Astronomical Images : The 'three points' and the irregular motion of the epicyle of the superior planets

Erasmus Reinhold

Astronomical Images

<p style='text-align: justify;'>This Parisian edition was copied from the first edition of the commentary of Peuerbach by Erasmus Reinhold, printed in Wittenberg by Hans Lufft in 1542. Subsequently, in 1556, Charles Perier published a new edition, copied from the revised edition printed by Lufft in 1553, which contained additions to the theory of the Sun (<i>[Theoricae] auctae novis scholiis in theoria Solis ab ipso autore</i>). This Reinhold diagram, which first appeared in 1542, has no model in the original (c. 1474) edition of Peuerbach's treatise, and no real equivalent in the subsequent editions. It was also introduced in the edition of Peuerbach's <i>Theoricae novae</i> printed in Wittenberg in 1551. It illustrates the irregular movement of the epicycles of the superior planets around their own centres. Peuerbach had demonstrated before that the movement of the eccentric that bears the centre of the epicycle is itself irregular in relation to its own centre, but regular in relation to the centre of the eccentric equant (see fol. 41r): the centre of the epicycle moves more slowly as it approaches the apogee of the eccentric, and faster as it approaches the perigee. Reinhold added that in relation to the centre of the World, which is more distant from the centre of the equant than the centre of the eccentric, the centre of the epicycle moves yet more slowly near the apogee and faster near the perigee. As for the rotation of the body of the planet on the epicycle, it is accomplished eastward (<i>secundum successionem</i> or <i>in consequentia</i>) in the superior part of the epicycle, and westward (<i>contra successionem</i> or <i>in antecedentia</i>) in the inferior part, whereas the contrary occurs in the case of the Moon (see fol. 29r). There is another difference between the epicycle of the Moon and that of the superior planets. The epicycle of the Moon is situated in the same plane as its eccentric deferent, whereas the epicycles of the superior planets rotate in planes other than those of their own eccentrics. Reinhold explains that the axis of this epicycle, which is necessarily perpendicular to the plane of the epicycle, is oblique in relation to the plane of the eccentric. This obliquity is visible in the three-dimensional diagram printed in Schreckenfuchs' 1556 commentary (see plate after p. 98). In addition, the rotation of the epicycle around its own centre is irregular. According to Peuerbach, this irregularity 'follows a rule: the body of the planet regularly moves away from the point of the mean apogee of the epicycle, wherever it is' (<i>haec tamen irregularitas hanc habet regulam, ut a puncto augis epicycli mediae, quicunque sit, corpus planetae regulariter elongetur</i>). The consequence is the same as in the case of the Moon: the mean and true apogees of the epicycle change perpetually, and the rotation of the epicycle is faster in the superior half of the eccentric, and slower in the inferior half. For a definition of the true and mean apogees of the epicycle, see Reisch (1503), sig. o3r; see also, concerning the Moon, Apian (1537), fol. 9r, and Reinhold (1553), fol. 29r. Reinhold notes that this rule for the irregular motion of the epicycle concerns all the planets that have an epicycle, as 'neither the eccentrics nor the epicycles of these planets regularly and uniformly move around their own centres, but rather in relation to other points' (<i>neque enim eccentrici neque epicycli horum planetarum super suis centris uniformiter ac aequabiliter incedunt, sed potius super aliis punctis</i>). However, there is a difference: the movement of the centre of the epicycle of the Moon (carried by the lunar eccentric) is regular in relation to the centre of the World, and the movement of this lunar epicycle is regular in relation to what is called the 'opposite point' (see entries on the motion of the Moon), whereas the eccentrics and the epicycles of the other planets (the Sun excepted) move regularly in relation to 'the same so-called centre of the equant' (<i>super eodem aequantis ut vocant centro</i>). The reasons for this hypothesis are revealed in Ptolemy's <i>Almagest</i> and Regiomontanus' <i>Epitome</i>. In this elementary textbook the explanations already given in the theory of the Moon will suffice. Reinhold there refers to the explanation of the 'three points': the mean and true apogees of the epicycle and the 'point of contact' (<i>punctum contactus</i>, corresponding to Peuerbach's <i>punctum cavitatis</i>); see fol. 31r. In the diagram the outermost circle represents the ecliptic, whose centre is the centre of the World (D). The interior circle (BAGF) is the eccentric to which the centre of the epicycle is attached. This epicycle is shown in eight successive positions. The vertical line is the line of the apogee (<i>linea augis</i>), passing through the apogee of the eccentric (B), the centre of the equant (H), the centre of the eccentric deferent (C) and the centre of the World (D). For each position of the epicycle (except when its centre is on the line of the apogee), the diagram shows three lines intersecting at the centre of the epicycle: the one that is drawn from the centre of the equant, H, ends up at M, indicating the mean apogee of the epicycle (<i>aux media epicycli</i>); the one drawn from the centre of the World, D, ends up at V, indicating the true apogee of the epicycle (<i>aux vera epicycli</i>); the third line, drawn from the centre of the eccentric deferent, C, ends up at P, the 'point of the cavity of the epicycle' (<i>punctum concavitatis</i>), or 'point of contact' (<i>punctum contactus</i>). Only when the centre of the epicycle is on line of the apogee (at B and G), do the three lines coincide. In every other position of the epicycle, the point of contact (P) 'is always in the middle position, between M and V, just as the centre of the eccentric is in the middle position between the two other centres'. A fourth point, N, is marked on the circumference of the epicycle. The legend does not mention it, but it certainly indicates the position of the body of the planet. Therefore, the arcs of circles drawn above the circumference of the epicycle measure the equation of the centre (<i>aequatio centri</i>), arc VM (between the true and mean apogees); the true argument (<i>argumentum verum</i>), arc VN (from the true apogee to the centre of the body of the planet); and the mean argument (<i>argumentum medium</i>), arc MN (from the mean apogee to the centre of the body of the planet). On the definition of these terms, see Reisch (1503), sig. o3r. On similar measurements concerning the Moon, see Reinhold (1553), fol. 35r. There is another line, ACF, perpendicular to the line of the apogee and passing through C, the centre of the eccentric. A and F are called <i>longitudines mediae</i>. The eccentric is thus divided into four equal portions: BA, AG, GF, and FB. BA and FB constitute the superior half of the eccentric, AG and GF the inferior half. The mean apogee (M) precedes the point of contact (P) in BA and AG, the first and second quarters. In the first, M moves away from P, going eastward (<i>secundum seriem Signorum</i>); in the second, it approaches it, going westward (<i>contra seriem</i>). The mean apogee follows the point of contact in GF and FB, the third and fourth quarters. In GF, M moves away from P, going westward; in FB it approaches it, going eastward. Whenever the mean apogee moves eastward (that is in the superior half of the eccentric, FA), the movement of the planet on the epicycle is faster, as in the case of the Moon, for then the mean apogee moves in the same direction as the planet (<i>apogion medium movetur in eandem partem, in quam planeta</i>). In the inferior half of the eccentric (AF), the mean apogee and the planet on the epicycle move in opposite directions, the latter being faster than the former. Thus, the movement of the planet on the epicycle is slowed a little, as in the case of the Moon. On the consequences of the respective position of the three points, see also fols. 52r and 54r. On the <i>longitudines mediae</i>, see fols. 54r, 56r, 57v. All quotations from Reinhold's commentary are translated or paraphrased by Isabelle Pantin.</p>