How did astronomers measure changes in Mars' position?

How does one begin to unravel the tangled motions on the celestial sphere?

The Earth, after all, is a moving observatory. The motion of our own planet skews our perception of the positions of the planetes (wandering stars) against the background of the very, very distant "fixed" stars.

For a variety of reasons, early astronomers discovered they could remove one degree of freedom by considering only measurements taken "at opposition." "Oppositions" are moments when a planet and the Earth lie on the same line in space with respect to the Sun. This enables an astronomer to chart the position of a planet with respect to a fixed reference point--which Kepler and Copernicus believed to be the Sun.

"Oppositions" for Mars follow in the animation below:

Kepler, however, realized that the process by which astronomers reckoned these critical measurements was flawed. We explore his ideas on this subject in the next section.

Here is an illustration in "three dimensions" of "oppositions:"