can use to compute the positions and velocities of the planets at some point in time. While highly empirical, those polynomial coefficients are generated by a highly theoretical model.

That highly theoretical model that JPL uses to generate their ephemeris model has its roots in Kepler's laws, not Ptolemy's astronomy. That gravitation is an inverse square law falls right out of Kepler's laws. The Ptolemaic model offers no such insight.

 

[DrStupid]

The Ptolemaic model offers no such insight.

This also applies to all empirical models, but that doesn't necessarily mean that they are not scientific. Of course celestial spheres are not scientific (because they are already falsified) but the original question is limited to the mathematical aspect and in science it is common practice to use mathematical models that are originally designed for completely different systems (e.g. Alfred Osborne used the Schrödinger equation for freak waves). Therefore it should also be acceptable to use epicycles for the approximation of orbits.

 

[D H]

This also applies to all empirical models, but that doesn't necessarily mean that they are not scientific.

It certainly does not apply to all empirical models. Some do provide very deep insight. I mentioned one such empirical model already: Kepler's laws. That gravitation is an inverse square law can be and was derived from Kepler's laws. Another is Planck's law, the original form of which was u(f,T) = af^3/(\exp(bf/T)-1). The a and b were empirical values. Later modifications to Planck's law related those empirical constants to Boltzmann's constant and to quantum mechanics.

Therefore it should also be acceptable to use epicycles for the approximation of orbits.

There is no computational advantage in doing so. Such a model is rather disadvantageous compared to other empirical and semi-empirical models of orbits.

 

[DrStupid]

There is no computational advantage in doing so.

There are scientists that do not agree in this opinion (e.g. http://articles.adsabs.harvard.edu//full/1979ApJ...234..275M/0000275.000.html)

 

[twofish-quant]

Does Ptolemy's theory of epicycles mathematically reduce to Kepler's 3 Laws in the limit of the number of epicycles tending to infinity? Thanks

No it doesn't.

You get parallax with Kepler. You don't with Ptolemy. (And in fact the lack of parallax is why Tycho rejected sun-centered models.)

 

[D H]

Kepler's laws were an approximation to Ptolemy's epicycles. How are Ptolemy's epicycles "wrong"?

Kepler's laws were anything but an approximation to Ptolemy's epicycles. Ptolemy's epicycles were laden with many problems. The discrepancies between model and reality became more apparent as observations improved over time. There is a huge difference between tracing out the path of a planet over time, as Ptolemy (and later Hanson) did, and tracing out the path of a planet in time, as Kepler did.

Yes, but today we would call it Fourier series.

Ptolemy's epicycles were not Fourier series. The frequencies in a Fourier series are highly constrained. Fourier series "work" because the underlying functions are orthogonal. This orthogonality does not arise if one picks random frequencies, which is how the Ptolemaic epicycles were built up over time. Ptolemy started with some gross behaviors that were known to the ancients. Others added on to these, but there was no Fourier formalism in their ad hoc additions.

There are scientists that do not agree in this opinion (e.g. http://articles.adsabs.harvard.edu//full/1979ApJ...234..275M/0000275.000.html)

First off, these scientists never once mentioned the term epicycles. They did not use Ptolemy's epicycles. They used Fourier analysis.

Secondly, they used Fourier analysis not because of their computational advantages but because they did not want any Keplerian assumptions to slip into their analyses. They wanted to test whether the observations were those of a Keplerian orbit, how much the observations deviated from such an orbit, and do so without making any a priori assumptions regarding Keplerian orbits.

Thirdly, those scientists are looking at a two dimensional problem; the third dimension (radial distance from the Earth) is more or less unobservable in the cited application. Fourier analysis works just fine on two dimensional problems. It doesn't work in three dimensions, and neither did Ptolemy's astronomy (at least not very well). The orbits of the planets in our solar system observably are a three dimensional problem.

Does Ptolemy's theory of epicycles mathematically reduce to Kepler's 3 Laws in the limit of the number of epicycles tending to infinity? Thanks

If you

• Throw out the falsified parts of Ptolemy's astronomy,
• Ignore that modeling the behavior of the solar system is a 3D problem,
• Ignore that Ptolemaic astronomy did not fair well in this regard,
• Ignore that there is big difference between describing the path over time and the path as a function of time,
• Ignore parallax,
• Throw out the techniques used by Ptolemy and those who followed to develop their epicycles, and
• Use Fourier analysis instead,

Then yes, "Ptolemy's theory of epicycles mathematically reduce to Kepler's 3 Laws in the limit of the number of epicycles tending to infinity." A whole lot of throwing out and ignoring is required to get to this point. In short, the answer to the question raised in the opening post is no, it doesn't.

Even if this were the case (which it isn't), this historical revisionism ignores that Kepler's laws were the key step that let Newton (and later, Einstein) see how gravitation is the mechanism that describes planetary motion. That insight pops right out of Kepler's laws. Ptolemy's astronomy offers no such insight.

 

[twofish-quant]

One other thing is that by the 16th century, the Ptolematic system was dead anyway. The main system in used then was the Tychoian system which was mathematically identical to the Coprernican system had the big advantage that you didn't have to explain the lack of parallax.