In the early 1600's, four world-systems competed for the favor of enlightened thinkers:
 (1) The ancient Ptolemaic system, in which the motions of the planets were combinations of uniform circular motions, moving around a fixed Earth.
 (2) The Copernican system, in which the motions of the planets were combinations of uniform circular motions, moving around a fixed Sun.
 (3) The Tychonic system, in which the motions of the planets were combinations of uniform circular motions, moving around a Sun which itself moved around the Earth in a combination of uniform circular motions.
 (4) The Keplerian version of the Copernican system, in which the motions of the planets were ellipses, moving around a fixed Sun.
  In each of the first three world-systems, the changing apparent motion of the Sun (faster in January and slower in July, relative to the stars) was explained by putting the Earth not at the center of a circular motion, but at a point called the eccentric, which was off-center, so that when the Sun was closer to the Earth in January, its "motion" appeared larger, and when it was further away in July, its "motion" appeared smaller. In Kepler's system, the changing motion of the Sun was explained by putting it a little off-center from the Earth's elliptical orbit, at a focus, and changing the speed of our orbital motion according to the Law of Areas, so that when we were closer to the Sun in January, not only did its motion appear larger because we were closer, but also because the Earth itself was actually moving faster.
  In the Ptolemaic world-system, the retrograde motions of the planets were explained as being due to extra circular motions called epicycles, whereas in the Tychonic system the retrograde motions were explained by having the other planets move around the Sun, and the Sun move around the Earth, and in the Copernican and Keplerian systems they were explained by having all of the planets, including the Earth, move around the Sun.
  The Keplerian system was the only world-system in which the motions were not some combination of uniform circular motions. Instead, the complex combinations of eccentrics, equants, epicycles and deferents were replaced with a single smooth elliptical motion for each planet. Sometimes that simplicity alone is stated as a reason to believe in the Keplerian system, but if you believe in a god of infinite power and wisdom there is no reason why he can't arrange things in as complex a way as he desires, and even if that complexity can be modeled by a simpler way it doesn't mean that the actual situation isn't truly complex. In fact, despite the condemnation of Galileo, in the centuries following that condemnation, many deeply religious members of the church, including priests themselves, performed astronomical observations, and pondered the question of the motions of the planetary bodies. This was deemed perfectly acceptable as long as it was realized that all mathematical representations of those motions were mere fictions which could hardly encompass the full majesty and glory of the Creator's universal plan.
  One such devout individual, famous in astronomical literature for a number of reasons, was Giovanni Domenico Cassini (the same Cassini of the Cassini division, and the Cassini spacecraft).
  Among other things, Cassini attempted to distinguish between the two major ways of describing the planetary motions (and in particular, the motion of the Sun) -- the Ptolemaic concept of uniform circular motion, with the Earth at the eccentric of the solar orbit, and the Keplerian concept of elliptical motion, with the Sun at a focus of the solar orbit. He was not concerned with whether the Earth or the Sun truly moved -- that was something that only the Pope could aver, and God could know -- but he realized, as had a number of others, that there was a way to distinguish between the geometry of the two models.
  To see how this works, you must remember that the Sun moves faster among the stars (eastward along the Ecliptic) in January, and slower in July -- specifically, 3.4% faster than its average motion in January, and 3.4% slower than its average motion in July. How are we to explain this changing motion?
  In the Ptolemaic system, faster than usual (or slower than usual) motions are achieved by placing the Earth not in the center of the uniform solar motion, but at a point called the eccentric (see figure 1 below). As the distance between the Sun and Earth changes, the angle that the Sun appears to be moving through changes as well. When the Sun is at its closest, in January (on the right side of the diagram), it appears to be going faster not because it really is going faster, but simply because it is closer. And on the opposite side of the orbit, in July (on the left side of the diagram), it appears to be going slower simply because it is further away. Each increase or decrease of 1% in its distance causes an approximately equal but opposite decrease or increase of 1% in the size of the Sun and in its apparent motion. So Ptolemy explained the apparent change in the Sun's rate of motion by placing the eccentric (and the Earth) about 3.4% from the center of the Sun's orbit around the Earth.

  In the Keplerian system things look very much the same, to a first approximation (as shown in figure 2, below). When the Sun is closer it and its motion appear larger, so it appears to be going faster than usual, just as in the Ptolemaic system. However, there is a critical difference. In Ptolemy's model the motion of the Sun around its circular orbit is absolutely uniform. In Kepler's model, as the radius vector between the Earth and Sun shrinks, the moving object moves faster and faster, according to the Law of Areas, in order to ensure that the area swept out by the radius vector remains constant. As a result, each 1% change in distance produces a 2% change in the apparent velocity -- 1% because we are closer (just as in the Ptolemaic system), plus an EXTRA 1% because the motion really is 1% faster than usual. So in the Ptolemaic system, to explain the changing solar motion the eccentric must be offset from the center of the orbit by 3.4%, whereas in the Keplerian system, the focus is offset only half as much, or 1.7%.

  In the mid-1600's, Cassini attempted to measure the "orbit" of the Sun, to see whether it changed by the smaller amount predicted by Kepler, or the larger amount predicted by Ptolemy. A measurement revealing the answer to this question would not reveal whether the Earth moved or the Sun, because Tycho's model could be changed, replacing the combination of circular motions with Keplerian ellipses, and produce exactly the same observations -- but at least it would show whether the motions were better represented by Ptolemy's calculations or Kepler's.
  In those days it wasn't possible to measure the distance of the Sun, but presuming the size of the Sun is approximately constant, Cassini could estimate relative changes in distance by seeing how its apparent size changes. If Ptolemy's ideas were correct, the distance and the size of the Sun would change by 3.4% either way from the average amount, whereas if Kepler's ideas were correct, the change in the distance and the apparent solar size would be only half that much.

The apparent size of the Sun at perihelion and aphelion

  Although the difference in the Sun's size caused by its change in distance is obvious in the diagram above, in which the two extremes of size are placed side by side, it is small enough that it was difficult to measure at all, let alone with the accuracy required to decide whether Kepler's or Ptolemy's calculations were the more accurate. However, in the city in which Cassini lived (Bologna, in Italy), there was a church (the Basilica of San Petronio) which contained a 'meridiana'. This was an astronomical calendar of sorts, in which an image of the Sun produced by a small hole in the southern wall of the church crossed a line on the floor of the church at local noon each day. Cassini proposed moving and greatly elongating the line, by moving the hole and meridiana to allow for more accurate astronomical observations, and despite the enormous cost involved (equivalent to more than a quarter million dollars in today's money), was able to convince those in authority to undertake the new construction (and even received a fee for his work on the plans). Using the new meridiana Cassini was able to show that the variation in the solar size was, beyond any doubt, just as predicted by Kepler. The orbit of the Sun around the Earth (in the Tychonic system), or of the Earth around the Sun (in the Keplerian/Copernican system), is indeed an ellipse with an eccentricity of 1.7%, NOT a circle with an eccentric offset of 3.4%. (The meridiana was also used for several other astronomical observations in subsequent centuries, none of which could have been as accurately accomplished in any other way.)

On the left, an image of the meridiana passing the Gothic pillars supporting the basilica, and on the right, of the solar image projected on the meridiana, to encourage you to read more about it.

  Although this did not establish that the Earth moved, it did show that Kepler's theories of planetary motion were most likely to be correct. Rather interestingly, however, Cassini himself did not actually adopt Kepler's theory of elliptical motion. Instead, he proposed that the orbits were a kind of curve called Cassinians, in which the PRODUCT of the distances to the foci is a constant (whereas in elliptical orbits, the SUM of the distances is a constant). So there is a certain irony involved in this story -- that the man who established the superiority of Kepler's theory of elliptical motions to Ptolemy's theory of uniform circular motions did not himself accept the theory as presented by Kepler. It would require Isaac Newton's great mathematical synthesis of the laws of terrestrial and heavenly motion, and his discovery of the universal nature of the force of gravity, to prove that elliptical motion was indeed the correct result.
  And of course this still does not address the question of what is really moving -- the Earth, or the Sun. Tycho had presumed that the answer to that lay in the measurement or non-existence of stellar parallax, but measuring stellar parallax was beyond his ability, and in fact well beyond the ability of astronomers for more than two hundred years after his death. As it happened, the proof that it was the Earth that moved came from a completely different and completely unexpected direction -- Bradley's discovery of stellar aberration.