Combining Gravitation and the Equant
In chapter 32 of the New Astronomy, Kepler hypothesizes the form of action of a physical principle. The footprints of this physical principle are the apparent existence of an equant and a bisected eccentricity in the motions of all of the planets. The true principle of universal gravitation operates in such a way as to cause planets to spend less time moving the same distance, in proportion to how close they are to the sun. In chapter 40, Kepler first proposes the notion of equal area-equal time to measure the sum of the infinite number of times the planet took to traverse infinitesimal arcs. He tests it in chapter 43, but it does not work to accurately give the position of Mars when its orbit is assumed to be circular.
But what if we were to combine the two ideas? What if we made a model that included both an equant, and the principle of gravitation? What would necessarily have to happen to the equant, were it to sweep equal angles in equal times? Experiment with this animation to find out:
As in a number of other animations, you can use a and z to change the speed and the spacebar to pause. You can also use s and x to change the eccentricity.