Astronomical Images : The mean motion and the equation of the eighth sphere

Peter Apian

Astronomical Images

This Venetian edition of Peuerbach's Theoricae novae was copied from Apian's 1528 edition, printed in Ingolstadt. Subsequently, the work went through several further editions. Apian's edition added new woodcuts as well as notations to some of those from earlier editions. Some errors in the woodcuts in the 1528 edition were repeated in this Venetian edition of 1537. This diagram is a new version of the fourth diagram of the section on the eighth sphere of the original (c. 1474) edition of Peuerbach's Theoricae novae. Apian has been inspired by a figure in the Fine edition (Paris, 1525) and has adopted part of its labelling, but has also added new features. However, his own diagram has almost no legend, whereas the Fine diagram has a complete one. Compared to the Fine diagram, the Apian diagram is reversed from left to right. After describing the proper movement of the eighth sphere, Peuerbach explains how it is measured: 'the mean motion of approach and recession of the eighth sphere is the arc of the small circle reckoned from the uppermost point of the quadrant eastward up to the beginning of Aries of the eighth sphere' (medius motus accessus et recessus octavae sphaerae est arcus circuli parvi a puncto supremo quartae secundum successionem Signorum usque ad caput Arietis octavae sphaerae computatus). As indicated below the diagram, 'AB is a portion of an arc of [the ecliptic] of the Prime Mover' (arcus AB est portio arcus [eclipticae] primi mobi[lis]). We know that the ecliptics of the ninth and tenth spheres are in the same plane and can be represented by the same circle. Both ecliptics can be called the 'fixed ecliptic'. Thanks to the symbols, we understand that A is the beginning of Aries of the tenth sphere (or Prime Mover) and B the beginning of Aries of the ninth and the centre of the small circle on which the beginning of Aries of the eighth sphere rotates. Thus AB represents the proper motion of the ninth sphere, called the 'motion of the apogees and of the fixed stars' (motus augium et stellarum fixarum) in the astronomical tables. 'C is the pole of the ecliptic of the ninth sphere' (C est polus eclip[ticae] non[ae] sphae[rae]). Three arcs of circle are drawn from C. The middle one passes through B, the first degree of Aries of the ninth sphere, and through D and E on the small circle. The other arcs from C pass through other points of the fixed ecliptic and of the small circle. The numbers inscribed on the diagram (0, [1], 2, 3) suggest that the mean motion of the eighth sphere is measured from D. Thus, D would mark the 'uppermost point of the quadrant' (punctum supremum quartae) from which the mean motion of approach and recession of the eighth sphere is measured on the small circle. The problem is that, in principle, the punctum supremum is the northern point. The order of the Signs, from west to east, is indicated by the movement of the first degree of Aries of the ninth sphere (from A to B). As the diagram has been inverted from left to right (and not upward), west is right and east left. In the Fine edition, as in all the editions that copied the diagram (notably the Wittenberg 1535 edition and the commentary of Reinhold, which correct the inversion), west is clearly left, east is right, the pole of the ecliptic, like point D, is north, and E is south. Although the Apian diagram is not so clear, we must probably consider that it illustrates the case when the first degree of Aries is at F; then the arc of the mean motion of the eighth sphere is DF. Arc DF corresponds to a graduated portion of the fixed ecliptic, also limited by the great circles passing through the poles of the fixed ecliptic and, on the one hand, B, the first degree of Aries of ecliptic 9, on the other hand, the first degree of Aries of ecliptic 8. This portion is the equation of the eighth sphere (aequatio octavae sphaerae), defined by Peuerbach as 'the arc of the ecliptic of the ninth sphere intercepted between the centre of the small circle and the great circle from the poles of the ecliptic of the ninth passing through the beginning of Aries of the eighth' (aequatio autem octavae sphaerae est arcus eclipticae nonae sphaerae centrum parvi circuli et circulum magnum a polis eclipticae nonae per caput Arietis octavae transeuntem interiacens). Some rules are added by Peuerbach: the equation is null when the mean motion of approach and recession is 0 degrees or 180 degrees; it is maximal when the mean motion is 90 degrees or 270 degrees. When the mean motion is less than 180 degrees, the equation is added to it; when it is more than 180 degrees, it is subtracted. For an improved version of the same diagram, see Reinhold (1553), fol. 170v. Translated quotations of Peuerbach's Theoricae are from Aiton (1987).