“How small a heap of grain we have gathered from this threshing!”
An Error in the previous chapters
When Kepler was using the sum-of-distances approach to measure time in chapters 48 and 49, he had divided the fictitious semicircle evenly, thus getting the distance of the planet from the sun at each degree of mean anomaly. Is this what Kepler did when he originally developed the use of sum-of-distances in chapter 40? Take a look at the area-time page to see an animation of adding up an infinite number of distances.
But now Kepler is doing something different.
What's the difference?
The area-time animations first introduced in Part III rely on the principle that the time taken for the planet to move equal amounts of arc, is measured by its distance from the sun. Each arc of motion is weighted equally, and a sum of the distances from the sun (or the area swept out, as an approximation) is an accurate measure of the time for the planet to have traversed those arcs. But now the distances are being added up for equal amounts of mean anomaly. This is how those equal amounts of mean anomaly would look on the eccentric actually traveled by the planet (exaggerated):
If we add distances from the sun in this fashion, we are over-counting the distances from the sun near aphelion (since there are more of them for the same amount of arc), and under-counting them near perihelion (since there are fewer).
Six new approaches
Kepler tries six other approaches to determining the equations, which this page does not go through. Unless you've already done chapters 51-60 and understand them pretty well, don't be too concerned with understanding every detail in this chapter. Plus, we have the Harmony of the World to work on!
Kepler's sixth method
Kepler's sixth method in this chapter introduces a new way of thinking of the epicycle:
“[W]e can here put aside the other persuasion concerning the planet's epicyclic motion, and take one step closer to the truth of the physical cause, leaving to the epicyclic mode nothing but a reciprocation on the diameter.”
That is, rather than thinking of an epicycle rotating about an empty point, think only of the planet’s distance from the sun increasing and decreasing along a radial line going through the sun. What could cause this “reciprocation,” as Kepler calls it? Stick through to chapter 57 to find out.