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Astronomical Images : Variations in speed of the motion of the Moon's epicycle

Peter Apian

Astronomical Images

<p style='text-align: justify;'>This Venetian edition of Peuerbach's <i>Theoricae novae </i>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 woodcut has no model in Peuerbach's original (c. 1474) edition. It is the last of three diagrams conceived by Apian to explain the motion of the epicycle of the Moon. Although simply called <i>Theorica velocitatis et tarditatis motus epicycli</i> ('theory of the slowness and fastness of the movement of the epicycle'), this diagram is intended, in reality, to fulfil two functions. First, in Peuerbach's words, it shows that 'the revolution of the epicycle about its own centre is faster when the centre of the epicycle is traversing the upper half of the eccentric and slower when it is traversing the lower half' (<i>revolutio epicycli circa centrum suum centro epicycli per superiorem eccentrici medietatem discurrente sit velocior, per inferiorem vero tardior</i>). '<i>Medietas superior eccentrici</i>' and '<i>med[ietas] infe[rior] ecc[entrici]</i>' are inscribed on the diagram. This increased velocity in the part near the apogee of the eccentric is presented by Peuerbach as a consequence of the fact, shown in the preceding diagram, that the true apogee of the epicycle 'is always between the mean apogee and the point of the cavity under which the true apogee usually is, when the centre of the epicycle is in the apogee or perigee of the deferent' - this 'point of the cavity' (<i>punctum cavitatis</i>, called <i>punctum contactus</i> by Erasmus Reinhold) being on the line drawn from the centre of the eccentric deferent and passing through the centre of the epicycle. The problem is that Peuerbach's text is not quite explicit and that no line drawn from c, the centre of the eccentric deferent, is shown on this diagram (except the axis of the deferent orbs of the apogee of the Moon, the vertical line). Secondly, the diagram also fulfils the function of Peuerbach's original <i>Theorica linearum et motuum Lunae</i> (see Peuerbach (1482), sig. e6v). As in the preceding diagram (fol. 9v), the outermost circle, whose centre is the centre of the World (d), represents most probably the zodiac (rather than the most exterior orb of the Moon). The middle circle is the eccentric deferent of the epicycle of Moon (<i>deferens epicyclum Lunae</i>) and the centre of the epicycle of the Moon is attached to it. This epicycle is shown in eight successive positions. The small interior circle is the circle described by the centre of the eccentric deferent (c) as it moves around the centre of the World (d). The vertical line is the axis of the deferent orbs of the apogee of the Moon. The point that is diametrically opposite to the centre of the deferent is the 'opposite point' (<i>punctum oppositum</i>). In Apian's 1528 edition, this 'opposite point' is labelled k, as in the preceding diagram; but here the lettering is incomprehensible. For each position of the epicycle (except when its centre is on the axis of the deferent orbs of the apogee of the Moon) the diagram shows two lines intersecting at the centre of the epicycle: the one that is drawn from the 'opposite point' marks point M on the circumference of the epicycle, indicating the mean apogee of the epicycle (<i>aux media epicycli</i>); the one drawn from point d, the centre of the World, marks point A on the circumference of the epicycle, indicating the true apogee of the epicycle (<i>aux vera epicycli</i>). In the preceding diagram, this same point of the true apogee was labelled V. For one position of the epicycle (in the left superior quarter of the diagram), M and A have been inverted. This diagram also shows several points, lines and features that were not in the preceding one. The body of the Moon is represented by black dots and the radius from the centre of the epicycle to the centre of the lunar body is drawn. Arcs of circles are drawn above the circumference of the epicycle from this radius to the lines of mean and true apogees, in order to measure, as it appears, the distance between the body of the Moon and the true and mean apogees of the epicycle. As the epicycle always moves eastward, this distance is sometimes less than 180 degrees, sometimes more than 180 degrees, and sometimes equal to 180 degrees, but the arc of circle is always drawn in the same way: in order to intercept the least possible distance between the points. Other arcs of circles are drawn above the circumference of the outermost circle, in order to measure the motion, or longitude, of the Moon: they represent arcs of the zodiac from the beginning of Aries (B). They are drawn only for one position of the Moon: that in the left superior quarter of the diagram (where the aforementioned inversion of letters has occurred). The line of the mean motion, or longitude, of the Moon (<i>linea medii motus</i>), is 'extended from the centre of the World to the zodiac through the centre of the epicycle'. The line of the true motion, or true longitude, of the Moon (<i>linea veri motus</i>), is 'extended from the centre of the World through the centre of the body of the Moon up to the zodiac'. The true motion or longitude of the Moon (<i>verus motus Lunae</i>) is the arc of the zodiac from the beginning of Aries up to the line of the true motion of the Moon. According to the legend (on the next page, not on the photograph), it is labelled BE, whereas the line of the true motion is dG. Perhaps this is only an apparent contradiction. For two arcs of circle are drawn above the outermost circle (one for the true motion, the other for the mean motion), though one would suffice; at the intersection with the line of the true motion the letter above the superior arc of circle is probably g, and between the circle and the inferior arc of circle we discern a small letter that must be e (though it looks like 'c'). In any case, g and e refer to exactly the same point (the intersection of the zodiac, or ecliptic, and the line of the true motion of the Moon). The mean motion, or longitude, of the Moon (<i>medius motus Lunae</i>), is the arc of the zodiac from the beginning of Aries to the point where the line of the mean motion intersects the eighth sphere. It is labelled BF in the legend (Bf on the diagram). The equation of the centre (<i>aequatio centri</i>) is 'the arc of the epicycle intercepting its true apogee and mean apogee': arc AM on the diagram. This equation is null when the centre of the epicycle is on the axis of the deferent orbs of the apogee of the Moon (as shown before); 'it is greatest when this centre is a little below the mean distances of the deferent' (<i>maxima vero cum ipsum fuerit modicum infra longitudines medias deferentis</i>), that is slightly below the line that intersects at right angles at the centre of the World the axis of the deferent orbs of the apogee of the Moon. Two epicycles are represented in this position. Their centres are on the line that intersects at right angles at the 'opposite point' the axis of the deferent orbs of the apogee of the Moon. The mean argument of the Moon (<i>argumentum Lunae medium</i>) 'is the arc of the epicycle, reckoned in the direction of the motion of the centre of the body of the Moon from the mean apogee of the epicycle up to the same lunar centre'. According to the legend, it is MH. A letter that looks like b but must mean h is effectively marked near the body of the Moon in the upper left quarter of the diagram. As in the drawing of that particular epicycle, M and A have been inverted; the lettering does not help much here. The true argument of the Moon (<i>argumentum Lunae verum</i>) 'extends from the true apogee up to the centre of the body of the Moon', that is from A to h if the lettering were accurate. The difference between the true and mean arguments is the equation of the centre (<i>aequatio centri</i>), described above. The equation of the argument of the Moon (<i>aequatio argumenti Lunae</i>) is 'the arc of the zodiac lying between the lines of mean and true motion or longitude'; it is labelled fg. When the true argument is less than 180 degrees, the line of mean motion precedes the line of true motion, all reckoned eastward (<i>in Signorum successione</i>). The equation is then subtracted from the mean motion or longitude; but when it is more than six Signs, the reverse occurs. This process would be easily visible on the diagram if the arcs of circle above the epicycle, seemingly meant to measure the true argument (the distance from A, the true apogee, to the centre of the body of the Moon, reckoned eastward) were correctly drawn. As mentioned above, all distances must be measured eastward according to the succession of the Signs, the direction of the rotation of the eccentric and of the epicycle. But the arcs of circle measure the distance between the apogees of the epicycle and the body of the Moon in both directions: they always indicate the smallest possible distance between A or M and the centre of the lunar body, so that we get the impression that the lunar argument is always less than 180 degrees, which does not make sense. According to Peuerbach, the equation of the argument (<i>aequatio argumenti Lunae</i>) is null when the centre of the body of the Moon is 'in the true apogee or perigee of the epicycle, wherever the centre of the epicycle then may be'. On the contrary, this equation of the argument is 'greatest when the centre of the epicycle is in the perigee of the eccentric and when in addition the Moon is on the line drawn from the centre of the World at a tangent to the circumference of the epicycle'. But, as a rule, the equation of the argument continually increases as the centre of the epicycle approaches the centre of the World. In Apian's diagram no tangent to the circumference of the epicycle is drawn and, as mentioned above, the arcs of circle seemingly meant to measure the argument create confusion. However, in later editions, as in the Wittenberg 1535 edition of the <i>Theoricae novae</i>, this confusion is rectified. Two diagrams instead of one are drawn. The first, <i>Thaeorica [sic] velocitatis et tarditatis motus epicycli</i> (sig. C3v), much resembles our Apian diagram, except that all that concerns the measurement of the Moon's longitude on the zodiac has been suppressed. Besides, for every position of the epicycle, the line from the centre of the eccentric deferent to the centre of the epicycle is drawn: this line ends in point P, the 'point of the cavity' (<i>punctum cavitatis</i>). Thus, what the diagram clearly shows is the change in position of P with respect to F (the centre of the body of the Moon), and to M, the mean argument of the Moon. A second diagram is then drawn (<i>Theorica motuum et linearum Lunae</i>, sig. C5v): it shows the lines of the true and mean motions or longitudes of the Moon, the arcs of circle that measure these longitudes on the zodiac, and one position of the epicycle of the Moon (the one in the left, that is eastern, superior quarter) with the lines of the true and mean apogees. The lettering is legible and accurate and we see immediately the equation of the argument, the equation of the centre, and the true and mean arguments. We also see that the line of the mean motion of the Moon coincides with the line of the true apogee. Therefore, the relationship between the measurement of the longitude of the Moon on the zodiac and the movement of the epicycle becomes obvious. In Reinhold's edition (1542), we also have two diagrams: one <i>Theorica velocitatis et tarditatis motus epicycli</i> (fol. 33r), and one <i>Schema linearum motuum et aequationum Lunae</i> (fol. 35r). They are new improved versions of the 1535 diagrams. 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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