Chapter 4: Copernicus and Ptolemy
“...Further, while he strives to outdo Ptolemy in the uniformity of motions, he is in turn outdone by him in the perfection of the planetary path. For, in Ptolemy the planet bodily traces out a perfect circle in the aethereal air. Copernicus, on the other hand, says in Book V ch. 4 that for him the path of a planet is not circular, but goes outside the circular path at the sides. This is easily demostrated in the present diagram.” (p. 136)
Copernicus was of the view that the equant had to be rejected in astronomy, because it introduced non-uniform circular motion. Click here to read more about Copernicus's thoughts on the matter. To avoid the equant, Copernicus had to use two epicycles, each moving uniformly, but with the smaller one moving at a double rate. This makes a very close match, although not a completely exact one. The smaller epicycle has a radius equal to half the distance between the center and the equant, while the larger has a radius equal to the distance between the observer and the center, plus half the distance from the center to the equant. Thus, when the double epicycle is created to match a bisected equant, the epicycle sizes are in a ratio of 1:3.
Here you can see the very close equivalence of Ptolemy's equant and Copernicus's double epicycle. There are two planets in this animation, one blue and one red. The blue one moves at what is perceived by the blue equant point to be a constant angular speed. The red planet moves according to Copernicus's double epicycle.
Experiment with this animation to find out where the two ways of placing the planet differ most, as observed by the sun.
In two experiments, Kepler finds the difference between these two models, when using Mars's eccentricity, to be 1'33" and 1'55", "a very small difference indeed." He writes: "Thus you see that, as far as the eccentric equation is concerned, there is a very slight difference preventing the two forms of hypothesis from being equivalent. (p.138)" This error is small enough to be within the bounds of observational error, being hardly observable, and conseqently does not prevent our considering the hypotheses equivalent.
Equivalence? How big is a minute?
Imagine a .7mm pencil lead, held eight feet away from you. That's a minute. Try taping a pencil to a wall and walking eight feet away. What device could you use from where you are standing to measure such a tiny angular distance?
It is also a width of about a quarter of a mm at the distance of a meter, or a hundreth of an inch at a distance of a yard.
It is the red dashed oval.
This paragraph deserves an image:
“Besides, should Copernicus retain the liberty of setting up the ratios of the epicycles, it can happen that the planet's path would come out twisted, higher before and after apogee than at apogee istelf, and lower before and after perigee than at perigee itself.” (p.137)
Here, the second epicycle is almost as large as the first, resulting in the planet being "higher before and after apogee than at apogee itself."