“[T]he faith that was pledged in chapter 6 and in many other places in this work, I have redeemed from all tincture of self-justification, and have shown that the eccentric of Mars cannot be referred to anything but the sun itself; and that, in addition, it is not only reason that stands with me, but the observations themselves, in my releasing the observations of Mars from the sun's mean motion and measuring them out by the apparent motion of the sun.” (p.528)
Liberation from the Mean Sun
Kepler speaks for himself here, so there is no need to simply rephrase what he says on this web page. One thing in this chapter that may cause confusion is on page 527, where he speaks of the alternative mean-sun line of apsides being made through H and F, creating a new center at I. He refers to Chapter 6, but it seems more correct to refer to a specific part of Chapter 5. In this section, pages 146-150, Kepler rejects making a mean-sun line of apsides through the same center determined by the apparent-sun. Rather, he connects the mean-sun and the equant of Mars, creating a new hypothesis that differs from the apparent-sun one by only 4'24". The reader may be surprised to see Kepler use "the center of uniform motion for ... the sun" (point H) here, rather than the center of the earth's orbit (point B), but remember what has been developed in Part III -- the bisected eccentricity for the first inequality has introduced point B, which didn't exist in the diagrams of Chapter 5. The mean sun is the equant of the Earth's orbit, not its center.
Recall that in chapter 51, the distances from the sun to Mars on opposite sides of the line of apsides were equal. On this diagram from p. 527, A is the true sun, B the center of the earth's orbit, and C the center of Mars's orbit. It is clear that if AE and AD were repeatedly found to be equal all along the orbit that line AC is indeed the line of apsides, and the mean sun is not on it! Changing the line of apsides from AC to HF would require the planet's position at D to be closer to the sun than its position at E, something which goes against the many distances determined in chapter 51.
Additionally, as we have seen since Chapter 44, Mars does not move in a circle, but, rather, the distances from the sun prove that is must approach the sun in the middle longitudes: it has an oval path. But the symmetry seen in Chapter 51 indicates that this physical path of the planet really is situated along AC as its line of apsides. Situating it along line FH would give it a lop-sided shape.