thing of this kind in the rather obscure explanations of his lunar hypothesis, which, never having been accepted, it would be useless to give an account of.* In Ptolemy's theory, Art. (115), the evection was the result of an apparent increase of the first lunar epicycle caused by its approaching the earth at quadratures; but, in this second method, it is the result of an actual change in the elements of the elliptic orbit. D’Arzachel, an Arabian astronomer, who observed in Spain about the year 1080, seems to have discovered the unequal motion of the apsides, but his discovery must have been lost sight of, for Horrocks, about 1640, re-discovered it 'in consequence of his attentive observations of the lunar 'diameter: he found that when the distance of the sun from 'the moon's apogee was about 45° or 225°, the apogee was 'more advanced by 25° than when that distance was about '135 or 315°. The apsides, therefore, of the moon's orbit 'were sometimes progressive and sometimes regressive, and required an equation of 12° 30', sometimes additive to their 'mean place and sometimes subtractive from it.'t เ Horrocks also made the eccentricity variable between the limits 06686 and 04362. The combination of these two suppositions was a means of avoiding the introduction of Ptolemy's eccentric or the second epicycle of Copernicus: their joint effect constitutes the evection. 6 * Après avoir établi les mouvemens et les époques de la lune, Boulliaud revient à l'explication de l'évection ou de la seconde inégalité. Si sa théorie n'a pas fait fortune, le nom du moins est resté. En même temps que la 'lune avance sur son cône autour de la terre, tout le système de la lune est 'déplacé; la terre emportant la lune, rejette loin d'elle l'apogée, et rap'proche d'autant le périgée; mais cette évection à des bornes fixées.' Delambre, Hist. de l'Ast. Mod., tom. 11. p. 157. + Small's Astronomical Discoveries of Kepler, p. 307. Variation. 121. After the discovery of the evection by Ptolemy, a period of fourteen centuries elapsed before any further addition was made to our knowledge of the moon's motions. Hipparchus's hypothesis was found sufficient for eclipses, and when corrected by Ptolemy's discovery, the agreement between the calculated and observed places was found to extend also to quadratures; any slight discrepancy being attributed to errors of observation or to the imperfection of instruments. But when Tycho Brahé (A.D. 1580) with superior instruments extended the range of his observations to all intermediate points, he found that another inequality manifested itself. Having computed the places of the moon for different parts of her orbit and compared them with observation, he perceived that she was always in advance of her computed place from syzygy to quadrature, and behind it from quadrature to syzygy; the maximum of this variation taking place in the octants, that is, in the points equally distant from syzygy and quadrature. The moon's velocity therefore, so far as this inequality was concerned, was greatest at new and full moon, and least at the first and third quarter.* * * 'It appears that Mohammed-Aboul-Wefa-al-Bouzdjani, an Arabian 'astronomer of the tenth century, who resided at Cairo, and observed at 'Bagdad in 975, discovered a third inequality of the moon, in addition to 'the two expounded by Ptolemy, the equation of the centre and the evec'tion. This third inequality, the variation, is usually supposed to have 'been discovered by Tycho Brahé, six centuries later...... In an almagest ' of Aboul-Wefa, a part of which exists in the Royal Library at Paris, 'after describing the two inequalities of the moon, he has a Section IX., "Of the third anomaly of the moon called Muhazal or Prosneusis" 'But this discovery of Aboul-Wefa appears to have excited no notice among his contemporaries and followers; at least it had been long quite 'forgotten, when Tycho Brahé, re-discovered the same lunar inequality.' Whewell's Hist. of Inductive Sciences, vol. 1. p. 243. ...... Tycho fixed the maximum of this inequality at 40′ 30′′. The value which results from modern observations is 39' 30". 122. We have already two epicycles, or one epicycle and an eccentric, to explain the first two inequalities: by the introduction of another epicycle or eccentric, the variation also might have been brought into the system; but Tycho adopted a different method:* like Ptolemy, he employed an eccentric for the evection, but for the first or elliptic inequality he employed a couple of epicycles, and this complicated combination, which it is needless further to describe, represented the change of distance better than Ptolemy's. To introduce the variation, he imagined the centre of the larger epicycle to librate backwards and forwards on the eccentric, to an extent of 40′ on each side of its mean position; this mean place itself advancing uniformly along the eccentric with the moon's mean motion in anomaly; and the libration was so adjusted, that the moon was in her mean place at syzygy and quadrature, and at her furthest distance from it in the octants, the period of a complete libration being half a synodical revolution. Annual Equation. 123. Tycho Brahé was also the discoverer of the fourth inequality, called the annual equation. This was connected * For a full description of Tycho's hypothesis, see Delambre, Hist. de l'Ast. Mod., tom. I. p. 162, and An Account of the Astronomical Discoveries of Kepler, by Robert Small, p. 135. I with the anomalistic motion of the sun, and did not, like the previous inequalities, depend on the position of the moon in her orbit. Having calculated the position of the moon corresponding to any given time, he found that the observed place was behind her computed one while the sun moved from perigee to apogee, and before it in the other half year. Tycho did not state this distinctly, but he made a correction which, though wrong in quantity and applied in an indirect manner, shewed that he had seen the necessity and understood the law of this inequality. He did not try to represent it by any new eccentric or epicycle, but he increased by (8m. 13s.) sin(sun's anomaly) the time which had served to calculate the moon's place ;* thus assuming that the true place, after that interval, would agree with the calculated one. Now, as the moon moves through 4' 30" in 8m. 13s., it is clear that adding (8m. 13s.) sin (sun's anomaly) to the time is the same thing as subtracting (4′ 30′′) sin(sun's anomaly) from the calculated longitude, which was therefore the correction virtually introduced by Tycho.† Modern observations shew the coefficient to be 11' 9". We have seen, Art. (75), how this inequality may be inferred from our equations. * That is, the equation of time which he used for the moon differed by that quantity from that used for the sun. + Horrocks (1639) made the correction in the same manner as Tycho, but so increased it that the corresponding coefficient was 11′ 51′′ instead of 4′ 30′′. Flamsteed was the first to apply the correction to the longitude instead of the time. Reduction. 124. The next inequality in longitude which we have to consider, is not an inequality in the same sense as the foregoing; that is, it does not arise from any irregularity in the motion of the moon herself in her orbit, but simply because that orbit is not in the same plane as that in which the longitudes are reckoned, so that even a regular motion in the one would be necessarily irregular when referred to the other. Thus if NMn be the moon's orbit and Nm the ecliptic, and if M the moon be referred to the ecliptic by the great circle Mm perpendicular to it, then MN and mN are 0°, 90°, 180°, 270°, and 360° simultaneously, but they M differ for all intermediate values: the difference between them is called the reduction. The difference between the longitude of the node and that of the moon in her orbit being known, that is the side NM of the right-angled spherical triangle NMm, and also the angle N the inclination of the two orbits, the side Nm may be calculated by the rules of spherical trigonometry, and the difference between it and NM, applied with a proper sign to the longitude in the orbit, gives the longitude in the ecliptic. Tycho was the first to make a table of the reduction instead of calculating the spherical triangle. His formula was reduction = tan" I sin 2L - tan*I sin4L, where I is the inclination of the orbit and I the longitude of the moon diminished by that of the node. The first term corresponds with the term - sin 2(gpt-y) of the expression for 0. |