Astronomical Images : Proportional parts of the Moon

Georg von Peuerbach

Astronomical Images

Georg Peuerbach (Georgius Aunpekh) was born in Peuerbach, near Linz. He studied at the University of Vienna, obtaining his BA in 1448 and MA in 1453. He held positions as court astrologer to the king of Hungary, and then to Emperor Frederick III. At the request of Cardinal Johannes Bessarion, Peuerbach began an abridgement of Ptolemy's Almagest, which was incomplete when he died in 1461. Peuerbach had also compiled Theoricae novae planetarum, a revision of the Theorica planetarum attributed to Gerard of Cremona. This originated as lectures given in Vienna in 1454, which were attended by Johannes Regiomontanus, who published the first edition in Nuremberg around 1474. This is the 1482 edition by Erhard Ratdolt, which contains copies of the original diagrams. As in the original edition, some woodcuts were coloured. Ratdolt was active in Venice and Augsburg, and was particularly interested in astronomical subjects. He also produced editions of Johannes Engel's Astrolabium planum in tabulis ascendens and G. Julius Hyginus' Poetica astronomica. Peuerbach's text was printed in a compilation that also included Johannes Sacrobosco's Sphaericum opusculum and Johannes Regiomontanus' Contra Cremonensia in planetarum theoricas delyramenta disputationes. This collection of astronomical treatises, and other similar ones, together comprised the main elementary texts available in the late fifteenth century. This woodcut, a slightly reduced copy of a diagram in the original edition of Peuerbach's treatise (c. 1474), is related to a section of the treatise that examines the consequences of the eccentricity of the Moon. It concerns a device for the calculation of the equation of the argument of the Moon (aequatio argumenti Lunae), measured on the zodiac (it is defined as the arc of the zodiac lying between the mean and true longitudes of the Moon), and dependent on the true argument of the Moon (argumentum Lunae verum), measured on the circumference of the epicycle, as it 'extends from the true apogee of the epicycle up to the centre of the body of the Moon'. 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 (measured on the epicycle) correspond unequal arcs of equations (measured on the zodiac): they are smaller near the apogee of the eccentric (and minimal at the apogee), larger near the perigee (and maximal at the perigee). The epicycle is called by Peuerbach 'the small circle' (circulus brevis), and the variation of the equations, for the same given argument, 'the variations of the diameter of the small circle' (diversitates diametri circuli brevis). In the diagram, the outermost circle is drawn around the centre of the World (c. mundi); its radius extends from this centre to the apogee of the deferent. The innermost circle is also drawn around the centre of the World, and its radius extends from this centre to the perigee of the deferent. The space between these two circles is divided into six orbs by five concentric equidistant circles. From the one next to the outermost circle to the innermost circle they are numbered from 10 to 60 along the vertical line that represents the axis of the deferent orbs of the apogee (linea augis). The circle that marks the exterior limit of the yellowed zone is the eccentric deferent of the epicycle (the centre of the lunar epicycle is attached to it, and rotates with it). Peuerbach explains that 'the line taken from the centre of the World to the apogee of the deferent is longer than the line extended from the same centre to the perigee. Moreover, the excess of the former over the latter, divided into sixty equal parts, is called proportional parts, and is twice the eccentricity' (Linea vero a centro mundi ad augem deferentis protracta, longior est linea ab eadem centro ad oppositum augis extensa. Excessus autem illius super istam divisus in 60 particulas aequales, minuta proportionalia dicitur, et duplus est ad excentricitatem). In other words, the portion of the linea augis divided into sixty proportional parts (proportionalia minuta) is twice the distance between the centre of the World and centre of the eccentric. The yellowed zone shows that at the apogee the eccentric deferent encompasses all the proportional minutes, that it encompasses none of them at the perigee, and that in other places it encompasses some of them: fewer near to the perigee and more, in proportion, near to the apogee. We see that when the centre of the epicycle is at the perigee, it is on line 60, and on line 0 at the apogee. Lines have been drawn from the centre of the World to the successive intersections of the eccentric deferent with the other lines: they mark the places where the centre of the epicycle is at 10, 20, 30, 40 and 50 proportional minutes. As for the use of this diagram, in Peuerbach's words, 'the equations of the arguments that are written in the tables are those that come about when the centre of the epicycle is in the apogee of the deferent' (Aequationes autem argumentorum, quae scriptae sunt in tabulis, sunt, quae contingunt, dum centrum epicycli in auge deferentis fuerit). 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 centrum Lunae (the angle formed, at the centre of the World, by the lines drawn respectively to the apogee and to the centre of the epicycle), the tables indicate the proportional parts; and if we know the true argument, they indicate the value of the diversitas diametri when the centre of the epicycle is at the perigee of the eccentric (that is when the proportional minutes are 60); then we calculate the diversitas diametri corresponding to the exact position of the centre of the epicycle, and this value is added to the equation of the argument taken from the tables. For a more detailed explanation, see Reinhold's 1553 edition, plate after fol. 35. Translated quotations of Peuerbach's Theoricae are from Aiton (1987).