Equivalence of Hypotheses
Claudius Ptolemy / Nicolaus Copernicus / Tycho Brahe
Ptolemy has everything go around the earth. Both the sun (yellow) and the center of Mars's epicycle (dark red) move around the earth, and the planet Mars (red) moves uniformly around the center of its epicycle (dark red). If you watch the gray line (Mars’s motion on its epicycle), you see that it appears to move at the same rate as the sun. It actually moves at exactly the rate as the “mean sun.” This is true for all the planets -- their epicycles are exactly the same as the motion of the mean sun -- quite a coincidence, isn’t it? Click here to see all the planets as Ptolemy prescribed their motion.
Copernicus has the earth (blue) and Mars (red) go about the sun (yellow). Or, at least Mars almost goes around the sun. Its motion is actually reckoned from the "mean sun" -- the center of the orbit of the earth, which is near to, but not the same as, the sun.
Brahe has the yellow sun go around the blue earth, and has red Mars go around the (mean) sun. The yellow sun moves uniformly, and the gray line connecting the sun with Mars moves in just the same way as the center of the epicycle in the Ptolemaic model moves around in its large circle.
If we take each of these animations, and then "zoom out," we can see where the lines of sight from the earth to Mars and from the earth to the sun appear on the infinitely-far-away zodiac. Kepler proves that the three are identical. Note that the retrograde motion of Mars always occurs when it is at opposition to the sun.
Here, a close-up view. The equivalence does not yet appear perfect, since the motion of the Copernican earth is considerable relative to the size of this animation.
Now, a zoomed-out view. Now, at an immense distance from the planets, you get a sense of where the lines of sight lie among the distant fixed stars. The three are equivalent.
These animations already include the equants and eccentricities for Mars and the Earth-Sun motion. Since they are very small, they are almost imperceptible in these animations. To convince the rigorous reader that the equivalence of hypotheses is more than skin-deep, another web page has been created of these animations, with the eccentricities highly exaggerated. Click here to view it.