Astronomical Images : Proportional parts of Mercury

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 woodcut is an improved version of the traditional figure of the proportional minutes of Mercury. As in the cases of the Moon and of the superior planets and Venus, the 'proportional parts' concern the calculation of the equation of the argument of the planet (aequatio argumenti planetae), defined as the arc of the zodiac lying between the true motion of the planet and the true motion of the epicycle. This equation of the argument is dependent on the true argument of the planet (argumentum verum planetae), measured on the circumference of the epicycle from the true apogee of the epicycle to the centre of the body of the planet. All these terms and notions are defined by Peuerbach in the chapter on the superior planets. They apply also to Venus and Mercury. As the distance of the centre of the epicycle from the centre of the World varies, the diameter of this epicycle, measured on the zodiac from the centre of the World, also varies. Therefore, to the same value of the true argument of the planet (measured on the epicycle) correspond unequal arcs measured on the zodiac: they are smaller when the planet is more distant from the Earth, larger when it is closer. The variations of the equations for the same given argument are called the 'variations of the diameter' (diversitates diametri), and the proportional parts are a device for the calculation of such variations: the difference between the real measure and the mean measure, given in the tables, being, depending on the case, added to or subtracted from the latter. It has previously been shown that the centre of the epicycle of Mercury is at the greatest distance from the centre of the World once a year (when the centre of the eccentric deferent is at the apogee of the 'small circle'), but that during the same period it is twice at its closest distance to it (when the centre of the eccentric deferent encounters the tangents to the small circle, drawn from the centre of the World). Peuerbach notes that 'the equations of arguments of Mercury that are written in the tables are those that occur when the centre of the epicycle is in the middle of its separation from the Earth. This happens when the centre of the epicycle is distant from [the line of] the apogee of the equant by two Signs, four degrees, and thirty minutes' (aequationes enim argumentorum Mercurii, quae in tabulis scribuntur, sunt quae contigunt dum centrum epicycli fuerit in mediocri eius a Terra remotione. Hoc autem accidit centro epicycli ab auge aequantis per duo Signa quattuor Gradus et triginta Minuta distante). As in the case of the superior planets and Venus, there are two main kinds of proportional parts of Mercury (whereas the Moon has only one kind): the 'remoter proportional parts are the excess of the maximum distance of the centre of the epicycle over its mean distance, divided into sixty equal parts' (minuta igitur proportionalia longiora sunt excessus remotionis centri epicycli maximae super mediocrem eius remotionem, in sexaginta partes aequales divisus); the 'closer proportional parts are defined as the excess of the mean distance of the centre of the epicycle over its minimum distance, similarly divided into sixty equal parts' (minuta proportionalia propiora dicuntur excessus remotionis centri epicycli mediocris super remotionem eius minimam, similiter in sexaginta partes aequales divisus). Therefore, there are also two kinds of 'variations of the diameter': the 'remoter variations' (diversitates diametri longiores) are the excesses of the equations when the centre of the epicycle is at its mean distance from the centre of the World over the equations when it is at its maximal distance; the 'closer variations' (diversitates diametri propiores) are the excesses of the equations when the centre of the epicycle is at its minimal distance over those when it is at its mean distance. But Mercury also has a third kind of proportional parts, due to the fact that the centre of its epicycle is at its minimal distance at two different points of the eccentric, and that, between these two points, its distance from the centre of the World increases as it approaches the line of the perigee of the equant. The diagram shows the difference between these three kinds of proportional minutes. They correspond to the three blackened zones, enclosed inside the oval figure (ovalis figurae ambitus, according to the legend), NIHBKM, that represents the path of the centre of the epicycle of Mercury on its mobile eccentric deferent. The 'small circle' is FSPDOT, the centre of the equant is D, the centre of the World, C. Two circles, delineated by a double line, represent the eccentric deferent when its centre is at the apogee of the small circle, at F, and when its centre is at the perigee, at D. The first circle is NLQA and the second (which is also the eccentric equant), LBA. The letter B, which marks the perigee of the equant (and of the eccentric deferent in its lowest position) is scarcely legible on the diagram: it has probably been printed near number 40 and is split in two by the vertical line. In any case, B is mentioned in the legend, and marked in the preceding diagram (plate after fol. 68). The points marking the mean distance (mediocris remotio) are I and M. At these two points the distance between the centre of the epicycle and the centre of the World (CI and CM) is equal to one radius of the eccentric deferent (in his 2 punctis IM, centrum epicycli distat a centro mundi quantitate semidiametri eccentrici). The circle passing by I, V and M, and centred on C, the centre of the World, is the outermost circle of the 'closer proportional parts' (minuta proportionalia propiora). Line IM crosses the small circle at T and S. If 'we place the centre of the deferent of the epicycle at S, the centre of the epicycle being at I, SI, the radius of the eccentric, and CI, the distance from the centre of the World, will be equal' (si â?¦ centrum deferentis epi[cyclum] ponimus in S, dum centrum epi[cycli] in I, erunt SI et CI aequales lineae h.e [= hoc est?], semidiameter eccentrici, et distantia a centro mundi). 'Similarly, if the centre of the deferent of the epicycle is at T, the centre of the epicycle being at M, TM and CM will be equal' (eodem modo si centrum def[erentis] epi[cyclum] ponatur in T, quando centrum epi[cycli] in M, erunt TM and CM lineae aequales). 'Line CN, the maximal distance, represents 69 of the parts of which the mean distance, CI or CM, is 60, and the minimal distance, CH or CK, about 55.34' (maxima igitur distantia seu linea CN 69 talium qualium mediocris distantia CI, vel CM, 60. Et talium minima distantia CH, vel CK, 55.34 fere). CB, the 'line of the perigee' (linea perigii), from the centre of the World to the perigee of the eccentric equant, represents 57 of the same parts. The excess of CN (from the centre of the World to the apogee of the eccentric centred on F) over CI, the mean distance, is NV, that has 9 of the same parts (differentia igitur seu excessus lineae CN, super lineam CI, id est, linea NV, existit earundem partium 9). V, on the diagram, marks the limit of the upper blackened zone. Thus, 9/60 is the value of one remoter minute (huius differentiae pars sexagesima vocatur minutum longius). The excess of CI, the mean distance, over CH, the minimal distance, that is line MR, represents about 4.26 of the same parts. R is on the circle, centred on C, passing through H and K, the points of the minimal distance; MR and IG delimit the two middle blackened zones. Thus 4.26/60 is the value of one closer minute. The excess of CB, the line of the perigee, over CH, the minimal distance, is represented by XR, that is 1.26 of the same parts. The proportion between MR (4.26 parts) and XR (1.26 parts) is approximately 60/26. This corresponds to the third kind of proportional parts (more precisely, to the second kind of closer proportional parts), and to the bottom blackened zone in the diagram. To conclude, when the centre of the epicycle moves along MNI, there are remoter proportional parts (from 60 at N, to 0 at I and M), and there are closer proportional parts along IHBKM. These closer parts are 0 at I and M, the points of mean distance, but 60 at H and K, and 40 at B. For more information on the calculation of the equation of the argument, see fols. 56r and 57v. Translated quotations of Peuerbach's Theoricae are from Aiton (1987). Quotations from Reinhold's commentary are translated or paraphrased by Isabelle Pantin.