Today, we describe the precession as a slow motion of rotation of the Earth’s axis – which remains nearly parallel to itself during the time of a revolution about the Sun – round a line parallel to the axis of the ecliptic. On the contrary, Copernicus, to explain the parallelism of the Earth’s axis, while moving about the Sun, must invoke an opposite annual rotation of the Earth’s axis around a line parallel to the axis of the ecliptic, which he names: "motion of the inclination" ("motus declinationis"). 

If these motions are equal, the Earth’s axis will keep a constant direction, but assuming that the opposite motion of the axis exceeds the annual motion of the Earth, the axis will slowly change its orientation. Due to this, the intersection of the celestial equator and of the ecliptic at a given point moves Westwards, i.e., in the opposite direction of the Earth’s motion about the Sun. 

But these are only the basic motions described qualitatively. When analysing in detail the periods of the motions ... Copernicus discovered he had to account, in some way, for a certain number of secondary motions of quite long periods as deduced from the observations made by ancient astronomers such as Timocharis (3^rd cent. B.C.) and by the variation of constants, the knowledge of which goes back to Hipparchus (2^nd cent. B.C.). They consist of:

Ptolemy, having at his disposal only 450 years of previous observations (lunar eclipses apart), accuracy of which was not always the best, could do nothing except establish parameters relevant to his own times, or maximum, to the period of 300 years from Hipparchus to himself ... this is the way he assumed the constancy of the solar parameters, of the precession ones and those of the obliquity of the ecliptic ... 

... but when, during the ninth century, the Islamic astronomers started to investigate these basic parameters, such as the length of the tropical year, the precession and the obliquity, they found out that significant changes had occurred from the time of Ptolemy. So, they had only two possibilities. They could base their theories exclusively upon their own observations, ignoring or considering erroneous the parameters of the Almagest, or they could try, in some way, to develop models showing long period variations in their basic parameters ...  

... through "Epitome" and "Theoricae novae planetarum" by Peurbach, Copernicus had a smattering of the theories attributed to Thabit and to az-Zarqal, and through the " Alfonsine Tables ", the Theoricae by Peurbach and the "De motu octavae sphaerae" (1522) by Johann Werner, he was able to learn other methods to calculate a variable precession, a tropical year and an obliquity [indeed, these authors, developed models with long period variations] ... 

... these are the principal subjects of Book III of "De revolutionibus", and as they present theories not debated in the Epitome and in the Almagest, the best of the originality of Copernicus in the invention of models and in the derivation of parameters is emphasized in Book III, which is also the one where he finds some of his worst difficulties. 

Due to the recent discovery that the lunar model of Copernicus was anticipated, about two hundred years before, by Ibn ash-Shatir of Damascus (1306-1375), nothing particularly original is to be found in Copernicus’s Lunar model. Still from the historian point of view, concerned with Copernicus’s sources and procedures, Book IV is much interesting as a highly synthetic and highly medieval exposition of methods and even of numerical parameters of Greek, Indian, Arabic and Latin origin, that found their worthiness into the "Alfonsine Tables" and the "Epitome", and then into the book, where it is said they mark a revolution in astronomy ... 

... ironically, the sole and unique sphere in which Copernicus was able to make corrections to essential mistakes made in the kinetic theories of the "ancients" had nothing to do with heliocentricity and with the motion of the Earth, but specifically referred to the geocentric motion of the Moon. Ptolemy’s model, for the second inequality, incurred in a serious problem: at the quadratures, the Moon moved to about one half its distance at conjunctions and oppositions, which was absolutely incompatible with the absence of a valuable variation in the apparent sizes. The model used by Ibn ash-Shatir and Copernicus overcame this difficulty, keeping the Moon at a reasonable distance from the Earth throughout its orbit .... 

... in Book IV of the "Almagest" Ptolemy explains the various Lunar periodic motions that the Lunar theory must take into consideration – the synodic, tropical, anomalistic and draconitic months - and draws the parameters for the first inequality, that depends upon the Moon’s distance from the apogee of its eccentric ... he also discovered that the Moon was subject to a second inequality depending upon the angular distance from the mean Sun.  

The effect of this inequality, as per Ptolemy’s Book V, is twofold. In the first place, the equation of the first inequality is varied in such a way that, whatever value it may have, it always increases when the Moon moves from the conjunction or from the opposition to quadrature, a correction that corresponds to the "evection" of the modern lunar theories. In the second place, the "proneusis" or inclination of the mean apogee of the lunar epicycle is equivalent to a two-monthly variation in the direction of the apsidal line of the lunar orbit and such that, starting from the conjunction, it diminishes in the first and third quadrant, and increases in the second and fourth. This is related, in a more remote way, to the modern description of the evection’s effects on the apsidal line and in the same way on the eccentricity, which depends on twice the Sun’s elongation from the lunar perigee, therefore being the period about 206 days ... 

... Ptolemy’s model was based on the principle that optically increased the effect of the lunar epicycle, making it move nearer to the observer at the quadratures The model of Ibn ash-Shatir and of Copernicus works on the alternative principle, enlarging the epicycle at the quadratures [with the use of a further epicycle] ...

Unlike the eclipses or the first visibility of the Moon, it is not easy to draw parameters, even for expert astronomers, that could have a serious interest to emphasize the problems that are inherent in the emerging theories or to perform improvements. 

At the time of Copernicus, the planetary theory was substantially the same as the one of Ptolemy’s in the "Almagest". Values were mostly unchanged, except for the necessary adjustments of the mean motions requested by the alterations of the mean motion of the Sun ... 

... nevertheless, Book V of "De revolutionibus" is technically the more adhering to Ptolemy. Its methods and its numerical parameters, with respect to the two previous books on the motions of the Earth and of the Moon, did not influence so much the astronomers that came after. Moreover, in the same way as in the past, Copernicus could only follow Ptolemy’s procedures to confirm, or slightly modify Ptolemy’s results. Only in the sidereal motion of the planetary apsidal lines, that appear considerably greater than their correct values, due to a combination of mistakes, some he made himself and some due to measurements made by Ptolemy, he added a few elements of his own to those of Ptolemy’s. 

Notwithstanding, Copernicus does not add anything to Ptolemy’s methods and only modifies his parameters slightly, two important steps aside from Ptolemy’s theory can be found in Book V. The first, the most well known, is the representation of the second anomaly of the superior planets by means of the motion of the Earth revolving about the mean Sun, and the anomaly of the inferior planets by the motion of the planet revolving about the mean Sun, in place of the epicycle used in Ptolemy’s geocentric model.  

At the time of Copernicus and for Copernicus himself, the importance of the innovation was that it permitted an unequivocal determination of the amplitude of the distance of the planets from the mean Sun, which is the main advantage of the heliocentric theory, over the geocentric theory, both in the Tychonic and in the Copernican form. 

The other step aside from the Ptolemaic theory is the use of models for the planets in which all the single motions are circular and uniform or, more physically speaking, each sphere revolves about an axis that passes through its centre, a principle violated by the models of Ptolemy, taken into consideration both from the geometrical and physical point of view. 

The physical principle of motion in the space was extremely important for Copernicus, and its violation by Ptolemy proved very much embarrassing for him... 

... the models of Copernicus in Book V with-keep guarded the uniform circular motion, as per his wish, but it is quite improbable they reflected the original thoughts of Copernicus ... 

... the discovery of the planetary theory made by the astronomers of the school of "Maragha", as they called it because of the relation some of them had with the observatory of Maragha in North-Western Iran, not only is of great importance in itself, but also demonstrated that the planetary theory of Copernicus originated from Arabia, transmigrated into Europe, by unknown ways, maybe through Byzantine sources and reached Italy in the 15^th century ... 

... for the superior planets he derives his elements using Ptolemy’s methods and thus finding that the eccentricity of Saturn and Jupiter remained constant since Ptolemy’s times; that the eccentricity of Mars had slightly diminished and that the apsidal lines showed an obvious displacement in the period of 1400 years. 

For the inferior planets, instead, he would not have been able to update constants autonomously, because he could not carry out the necessary observations required by Ptolemy’s methods [because of the latitude at which he lived]. Therefore, the best he could do was to directly use the observations of Ptolemy, adapting Ptolemy’s elements to his models and making some hypothetical adjustments in the eccentricity of Venus and in the direction of the apsidal line of Mercury.  

The difficulties and the boredom, in the simple computation of these derivations and in the preparation of the text and tables of Book V, were outrageous. Since it was not possible to cumulate the data of the following years, it was, therefore, not possible to demonstrate if the elements of Copernicus were accurate, a judgement that, even now, after studying Book V in detail, is difficult to give ... 

bibliography: N.M. Swerdlow & O. Neugebauer - "Mathematical Astronomy in Copernicus's De Revolutionibus" - Springer-Verlag