Astronomical Images : Variations in speed of the motion of the Moon's epicycle

Erasmus Reinhold

Astronomical Images

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 ([Theoricae] auctae novis scholiis in theoria Solis ab ipso autore). This diagram was first used in the edition of the Theoricae novae printed in Wittenberg in 1535. It has no equivalent in the original (c. 1474) edition of the treatise, but the edition procured by Peter Apian (Ingolstadt, 1528) contains a similar, though much less accurate, figure. It is the third diagram conceived to explain the motion of the epicycle of the Moon (see also fols. 29r, 31r and 35r). This diagram, called Theorica velocitatis et tarditatis motus epicycli ('theory of the slowness and fastness of the movement of the epicycle'), shows that in Peuerbach's words, 'the revolution of the epicycle about its own centre is faster when the centre of the epicycle is traversing the upper half of the eccentric, and slower when it is traversing the lower half' (revolutio epicycli circa centrum suum centro epicycli per superiorem eccentrici medietatem discurrente sit velocior, per inferiorem vero tardior). 'Superior medietas ecce[ntrici]' and 'inferior medietas' are inscribed on the diagram. This increased velocity in the part near the apogee of the eccentric is presented by Peuerbach as a consequence of the fact, shown in the preceding diagram (fol. 31r), that the true apogee of the epicycle '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' - this 'point of the cavity' (punctum cavitatis, also called punctum contactus by Erasmus Reinhold) being on the line drawn from the centre of the eccentric deferent and passing through the centre of the epicycle. As in the preceding diagram (fol. 31r), the outermost circle, whose centre is the centre of the World (T), represents the zodiac. The middle circle is the eccentric deferent of the epicycle of Moon (deferens epicyclum Lunae) and the centre of the epicycle of the Moon is attached to it. This epicycle is shown in nine 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). The vertical line is the axis of the deferent orbs of the apogee of the Moon. Point V, diametrically opposite to the centre of the deferent, is the 'opposite point' (punctum oppositum). 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 two lines intersecting at the centre of the epicycle: the one that is drawn from the 'opposite point' marks point M on the circumference of the epicycle, indicating the mean apogee of the epicycle (aux media epicycli); the one drawn from S, the centre of the eccentric deferent, marks point P on the circumference of the epicycle, indicating the 'point of the cavity'. The body of the Moon is represented by black dots, and the radius from the centre of the epicycle to the centre of the lunar body (F) is drawn. In order to measure the distance between these points, arcs of circles are drawn above the circumference of the epicycle from F to P, the 'point of the cavity', in the left (eastern) part of the diagram, and from F to M, the mean apogee of the epicycle, in the right (western) part. According to the legend, lines MF are 'equal arcs of the epicycle measuring the distance between the Moon and the mean apogee; we will soon call this arc 'mean argument'' (MF sunt etiam arcus aequales epicycle, quo distat Luna ab auge media, et mox vocabitur argumentum medium). What the diagram clearly shows is the change in position of P with respect to F (the centre of the body of the Moon), and to M, the mean argument of the Moon. Reinhold remarks that in some positions of the epicycle, P (punctum contactus) is less distant than M, the mean apogee, from F, the centre of the body of the Moon. Elsewhere, P and M coincide (when the centre of the epicycle is on the axis of the lunar apogee and perigee). Elsewhere, finally, P is more distant than M from F, and that is where the movement of the Moon on its epicycle is rather rapid (vides itaque punctum contactus P, alicubi minus distare ab F loco Lunae, quam ab eodem moco differt aux media M; alicubi haec duo puncta augis mediae et contactus coincidere ... Alicubi denique punctum P, longiori abesse intervallo, quam augem mediam a loco Lunae, ubi motus Lunae in epicyclo admodum velox existit). The diagram shows that, in this respect, the difference between the left (eastern) part of the eccentric (prima medietas eccentrici) and the right (western) part (altera medietas) is as crucial as the difference between the part around the apogee (superior medietas eccentrici) and the part around the perigee (inferior medietas). Reinhold clearly states in his commentary (fol. 32r) that in the eastern part of the eccentric the 'point of contact' precedes the mean apogee (praecedit punctum contactus, sequitur aux media), given that in the first (left superior) quarter the mean apogee recedes from the point of contact against the series of the Signs, that is westward ([aux media] in 1 quarta recendens a puncto contactus contra seriem Signorum), whereas in the second (left inferior) quarter, the same point 'returns' eastward to the point of contact (in 2 quarta revertens ad punctum contactus secundum seriem Signorum). In the western part of the eccentric (from the perigee to the apogee), the mean apogee precedes the point of contact (sequitur punctum contactus, praecurrit aux media), given that in the third (right inferior) quarter the mean apogee again recedes from the point of contact following the series of the Signs, that is going eastward ([aux media] in 3 quarta recendens iterum a puncto contactus secundum Signorum seriem), whereas in the last (right western superior) quarter, the mean apogee 'returns' westward to the point of contact (in ultima quarta regrediens ad punctum contactus contra seriem Signorum). Thus, in the superior part of the eccentric, the mean apogee moves westward ('against the order of the Signs'), and in the inferior part it moves eastward ('according to the sequence of the Signs'). Since 'wherever the mean apogee moves against the series of the Signs, the movement of the Moon on the epicycle is faster' (ubicunque medium apogion movetur contra seriem Signorum, motus Lunae in epicyclo sit velocior), the conclusion is obvious: in the inferior part of the eccentric, the movement of the Moon on the epicycle is significantly slowed (motus Lunae in epicyclo hic nonnihil retardatur, fol. 32v). Translated quotations of Peuerbach's Theoricae are from Aiton (1987). Quotations from Reinhold's commentary are translated or paraphrased by Isabelle Pantin.