Astronomical Images : Motions of the centre of the eccentric of Mercury and its epicycle

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, first printed in the first edition of the commentary of Reinhold (1542), is a new version of a traditional figure. Its first model was the <i>Theorica motuum</i> in the chapter on Mercury of the first edition of the <i>Theoricae novae</i> (see Peuerbach, 1482, sig. f6r). In his edition of Peuerbach (Paris, 1525), Oronce Fine conceived an improved version of it, which was copied in the Wittenberg 1535 edition. Then Reinhold afforded significant improvements by using ovoid lines and drawing what he called 'irregular figures'. According to its title, <i>Theorica omnem fere varietatem motus centri epicycli et apogii eccentrici Mercurii ostendens</i>, it 'shows almost all the variations of the motion of the centre of the epicycle and of the apogee of the eccentric of Mercury'. These variations are the consequences of the movement of the centre of the eccentric of Mercury on a 'small circle'. As in the preceding diagram (fol. 68r), this small circle is lettered FODP. The circle numbered from 1 to 12 is centred on D, the centre of the equant. The oval line GIHK (Reinhold calls it, in Greek, <i>conchoeides</i>) shows how the eccentric circle is transformed, as its centre circulates on the small circle. We see that DA is equal to DB, the radius of the eccentric, when its centre is at the perigee of the small circle (at the centre of the eccentric equant), and also to FN, the radius of the eccentric, when its centre is at the apogee of the small circle. So, when the centre of the eccentric is at F, the centre of the epicycle is at N, the most distant point from the centre of the World. When it is at D, at the perigee of the small circle, the centre of the epicycle is at B, its closest position to the centre of the World. And in both positions, F or D, the line marking its apogee (drawn from the centre of the World, C, and passing through the centre of the deferent of the epicycle) coincides with the line of the apogee of the equant. In Peuerbach's words, when the centre of the epicycle is 'in the apogee of the deferent' (<i>in auge deferentis</i>), it is also 'in the apogee of the equant' (<i>in auge aequantis</i>). The epicycle is represented with its centre at K and at H. The ovoid forms, at the top and at the bottom of the diagram, are a three-dimensional representation of the movement of the apogee and of the perigee of the eccentric deferent of Mercury. Lines CP and CO are tangential to FODP. They show the limits that the line marking the apogee of the eccentric will never pass beyond, westward or eastward. As in the preceding diagram, the small circle is divided into parts. The first and last, FP and OF, are four Signs long, the second and third, PD and DO, two Signs long. Other letters, e and R, have been added, to mark intermediary positions of the centre of the eccentric. According to Peuerbach, 'when the centre of the deferent moves by the motion of the two secondary orbs from the apogee of its circle toward the west, the centre of the epicycle will move by the motion of the deferent � just the same amount toward the east' (<i>cum centrum deferentis per motum orbium duorum secundorum movebitur ab auge sui circuli versus Occidentem, centrum epicycli per motum deferentis movebitur tantumdem versus Orientem</i>). The 'secondary orbs' are the 'deferent orbs of the apogee of the eccentric', contiguous to the eccentric deferent orb. On their movement westward on the centre of the small circle and on the movement of the eccentric deferent eastward but with the same speed (the period of both movements being one mean solar year), see entries on the orbs and axes of Mercury. By virtue of these movements, the centre of the eccentric deferent moves westward from F, while the line of the apogee of the eccentric recedes from the line of the apogee of the equant. Reinhold explains that 'when the centre of the deferent is at R, the diameter of the eccentric passing through the centre of the World is SRCT. Therefore, the apogee of the eccentric is at point S of the ovoid figure NSGAI. And the perigee of the eccentric is at point T' (<i>dumque idem centrum in puncto R, diameter eccentrici transiens per centrum mundi SRTC. Itaque apogion eccentrici in puncto S figurae</i> [in Greek:] menoeidous<i> NSGAI. Perigion autem eccentrici in puncto T</i>). When the centre is at P (which is 120 degrees distant from F), 'the centre of the epicycle is on the line drawn from the centre of the deferent, passing through the centre of the equant, that is in H' (<i>centrum epic[ycli] existit in linea a centro def[erentis] per centrum aequantis ejecta, videlicet in puncto H</i>). H is then 120 degrees distant from the apogee of the equant, eastward. We have already seen that the line marking the apogee of the eccentric cannot pass beyond the tangents to the small circle (fol. 68r). As the centre of the eccentric is at P, the diameter of the eccentric, passing through the centre of the World, is line GPL (that is also the western tangent to the small circle). The apogee of the eccentric is at G, and 'its perigee is P [a misprint for L], where [the diameter] is tangent to the two irregular figures' (<i>perigion ejusdem P</i> [= L] <i>in contactu duarum irregularium schematum</i>). At point H, 'the centre of the epicycle is at its nearest to the Earth, although it is not opposite to the apogee of the deferent, or on the tangent [to the small circle]' (<i>centrum epic[ycli] proximum est Terris, etsi non in opposito apogii deferentis, sive in linea contingente</i>). When the centre of the deferent approaches the centre of the equant (D), at the perigee of the small circle, 'then, the apogee and the perigee of the eccentric move eastward, in the direction of the apogee of the equant, the apogee of the eccentric describing a line combining [right and curve] GXA, while the perigee describes the curved line LYB' (<i>apogion et perigion eccentr[trici] moventur iam in consequentia versus apogion aequantis, apogion quidem ecc[entrici] describens mixtam lineam GXA, perigion autem curvam lineam LYB</i>). When the centre of the eccentric is at V, the apogee of the eccentric is at X, and the perigee at Y. Meanwhile, the centre of the epicycle progressively recedes from the centre of the World, as it approaches the perigee of the eccentric equant. When the centre of the deferent is at D, the centre of the epicycle is at B, at the perigee of the eccentric deferent and of the eccentric equant. For then the eccentric deferent and the eccentric equant are united, both represented by the circle with numbers. In that position the centre of the epicycle, though at the perigee of both circles, is more distant from the centre of the World than when it is at H. When the centre of the deferent is at point A (30 degrees eastward from D), the apogee and perigee of the eccentric are b and d. Then it arrives at O, 120 degrees from F. The centre of the epicycle is at K, also 120 degrees from the line of the apogee of the equant, but measured against the sequence of the Signs. The apogee of the deferent is at I, at its greatest distance from the line of the apogee of the equant, and the perigee at M, at its greatest distance from the same line. The centre of the epicycle, at K, is again at its nearest to the Earth. Peuerbach later enumerates the consequences of these motions, notably that 'the centre of the epicycle of Mercury � does not, as in the cases of the other planets, describe the circular circumference of the deferent, but rather the periphery of a figure that resembles a plane oval' (<i>centrum epicycli Mercurii � non ut in aliis planetis sit, circumferentiam deferentis circularem, sed potius figurae habentis similitudinem cum plana ovali peripheriam describere</i>). Reinhold remarks that centre of the epicycle of the Moon also describes an irregular figure, for in this case also the centre of the epicycle and the centre of the deferent move in contrary directions but at the same speed, but 'the centre of the epicycle of the Moon delineates rather a [in Greek:] <i>phakoeides</i>, that is lenticular figure, while that of Mercury is rather [in Greek:] <i>ôôdes</i>, that is resembling an egg' (<i>centrum epicycli Lunae potius deliniare schema </i>[in Greek:] phakoeides<i>, id est lenticulare, Mercurii contra potius </i>[in Greek:] ôôdes<i>, id est, ovi speciem gerens</i>), and this oval figure, he adds, 'we have represented in the above diagram and in the next diagram of the proportional parts' (<i>haec ovalis figura � quam nos pinximus et in praecedenti figura, et in sequenti minutorum proportionalium</i>). On this following diagram, see fol. 78r. 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>