Astronomical Images : The equation of the centre in the epicycle 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 has no model in the original (c. 1474) edition of Peuerbach's treatise. The woodcut was made for the first Wittenberg edition of the <i>Theoricae novae</i> (1535). It was modelled on a figure in the Fine edition (Paris, 1525), fol. 18v. In some copies of the 1553 edition of Reinhold's commentary, as in this Trinity College, Cambridge, copy, the woodcut is inverted (point B ought to be at the top of the diagram, and the zodiacal signs Taurus, Aries, Pisces, Aquarius, Capricorn, and Sagittarius, in the left part). In other copies, this mistake has been rectified. In any case, the order of succession of the Signs is a problem, as in the text and the legend the 'first half of the eccentric' is the left half, the centre of the epicycle moves westward on the diagram, against the succession of the Signs, which is absurd. According to Peuerbach, the 'equation of the centre in the epicycle' (<i>aequatio centri in epicyclo</i>) is the arc of the epicycle between its true and mean apogee: N and P, and C and R, in the diagram (the former being defined by the line drawn from the centre of the World, D, passing through the centre of the epicycle, the latter by the line drawn from the centre of the equant, H, also passing through the centre of the epicycle). This equation is null when the centre of the epicycle is at the apogee (B) of the eccentric or at the perigee, but it is maximal when the centre of the epicycle is at the <i>longitudines mediae</i> (on these points on the eccentric, defined by the line perpendicular to the line of the apogee and passing through C, the centre of the eccentric, see fol. 46r; for a slightly different definition of the <i>longitudines mediae</i>, see fol. 57v). There is the same proportion between the 'equation of the centre in the zodiac' (<i>aequatio centri in zodiaco</i>) and the whole ecliptic as between the 'equation of the centre in the epicycle' (<i>aequatio centri in epicyclo</i>) and the whole circumference of the epicycle, because these two equations are defined by equal angles (<i>angulus aequationis super centro mundi</i> and <i>angulus aequationis super centro epicycli</i>), delimited by parallel lines (see fol. 52r). Consequently, both equations have the same value in the astronomical tables. When the centre of the epicycle is at I, the equation of the centre in the epicycle is arc PN of the epicycle; when it is at F, the equation of the centre is arc CR. The angle of the equation of the centre in the epicycle (<i>angulus aequationis centri in epicyclo</i>) NIP is equal to angle HID (according to Euclid, <i>Elements</i> I.15). This angle HID is also equal to IDG (as shown before, according to Euclid, <i>Elements</i> I.19). Therefore NIP is equal to NDG, the angle of the equation of the centre in the zodiac (<i>angulus aequationis centri in zodiaco</i>). But equal angles do not correspond to equal arcs if the circumferences of the circles are unequal; the difference is proportional to the difference between the circles. The same diagram explains the computation of the true argument. If the planet is at Q (a letter that looks like P in the diagram) or at S, the mean argument will be RQ or PNTS, the true argument CRQ or NTS. 'The practical rule is also evident. When the centre of the epicycle is at F, that is in the first half of the eccentric, as the line of the mean motion, DO, precedes the line of the true motion of the epicycle, DC, the equation of the centre in the zodiac, CO, must be subtracted, in order to obtain the true motion of the epicycle or the true centre' (<i>Patet etiam regula practica. Quando enim centro epic[cycli] in F id est, prima medietate eccentrici, quia linea medii motus DO praecedit lineam veri motus epicycli DC ideo aequatio centri in zodiac CO est subtrahenda, ut relinquatur vel verus motus epicycli vel centrum verum</i>). The reverse occurs in the last half of the eccentric. When the centre of the epicycle is at F, the equation of the centre, CR, must be added to the mean argument RQ to obtain the true argument CRQ. The reverse occurs when the centre of the epicycle is at I. The reason is easy to understand: 'as in the first half of the eccentric, the line of the mean motion, like a mean <i>terminus ad quem</i>, precedes the line of the true motion that is like a true <i>terminus ad quem</i>, so the mean apogee, like a mean <i>terminus a quo</i>, precedes the true apogee' (<i>sicut in prima medietate eccentrici linea medii motus tanquam terminus ad quem, ita apogion medium tanquam medius terminus a quo, antecedit verum apogion</i>). Or else 'as in this first half of the eccentric the line of the mean motion of the planet is more distant from the apogee of the eccentric [than the line of the true motion], so, conversely, the planet is more distant not from the mean but from the true apogee' (<i>sicut in hac prima medietate eccen[trici] linea medii motus planetae longius distat ab apogio eccentrici, ita econtra planeta longius abest non a medio apogio, sed a vero</i>). Quotations from Reinhold's commentary are translated or paraphrased by Isabelle Pantin.</p>