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Astronomical Images : Mean and true apogees of the epicycle of the Moon

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 diagram was first used in the 1542 edition of Reinhold's commentary. It has no equivalent in the original (c. 1474) edition of the <i>Theoricae novae</i>, but the edition procured by Peter Apian (Ingolstadt, 1528) and that printed in Wittenberg in 1535 contain similar, though less accurate, figures. In the 1551 Wittenberg edition of the <i>Theoricae novae</i>, this Reinhold diagram has been introduced. As with the preceding diagram in Reinhold's edition (fol. 29r), this one is principally concerned with the distinction between the mean and true apogees of the epicycle of the Moon, showing that they change continually as the epicycle follows the rotation of the eccentric deferent. The changing place of a third point, <i>punctum cavitatis</i> or <i>punctum contactus</i>, is also shown. The outermost circle, whose centre is the centre of the World (T), represents the ecliptic. The three middle circles are the two circles that delimit the eccentric deferent orb of the epicycle of the Moon (<i>deferens epicyclum Lunae</i>), and the middle circle (ABCDEF) to which the centre of the epicycle of the Moon is attached. This epicycle is shown in six successive positions. The small interior circle is the circle described by the centre of the eccentric deferent (S) as it moves around the centre of the World (T). Point V, diametrically opposite to the centre of the eccentric deferent, is the 'opposite point' (<i>punctum oppositum</i>). The vertical line (ASTV) is the axis of the deferent orbs of the apogee of the Moon. For each position of the epicycle (except when its centre is on the axis of the deferent orbs of the apogee of the Moon), the diagram shows three lines intersecting at the centre of the epicycle. These lines were already drawn, or partially drawn, in the preceding diagram (fol. 29r): the one that is drawn from the 'opposite point', V, 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, T, ends up at V, indicating the true apogee of the epicycle (<i>aux vera epicycli</i>). The letters M and V are not those of the preceding Reinhold diagram (fol. 29r), but those of the corresponding diagram in Apian's edition (fol. 9v) and of the 1535 Wittenberg edition. Although the diagram does not quite show it clearly, M and V are marked on the circumference of the epicycle. A third line, drawn from the centre of the eccentric deferent, S, ends up at P, the 'point of the cavity of the epicycle' (<i>punctum concavitatis</i>). In the Apian and Wittenberg diagrams this line is also drawn, but the point on the circumference of the epicycle is not labelled. This third line illustrates the fact that 'no identical point of the cavity in which the epicycle is situated remains continuously over the mean or true apogee of the epicycle' (<i>nullum idem punctum concavitatis, in qua epicyclus situatur, continue super auge epicycli media sive vera maneat</i>). Only when the centre of the epicycle is on the axis of the deferent orbs of the apogee of the Moon does the line drawn from the centre of the eccentric deferent (S) coincide with the lines drawn from T and from V. In every other position of the epicycle, the true apogee (V) 'is always between the mean apogee and the point of the cavity under which the true apogee usually is, when the centre of the epicycle is in the apogee or perigee of the deferent' (<i>aux vera semper â?¦ sit inter augem mediam et punctum concavitatis sub quo aux vera, dum centrum epicycli in auge deferentis vel opposite fuerit, esse solet</i>). Thus, the line drawn from S to the centre of the epicycle always indicates the 'point of the cavity of the epicycle' under which the true and mean apogee of the epicycle are situated when these apogees are on the axis of the deferent orbs of the apogee of the Moon; and, as shown on the diagram, line TV is always (except in the two mentioned cases) between line VM and line SP. Reinhold explains: 'We understand that the plane of the epicycle remains and rotates in some cavity of the plane of the eccentric that is, by itself, immobile, as it is only carried by the movement of the eccentric. If we attribute to this plane of the eccentric as much thickness, or width in the direction of its centre, as the diameter of the epicycle, then the circumference of the epicycle will necessarily touch the concave surface of the superior part of the deferent of the apogee of the eccentric at only one point, according to Euclid, <i>Elements</i> III.2, etc. That is also why this point of contact can be called the 'point of the cavity', because it is superimposed onto the true and mean apogees of the epicycle, when the centre of the epicycle is at the apogee or perigee of the eccentric' (<i>Intelligimus autem superficiem planam epicycli existere ac rotari in quodam concavo superficiei planae eccentrici, quod per se est immobile, quia tantum ad motum eccentrici circumfertur. Huic item plano eccentrici, si tantam tribuimus vel crassitiem, vel latitudinem versus centrum, quantus est diameter epicycli, necesse est circumferentiam epicycli contingere superficiem concavam superioris deferentis augem eccen[trici] in uno tantum puncto, per ii. tertii ele. etc. Quare etiam punctum contactus vocari potest illud punctum concavitatis, quod super auge vera ac media epi[cycli] collocatur, dum centrum epic[ycli] habet apogion aut perigion eccentrici</i>). Translated quotations of Peuerbach's <i>Theoricae</i> are from Aiton (1987). Quotations from Reinhold's commentary are translated or paraphrased by Isabelle Pantin.</p>


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