Hornblower

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.

http://science.larouchepac.com/kepler/newastronomy/part1/copernicus.html

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

http://science.larouchepac.com/kepler/newastronomy/part1/copernicus.html

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