In early April of 1602, Kepler took what was to prove a fateful step in working out his Mars theory. He had already formulated a prototype of the “area law,” which we now know as his Second Law. This, he thought, would be a more accurate replacement for Ptolemy’s equant, which he saw as a stand-in for a dynamic principle. But, he wondered, would it really be more accurate than Ptolemy’s geometrical model?
To answer this question, Kepler conducted a simple and methodical test, comparing positions and times generated using the area law on a circular orbit with the empirically correct positions and times. The test covers two sides of a single sheet of paper (folium P356 of the Mars Notebook), and by the end of the inquiry Kepler had concluded that the test failed for a circular orbit, but would succeed if the orbit were made very slightly oval. Rather than abandoning the area law, Kepler took the radical step of developing the physics and geometry of the orbit, now presumed to be some kind of oval.
Using photographs of the manuscript page, Kepler’s reasoning will be summarized step-by-step. We will see, as if looking over his shoulder, how he arrived at the crucial discovery of the nine arc-minute discrepancy that led to the abandonment of circularity in planetary theory.
