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Astronomical Images : Proportional parts 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 woodcut is an improved version of the traditional figure of the proportional parts of the superior planets (see Apian (1537), fol. 17r). As in the case of the Moon, the proportional parts concern the calculation of the equation of the argument of the planet (<i>aequatio argumenti planetae</i>), 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 (<i>argumentum verum planetae</i>), measured on the circumference of the epicycle from the true apogee of the epicycle to the centre of the body of the planet. 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 of equations (measured on the zodiac): 'greater when the centre of the epicycle is at the perigee of the deferent than when it is at the mean distances of the deferent. In the latter case they are again greater than when the centre of the epicycle is at the apogee of the deferent'. Still as in the case of the Moon, the variations of the equations for the same given argument are called 'the variations of the diameter [of the small circle or epicycle]' (<i>diversitates diametri</i>). In the case of the superior planets, there are two kinds of 'variations of the diameter': the 'remoter or outer variations' (<i>diversitates diametri longiores</i>) are the excesses of the equations when the centre of the epicycle is at the mean longitudes (<i>longitudines mediae</i>) of the eccentric over the equations when it is at the apogee; the 'closer or inner variations' (<i>diversitates diametri propiores</i>) are the excesses of the equations when the centre of the epicycle is at the perigee over those when it is at the mean longitudes of the eccentric. The 'remoter or outer proportional parts' (<i>minuta proportionalia longiora</i>) are the excesses of the line taken from the centre of the World to the apogee of the deferent over the line extended from the same centre to the mean longitude, divided into sixty equal parts. The 'closer or inner proportional parts' (<i>minuta proportionalia propiora</i>) are the excesses of the line taken from the centre of the World to the mean longitude over the line extended from the same centre to the perigee, divided into sixty equal parts. The diagram shows the difference between these two kinds of proportional parts. It is encircled by a double line, drawn around the centre of the World (D). The radius of the interior line extends from this centre to the exterior limit of the eccentric deferent orb at its apogee. The outermost line represents the ecliptic. The innermost circle in the diagram is the interior limit of the eccentric orb. The exterior limit of the same orb and the middle circle that bears the centre of the epicycle are also drawn. The epicycle is shown in five different positions. The vertical line passes through the apogee of the eccentric circle (B), the centre of the equant (H), the centre of the eccentric (C), the centre of the World (D), the perigee of the eccentric circle (G), and another point, P. P, according to the legend, is equally distant from C and from D. 'LP [or rather KL] is the perpendicular drawn from this median point P to points K and L, on the eccentric circle' (<i>LP cathetos, seu orthogonalis linea, ejecta ex puncta medio P ad puncta K, and L, circumferentiae eccentrici</i>). 'DL is the line of the mean longitude properly speaking, for it is equal to the radius of the eccentric CL, according to Euclid, <i>Elements</i> I.4' (<i>DL linea longitudinis mediae proprie loquendo. Aequalis enim est semidiametro eccentrici CL, juxta quartam primi Elementorum</i>). For a slightly different definition of the <i>longitudines mediae</i>, see fols. 46r, 54r, 56r. Line DB, the 'line of the apogee' (drawn from the centre of the World to the apogee of the eccentric circle), is equal to DR (as both lines are the radius of an imagined circle). Line DG, the 'line of the perigee' (drawn from the centre of the World to the perigee of the eccentric circle), is equal to DS (for the same reason). The difference between DB, or DR, and DL (the line of the mean longitude properly speaking), is LR. The difference between DG, or DS, and DL is LS. LR and LS are each divided into sixty equal parts. We see on the diagram that DK and DL (drawn from the centre of the World to the points that determine the mean longitudes of the eccentric) and their extension separate the upper zone of the <i>minuta proportionalia longiora</i> from the lower zone of the <i>minuta proportionalia propiora</i>, each divided by concentric circles indicating the division into sixty parts or minutes. At its apogee, in B, the eccentric circle encompasses all the 'remoter or outer proportional minutes', at its perigee, in G, it encompasses none of the 'closer or inner proportional minutes', and in other places (as in L, I, and E) it encompasses some of them (either 'inner' or 'outer'): fewer near to the perigee, and more, in proportion, near to the apogee. For each position of the epicycle, tangents to its circumference are drawn from the centre of the World: lines DN. Line DM, in each case, is the line of the true motion of the epicycle (drawn from the centre of the World and passing through the centre of the epicycle). Arc NM of the ecliptic is the maximal equation of the argument for each position of the epicycle (<i>NM arcus zodiaci aequatio argumenti maxima, ad quemvis situm epicycli</i>). According to the legend, 'the sequence of the Signs must be understood as going from the right to the left, as in the sequence KBL' (<i>series Signorum intelligatur a dextra versus sinistram secundum litteras KBL</i>); we must probably read as LBK. When the centre of the epicycle is at L (one point of mean longitude), MN, the equation of the argument, is larger than is MN when the centre of the epicycle is at B (the apogee of the eccentric circle). The difference is arc NO, at each side, which Peuerbach calls 'remoter variation of the diameter' (<i>dum igitur centrum epicycli tenet punctum longitudinis mediae L aequatio argumenti MN, major est arcu MN dum centrum epic[ycli] in apogio ecc. B, quantitate arcui NO, utrinque, quem vocat autor diversitatem diametri longiorem</i>). The same equation of the argument, MN, when the centre of the epicycle is at a point of mean longitude, is smaller than is MN when the centre of the epicycle is at G (the perigee of the eccentric circle). The difference is arc NO, still at each side, called 'closer variation of the diameter' (<i>diversitas diametri proprior</i>). In Peuerbach's words, 'the equations of the arguments, which are written in the tables, occur when the centre of the epicycle is in the mean longitude, or distance, of the deferent' (<i>aequationes autem argumentorum, quae scribuntur in tabulis, contingunt centro epicycli in longitudine deferentis media constituto</i>). The tables do not give the values of the equation when the centre of the epicycle is in other places, but it is possible to reckon them. If we know the true centre (<i>centrum verum</i>) of the planet, the tables indicate the proportional parts; and if we know the true argument (<i>argumentum verum</i>) of the planet, we can calculate the value of the <i>diversitas diametri</i> (either <i>longior</i> or <i>propior</i>) corresponding to the exact position of the centre of the epicycle. 'The proportional part of this variation â?¦ is added to the equation of argument found in the tables, or subtracted from it' (<i>cuius diversitatis pars proportionalis â?¦ cum aequatione argumenti in tabula reperta addenda est, vela b ea minuenda</i>). It is added when the planet is at or near the perigee (where the equations of the argument are greater than those at the mean longitudes), that is 'if the variation is closer' (<i>si diversitas propior fuerit</i>), and it is subtracted near the apogee (where the equations of the argument are less than those at the mean longitudes), that is 'if it is remoter' (<i>si longior</i>). Then we obtain 'the equation of the argument, both true and corrected' (<i>aequatio argumenti vera et aequata</i>), corresponding to the exact position of the centre of the epicycle. On the equation of the argument, see also fol. 56r. 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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