Where are the Nodes?
Since Kepler now knows the eccentricity, the radius and therefore the size of the Martian ellipse, he can now take any mean anomaly, and tell you at that time what position Mars will appear in. But what is the orientation of the Martian ellipse? Kepler must first re-investigate the tilted pathway to completely conquer Mars.
He therefore takes his ellipse, and finds how it is tilted with respect to the ecliptic. This is broken into two tasks: first he will determine where the nodes are, and then in 62 he will find the angle of tilt, or the maximum inclination. With this new data, he can compare his new latitudinal findings with his work in part 2, when he knew very little about the nature of the Mars orbit.
The beginning of the chapter starts with four Earth-based observations of Mars spaced every 687 days (one Mars year). Because the observations do not tell you what you are actually seeing, he has to use his mind to determine how much the altitude, parallax, refraction, etc. are bending his perceived reality. After correction for these factors, he creates the following latitudinal table for Mars observations:
These latitudes correspond to an inclination of 1½′. Since Mars is returning to the same place every 687 days with the same inclination of 1½', then why are the latitudes growing? Observe the angle between the blue line emanating from Earth toward Mars, and the other from Earth along the ecliptic. This angle is called the latitude.
The same effect you are observing takes place when approaching a tall city building: as you get closer you have to keep looking up higher and higher to see the top, thereby increasing the angle between the ground and the line of vision. If earth were a human walking along the ecliptic, as it got closer to Mars it would have to keep looking up higher and higher, therefore increasing the angle between the ecliptic and the line of vision. In this diagram, Earth is not approaching a tall building, but a cliff, so watch your step.
With an inclination of 1½', Kepler says the distance to the node is about 40', but using the method of Chapter 60, he now calculates the distance to be about 37', read on.
Kepler then begins, as he said he would do in the title of the chapter, “An examination of the position of the nodes.” As is seen below, he starts with the ascending node.
“We will nevertheless accomplish our aim more accurately using the year 1595. For while on October 28 at 12h the latitude was 4½′ south, six days later, on the following November 3, at the same time, the latitude was 19′ 45″ north. Therefore, over 6 days the latitude was changed by 24′. So it changed 4′ per day. And since on October 28 at 12h its eccentric position was 16° 8 1/3′ Taurus, and the remaining latitude was 4½′, this would be traversed in one day and one eighth, after which time 37′ would be added to Mars’s position. Therefore, the node will be at 16° 45 2/5′ Taurus, at the beginning of November of 1595.”
Looking only at Kepler's text, you see how easy he considers this task. Once he can determine when the planet is at the node, he can simply calculate where it will be, using area-time and the ellipse. Kepler does not go through any of the calculatios himself in this chapter, but the steps are presented at the bottom of the above diagram. If they have left you somewhat confused, click here for a pdf of all the calculations involved.
I leave you to trust in Kepler, for his examination of the descending node!