Astronomical Images : Motions of Mercury, and the proportional parts of Mercury

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 1490 Venetian edition by Octavianus Scotus, which contains some woodcuts that were coloured. The top diagram copies a figure of the first edition. It shows the motions of the centre of the eccentric of Mercury, of the centre of its epicycle, and of the apogee and perigee of its eccentric. In order to fit here, it has been rotated (the top is now towards the right and the bottom towards the left), thus becoming scarcely legible. For a complete explanation of this diagram, see sig. f6r in our 1482 edition of Peuerbach. The bottom diagram is also a slightly reduced copy of a figure of the first edition. It shows the 'proportional parts' 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 been previously shown that the centre of the epicycle of Mercury is at its 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 coloured zones enclosed inside the sort of oval figure that represents the path of the centre of the epicycle of Mercury on its mobile eccentric deferent. The draughtsman has not delineated an oval, but simply joined two arcs of a very large circle (as in the first edition). The image has not been coloured by hand, as in the first edition, but probably stencilled, and the coloured parts do not correspond exactly to the drawn lines. In any case, these coloured parts are located in three zones: the upper zone of the 'remoter proportional parts', the middle zone of the first kind of 'closer proportional parts' (right and left), and the bottom zone of the other kind of 'closer parts'. The diagram is extremely simplified. Indeed, it needs the preceding diagram (here printed above) to be understood, though the comparison is not easy: not only has the above diagram been turned, but there are no common labels, no legends, and the first diagram does not try to represent the oval path of the centre of the epicycle. The points marking the limit between the upper coloured zone and the middle ones are the points where the centre of the epicycle is at its mean distance from the centre of the World. A circle has been drawn, centred on the centre of the World (c. mundi), whose radius is equal to this mean distance. The distance between the centre of the World and the upper point of the path of the centre of the epicycle is the radius of another circle, of which only a portion has been drawn. This portion and the corresponding portion of the first circle (or circle of the mean distance) delimit the area of the 'remoter proportional parts', divided into four strips (instead of six, as it ought to be). In any case, the reader understands that it is divided into 60 parts. A third circle is partially drawn. Its radius extends from the centre of the World to the lowest points, left and right of the middle coloured zone, which represent the points where the centre of the epicycle is closest to the centre of the World. This arc and the corresponding portion of the first circle (or circle of the mean distance) delimit the area of the 'closer proportional parts', also divided into four strips (instead of the required six). We see that when the centre of the epicycle is at the apogee of its path, this apogee encompasses all the remoter proportional parts (that is 60), while before and after this point, the encompassed proportional parts progressively decrease until they become null (at the points of mean distance). Then, other sorts of 'parts' come into play: the 'closer proportional parts', which are null at the points of mean distance and amount to 60 at the points where the centre of the epicycle is closest to the centre of the World. At the bottom of the path, where the centre of the epicycle is more distant than at the closest distance (but less than at the mean distance), the third kind of 'closer proportional parts' appear. If the circles that measure the proportional parts were numbered, the upper portion of circle (passing through the apogee) would be 60, the circle of the mean distance 0, and the circle of the closest distance, again, 60. What can be puzzling for us in this sort of diagram is that the 'remoter parts' and the 'closer parts' follow different rules. The significant remoter parts are inside the path of the planet, whereas the significant closer parts are outside it. For an improved diagram of the proportional minutes of Mercury, and more detailed comments, see Reinhold (1553), fol. 78r. For more information on the calculation of the equation of the argument, see Reinhold (1553), fols. 56r and 57v. Translated quotations of Peuerbach's Theoricae are from Aiton (1987).