View Full Version : Comparison of Ptolemy, Copernicus and Kepler orbital models

2013-Apr-11, 01:27 AM
See this animation in which Ptolemy used an eccentric circle with an equant while Copernicus used epicycles in their respective quests to account for motion we now know goes around a Keplerian ellipse.


For decades I have wanted to see something like this, as I find the Almagest and the Revolutions to be difficult to follow. This animation is a wish come true, and I heartily thank the author for creating and posting it.

The eccentricity and the location of the equant can be adjusted independently in this exercise. I did a sample calculation which showed that if the equant and the central body (Earth or Sun as the case may be) are equidistant from the center of the Ptolemaic circle, the radius vector from the central body to the satellite sweeps out area at the same rate at perigee and apogee. That should be the best first approximation of the Keplerian ellipse, and it is my educated guess that Ptolemy placed the equant at or near such a location.

At large eccentricity, upwards of 0.4, I could see clearly that the Copernican orbit was elongated horizontally from the Ptolemaic circle, whereas the Keplerian ellipse would be compressed horizontally. So far the Ptolemaic construction appears to be more accurate, although I cannot rule out the possibility that Copernicus may have tweaked his choices of epicycle radii to get the average accuracy of his results comparable to Ptolemy’s results.

In my opinion it is not belittling Copernicus to point out the apparent flaw in his method. He was thinking like a physicist and restoring physics to its place in astronomy, whereas Ptolemy and his immediate Greek predecessors had largely abandoned it and were content with purely mathematical lines of thought. The equant method is mathematically elegant and gives good agreement with observations at low and moderate eccentricity, but it is clunky mechanically. Copernicus, in taking a step backward into returning to combinations of uniform circular movements, constructed a model that could be envisioned in terms of a small potter’s wheel with its bearing riding on the edge of a larger wheel. With good enough bearings such an apparatus would run in the manner of the animation. This model provided Kepler, who believed in it, a basis for his own work.

Kepler started as a classical idealist, but he eventually abandoned the ideal circle when all attempts at reconciling it with Tycho’s precise measurements of Mars failed. He could have kludged it into agreement with more epicycles but decided to try simple alternative figures that could be calculated analytically. Lo and behold, a suitably constructed ellipse, differing only slightly from a circle at low eccentricity, worked well. The stage was set for Newton to explain the motion mechanically.

Let me add that Kepler was not entirely happy with his elliptical model, being unable to make full mechanical sense of it. I have seen him quoted as describing it as “one more cart-load of dung as the price for ridding the system of a vaster amount of dung.”1 Nevertheless he did not reject it on these grounds, and is to be heartily commended for that.

1Toulmin and Goodfield, The Fabric of the Heavens, p. 247, ca. 1960

2013-Apr-11, 06:02 AM
I almost cannot believe the quote. Far too human for a legend. But Newton finally made perfect sense of it, right? Today, isn't it a little harder to discover something so profound (I mean considering prominence of epicycles at the time)?

Perhaps now, we are doing our own epicycles. Believing we know, we invent things to keep it right.

2013-Apr-11, 09:35 AM
Perhaps now, we are doing our own epicycles. Believing we know, we invent things to keep it right.
I believe that's how it works.

2013-Apr-11, 09:58 AM
I believe that's how it works.

Yup. Nice essay. We should always be cautious when adding parameters to make a mathematical model consistent with reality. Without a solid, testable, bases for the new parameter, it is just an epicycle, even if you give it a new fancy kname.

2013-Apr-16, 02:14 PM
A quote from the other Copernicus thread:

It is an interesting scientific question as to how such metaphysical thinking may have persisted at the hinge between ancient and modern, and how this supposedly 'pure' ancient rational geometry gave rise to the purely empirical modern knowledge of planetary movement. The omitted discussion of Pythagoras in the precession chapter of On The Revolutions raises a comparable question to his use of circular motion: is this material grounded more in parsimony or perfection? Further, we can ask why did Copernicus choose to first write this rather mythological text and then leave it out, only for it to be reinserted three hundred years later? It seems inevitable that there were broad oral traditions of which Copernicus was aware but which he did not commit to paper.

My bold. That is a very good question. My conclusion is that Copernicus was a purist here, while Ptolemy was more parsimonious in resorting to an eccentric deferent with an equant to fit the observations to what we now know is a Keplerian ellipse. Please note that Copernicus used a centered deferent and two epicycles to fit the observed eccentricity and variable speed around the orbit. He could have achieved the same vector resultant by placing the deferent off center and using just the smaller epicycle, but it appears to me that he was after a sense of order and consistency rather than superficial simplicity. He used this same construction for all of the planets, while Ptolemy used one sort of construction for Mercury and Venus, which stay close to the Sun in angular position, and a different one for the other planets, which can have all possible elongations from the Sun. His treatment of the Sun as the common center and Earth as just another planet made this possible.

Let’s look back at Aristotle’s model, in which everything went around the presumed stationary Earth in nested combinations of concentric circles. His opinion on the Earth’s stationary state was not unanimous, but he had inferred it on the basis of what we now know were flawed exercises in physics which were the best he could do without some advances in mathematics that still were far in the future. Enough appropriate progress had been done in the meantime to assure Copernicus that neither observation nor logic could argue against a rotating and orbiting Earth. His Sun-centered model achieves a conceptual unity that Aristotle might well have endorsed if he could have been time-warped to the 16th century and had the benefit of early Renaissance mathematical progress.