Chapter 62On Kepler’s PremisesIn Part IV, Kepler proved the method for determining the distance of the planet at any time because he had proved the nature of the elliptical orbits. Therefore, he is able to compare the distances of the planets Earth and Mars from the Sun. With this in his toolbox, Kepler revisits what he attempted in chapter 12, and seeks out a precise geometric proof for determining the maximum inclination of Mars. First, with help of chapters 59 & 60, Kepler uses the equated anomaly of 166d.36m. to find the distance of Mars to the sun at 138,556, and from Earth to the sun at 100,666, and the estimated latitude of 6d 2m 30s. As the animation below will show, the sine of an angle equals the sine of its complement, so the law of sines will allow us to determine the inclination from the sun. But! Kepler is not satisfied with this (he never is), and needs a method that eliminates all suspicion. “In undertaking this, I shall also present a more universally applicable demonstration of the ratio between the inclination and the observed latitude.” (p. 608) Kepler recognized a problem that any rigorous scientist will find. The triangle AEK, whose angles AEK and EAK are neither measurements of longitude, latitude, or any other general convention, but are merely angular distances. How can we be sure that these angles will not merely give us a rough approximation instead of the true inclination which we seek? Below is an animation that shows how Kepler resolves the problem. Kepler calls this the minor premise. As you might have noticed, the end of the animation touches on also the major premise. Using this diagram [insert major premise diagram], Kepler demonstrates the most general, and universal methods of geometric proof, so that we are left without any doubts. This should be enough for anybody with a copy of the New Astronomy to tackle this proof. At this point, Kepler is able to find the inclination for all of the opposition points, and he constructs a table.
