Determining the Planetary Characteristics
It seems that we must start anew. The three positions of Mars used in the previous chapter gave an orbit whose aphelion disagreed significantly with the aphelion found earlier in the vicarious hypothesis.
“This is the source of our suspicion that the planet’s path is not a circle.”
Therefore, the distance of Mars cannot be known by knowing its distance at other parts of the orbit: each must be investigated in turn. Kepler now focuses on the aphelial and perihelial distances to get a true measure of the eccentricity.
Here, position ι is the aphelion, and θ is the perihelion. The larger circle is the path of Mars, while the smaller is the path of our Earth. By using measurements separated temporally by whole numbers of Mars years (when the earth was at δ, ε, κ, λ, and γ), Kepler can measure the aphelial distance. He can check whether the distance is right by calculating the heliocentric longitude of Mars based on the proposed distance, and seeing whether it comes out to be the same for all five observations. If not, he must adjust the distance. With this method, he concludes that the aphelial distance is 166,780, and that the perihelial distance is 138,500.
This gives an eccentricity of 14,140, or 9264 when the Martian orbit is taken as 100,000. This agrees with the bisection of the vicarious hypothesis, which gave 9282.
The bisection which Kepler proved for the Earth in Part III, and which he took as a fundamental truth in chapter 32 to enable him to hypothesize the species of the Sun moving the planets, has now been fully demonstrated for Mars as well.