85. Precession being dealt with, the greater part of the remainder of the third book is devoted to a discussion in detail of the apparent annual motion of the sun round the earth, corresponding to the real annual motion of the earth round the sun. The geometrical theory of the _Almagest_ was capable of being immediately applied to the new system, and Coppernicus, like Ptolemy, uses an eccentric. He makes the calculations afresh, arrives at a smaller and more accurate value of the eccentricity (about 1∕31 instead of 1∕24), fixes the position of the apogee and perigee (chapter II., § 39), or rather of the equivalent =aphelion= and =perihelion= (_i.e._ the points in the earth’s orbit where it is respectively farthest from and nearest to the sun), and thus verifies Albategnius’s discovery (chapter III., § 59) of the motion of the line of apses. The theory of the earth’s motion is worked out in some detail, and tables are given whereby the apparent place of the sun at any time can be easily computed. The fourth book deals with the theory of the moon. As has been already noticed, the moon was the only celestial body the position of which in the universe was substantially unchanged by Coppernicus, and it might hence have been expected that little alteration would have been required in the traditional theory. Actually, however, there is scarcely any part of the subject in which Coppernicus did more to diminish the discrepancies between theory and observation. He rejects Ptolemy’s equant (chapter II., § 51), partly on the ground that it produces an irregular motion unsuitable for the heavenly bodies, partly on the more substantial ground that, as already pointed out (chapter II., § 48), Ptolemy’s theory makes the apparent size of the moon at times twice as great as at others. By an arrangement of epicycles Coppernicus succeeded in representing the chief irregularities in the moon’s motion, including evection, but without Ptolemy’s prosneusis (chapter II., § 48) or Abul Wafa’s inequality (chapter III., § 60), while he made the changes in the moon’s distance, and consequently in its apparent size, not very much greater than those which actually take place, the difference being imperceptible by the rough methods of observation which he used.[54] In discussing the distances and sizes of the sun and moon Coppernicus follows Ptolemy closely (chapter II., § 49; cf. also fig. 20); he arrives at substantially the same estimate of the distance of the moon, but makes the sun’s distance 1,500 times the earth’s radius, thus improving to some extent on the traditional estimate, which was based on Ptolemy’s. He also develops in some detail the effect of parallax on the apparent place of the moon, and the variations in the apparent size, owing to the variations in distance: and the book ends with a discussion of eclipses. 86. The last two books (V. and VI.) deal at length with the motion of the planets.