Chapter 56
While I was anxiously turning this thought over in my mind, reflecting that absolutely nothing was accomplished by chapter 45, and consequently my triumph over Mars was futile, quite by chance I hit upon the secant of the angle 5°18', which is the measure of the greatest optical equation. And when I saw that this was 100,429, it was as if I was awakened form sleep to see a new light... (p.543)
Kepler has now developed the way to get the correct distances of the planet from the sun, by using a technique that ends up cutting off half as much of a lunule as before. Return to the diagram for the proposed circular orbit from chaper 40. The planet used to be at V, but now, the planet will still be in the direction of V, but will have a length not equal to AV, but rather to RV, where R is the perpendicular from A to the line from V through center A. You can see that a new, smaller red spot has been made close to V. This is the corrected location of the planet. Note that there is no longer an epicycle: the planet instead reciprocates along the line of distance.
(If you print this out and try it with a compass, you'll find a slight exaggeration in this diagram: Mars should actually be closer to V.)
An animation of this process. The two blue lines are always the same length.
Note that the path is not perfectly elliptical, but is what Kepler will call, in chapter 58, "puff cheeked."
Click here for the same animation with an absurdly large eccentricity. It actually stops looking like an ellipse and appears even more puff-cheeked!
Does this technique create the correct lengths? Presented here is the table at the conclusion of chapter 56, where Kepler announces success: the results of this method match the true Mars-sun distances as determined in chapter 51.
And that which had tormented us for a long time in chapter 39 now surrenders to us in the statement of the truth we have perceived. (p.544)