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Astronomical Images : The variation of the intersection of the ecliptic of the eighth sphere with the equator

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 is a reuse or a close copy of a woodcut in the 1535 Wittenberg edition of Peuerbach (in the Wittenberg edition, the letter D is similarly inverted), itself a close copy of a figure in the Apian edition of Peuerbach (Ingolstadt, 1528; see fol. 35v in our 1537 edition). In the Reinhold edition, not only has the diagram not been revised and corrected, but it is neither explained by a legend, nor accompanied by <i>scholia</i> (in the 1535 Wittenberg edition, there is a legend, though an incomplete one). The preceding <i>scholion</i> never refers to it. This <i>scholion</i> insists, rather, on the insoluble problems arising from the theory of the eighth sphere: as it multiplies the 'first degree of Aries', the astronomers fight amongst themselves about the meaning of the point from which the astronomical tables measure the movements and positions of the fixed stars and the planets - is it the first degree of Aries of the Prime Mover, or this mobile and elusive intersection of the vernal equinox? (<i>Ingens pugna et controversia oritur, utrum illud punctum, a quo tabulae motus numerant seu loca errantium et inerrantium syderum, sit caput Arietis primi mobilis, an vero ipsa mobilis et vaga intersectio verni aequinoctii</i>). Reinhold's non-commitmental attitude towards the diagram is a symptom of his perplexity and scepticism. The diagram illustrates the consequences of the proper movement of the eighth sphere, also called the 'third motion'. As the first degrees of Aries and Libra of ecliptic 8, also called the 'mobile ecliptic' (<i>ecliptica mobilis</i>), describe small circles around the corresponding points of ecliptic 9, also called the 'fixed ecliptic' (<i>ecliptica fixa</i>), 'it happens that the ecliptic of the eighth sphere, as its situation varies, intersects successively in various of its parts the equator of the Prime Mover [or tenth sphere]' (<i>contingit itaque ut ecliptica octavae sphaerae sub diversa eius habitudine successive in diversis suis partibus aequinoctialem primi mobilis intersecet</i>). That has important consequences, notably for the variation of the actual vernal and autumn equinox points, which are always at the intersection between the plane of the ecliptic of the eighth sphere and the plane of the equator, and of the declination of ecliptic 8 in relation to the equator. As the centres of the small circles complete their revolution on the ecliptic of the ninth sphere or fixed ecliptic (which takes 49,000 years), 'any point whatever of the ecliptic of the eighth sphere will cut the equator near the beginning of Aries and also near the beginning of Libra' (<i>quilibet punctus eclipticae octavae sphaerae aequinoctialem prope caput Arietis, atque etiam prope caput Librae primi mobilis secuerit</i>). But this movement is not regular, as the first degrees of Aries and Libra of the ecliptic of the eighth sphere rotate around the corresponding points of the ninth sphere (with a period of 7000 years), which also causes the variability of the greatest declinations of the ecliptic of the eighth sphere and of the equinoxes. 'Since [â?¦] the orbs carrying the apogee of the Sun move about the axis of the ecliptic of the eighth sphere, according to the motion of the same sphere, and the orb carrying the Sun moves about an axis parallel to that axis, it will necessarily follow that the centre of the body of the Sun is always found in the plane of the eighth sphere. But this plane is often, indeed frequently, outside the beginning of Aries of the Prime Mover' (<i>ex quo namque, sicut supra dictum est, orbes augem Solis deferentes super axe eclipticae octavae sphaerae ad motum eiusdem sphaerae moventur, et orbis Solem deferens super axe praedicto axi aequidistante, necessario sequetur, ut centrum corporis Solaris semper in superficie eclipticae octavae sphaerae reperiatur. Haec autem superficies saepe, imo frequenter est extra caput Arietis primi mobilis</i>). The diagram shows this variation of the angle of intersection between the plane of the mobile ecliptic (or ecliptic of the eighth sphere) and that of the fixed ecliptic (the plane of the ecliptic of the ninth and tenth spheres), and its consequences on the intersection of the mobile ecliptic with the equator and tropics of the tenth sphere, or Prime Mover. As in the preceding diagram (see fol. 164v), the ecliptic of the ninth sphere, or fixed ecliptic (circle ABCD), is represented between two other intersecting great circles: the ecliptic of the eighth sphere, or mobile ecliptic, shown in the two positions where it is most separated from the fixed ecliptic (circle HRMP and circle FOKT). Points C and A (the first degrees of Aries and Libra of the ecliptic of the ninth sphere) are the centres of the small circles, FGHI and KLMN, described by the corresponding points on the ecliptic of the eighth sphere. The lines representing the great circles that pass through the first degrees of Aries and Libra and the poles of the fixed ecliptic are also drawn: FAMFCH (for ecliptic 9, and also ecliptic 8 when it is joined with ecliptic 9; see fol. 168v), and NI and LG (for ecliptic 8, when it is most separated from ecliptic 9). The principal new features of the diagram are the equator (<i>aequinoctialis</i>, labelled OCZXQR) and the tropics, each marked '<i>tropic[us]</i>'. We should probably understand (in the absence of a legend we can only guess) that the two tropics do not represent the tropic of Cancer and the tropic of Capricorn, but one exemplary tropic corresponding to different declinations of the ecliptic: as the declination increases, the circumference of the tropics diminishes. We also see that in the diagram line KACH, which represents the great circle passing through the first degrees of Aries and Libra of ecliptic 9, also called the colure of the equinoxes, does not divide the equator into two equal parts (as it does in reality). This is due to the perspective (or the lack of perspective). AA (the letter is repeated, for 'A' is used for the first degree of Libra of ecliptic 9) probably represents the poles of the diurnal movement (or poles of the equator), and AC, the colure of the equinoxes, ought to pass not only through BB (which probably represents the poles of ecliptic 9) but also through AA. In the small circle of Libra, six dots have been added to indicate more positions of the first degree of Libra from K to M and arcs of circles have been drawn from the dots to the equator: each of them represent a portion of ecliptic 8 (in the corresponding position) and its intersection with the equator. It is also evident that when the first degree of Libra is at K, the intersection is at Q, when it is at M, at R. At L and at N, the intersection is at A, for ecliptic 8 and ecliptic 9 are joined. Of course, the same is true for the small circle of Aries, though no additional dots and arcs of circle have been drawn on it. Three lines intersecting at AA are drawn: they represent the great circles on which the declination of ecliptic 8 can be measured (they are partly graduated to this effect). VBAABBD probably represents the colure of the solstices, perpendicular to the colure of the equinoxes and passing through the poles of the equator and the first degrees of Cancer and of Capricorn of ecliptic 9. VB, the graduated portion of this line indicates the declination of ecliptic 9 and also of ecliptic 8 when it is joined to it (when its first degree of Aries is at G or I). XVAAO and ZTAAR probably represent the colure of the solstices of ecliptic 8, in the two positions where it is most separated from ecliptic 9 (R and O are the intersections of ecliptic 8 and the equator when the first degree of Aries 8 is at H and F). XV and ZT measure the maximal declination of ecliptic 8. We may observe that, in this case, the same letter (V) has been used for two different points. In any case, V and T, which represent two intersections of ecliptic 8 with the tropic at the solstitial points (the first degrees of Cancer and of Capricorn) are on the smaller tropic (corresponding to the maximal declination), whereas B, the intersection of ecliptic 9 with the tropic, is on the larger tropic. 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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