On referring the ecliptic position to the circle of Mars
When Kepler, reviewing Tycho's observations, writes in Chapter 10, that “on December 28 at 11h30m, they placed Mars at 16°47' Cancer by observation,” he does not mean that Mars was actually seen on the ecliptic at 16°47' Cancer. As Kepler will demonstrate in Chapter 12, Mars is almost never seen on the ecliptic, since it nearly always has an observable “latitude,” or height above or below (north or south of) the ecliptic. So, in creating a hypothesis for the longitudinal motion of Mars, it is necessary to have some way of taking the actual observed position of Mars against the stars and referring it to a position on the ecliptic. Here's how Kepler says this is typically done:
When asked what is the ecliptic position of a planet, astronomers define it thus: it is that point on the ecliptic at which the circle of latitude (at right angles to the ecliptic) passing through the sidereal position of the planet's body intersects the ecliptic. (p.192)
But, consider this: Mars moves along a pathway inclined to the ecliptic. Do the ecliptic positions of Mars move in the same way as the planet itself on its path? Here, with an exaggerated inclination of 40°, we have uniform motion along Mars's circle, with the corresponding ecliptic positions as well. The blue trail marks correspond to 10° of Mars's motion. Here are two views of the same animation:
Now, let's take a view of only the ecliptic. This animation was created with a Mars inclination of 60°:
Clearly, these ecliptic positions are not evenly spaced.
Consequently, the arc between the planet's position on its orbit and the nearest node is always greater than that between its ecliptic position and the same node.
To accurately measure the movement of the planet itself, this artifact of projection onto the ecliptic must be corrected, which means extending the distance the planet has moved along the ecliptic from the node. We wouldn't want to make the planet appear to speed up and slow down just by our method of observing it!
Making a Correction
Kepler demolishes the unsatisfactory method that had been employed to perform the correction. Writing about Tycho Brahe:
[T]hose who constructed the tables thought that the planet is not exactly at opposition to the sun unless AC (the observed distance of the planet from the node) is equal to arc AB, the elongation of the place opposite the sun from the same node. (pp.193-194)
Here is an animation (a work in progress!) with the earth (H) and Mars (I) moving around the sun (G). The earth's position in the ecliptic (E) is a right angle leading to the sun's view of Mars (F) and the earth's view of Mars (C). Point (B) is constructed on the ecliptic to make AB equal to AC in length. This animation should help to make the spherical geometry clear. It currently has the Earth and Mars moving at the same speed, and in the wrong direction.
Some say that the planet should be said to be in opposition when the point B is opposite the sun, but this would remove the reason for looking at opposition in the first place -- to remove the second inequality!
Using AB is not correct. Although it is true that Mars has moved further on its eccentric than spot E has along the ecliptic, AB is too long! The planet is actually moving along arc AF as seen from the sun, which is a shorter arc than AC -- its arc as seen by the earth.
"This is contrary to what they proposed to do... The arc AC has nothing to do with the first inequality."
The true correction that ought to be made is actually quite small -- Kepler puts it at about a minute at 45° (where it reaches a maximum). Incorrectly using AB (which is equal to AC) creates an excess of length that "can be in error by as much as 7 and 9 minutes." This incorrect correction was made by Tycho in the table in chapter 8.
Here is the animation again, with Earth and Mars moving at their proper speeds. Now you can see both proposed opposition times. The line from the sun through the earth intersects E, then B. The E intersection is the real apparent opposition, and the B intersection is the false correction attacked by Kepler. This animation also makes it clear why AC has nothing to do with anything, its lengthening being an artifact of Mars's latitude as perceived from the Earth, which is moving quite close to Mars, and thus lengthening EC and also AC.
Making the Correct Correction
Kepler writes that the perpendicular dropped to the ecliptic is a fine method for recording observations, but when, on the other hand, we compute the planet in its own hypothesis, we are concerned with the exact path of the planet, and not with the ecliptic to which it is inclined. So that would mean dropping a perpendicular up from the ecliptic to Mars's eccentric, instead of the other way around, or making AFE right, rather than AEF. Ask yourself, "If I were on Mars, how would I measure the moment of opposition? How would a Martian refer the position of the Earth to the Martian ecliptic?"
Given the small inclination between the ecliptic and the orbit of Mars of only 1°50', the difference between the two "accordingly does not exceed one minute." Here, with an inclination of 40°, is an animation to compare the two: (1) in blue, the position of Mars as dropped onto our ecliptic, and (2) in green, the position on the ecliptic that would drop perpendicularly to the Martian ecliptic -- the earth position that a Martian would consider to be in conjunction with the sun.
On the left, the inclination is 40°, and on the right, 15°.
Click here for an interactive version of this last animation. (Very helpful!)