same as at the first observation: the interval of these two times is the sidereal year. In the present problem, as well as in the former, it is best to take an interval of several years by apply ing the rule to some observations which were made at the distance of nearly one hundred years, the length of the sidereal year is now set down at 365d 6h 9m 125. 57. From comparing the results of the last two articles, it appears that the sun returns to the equinox every year before it returns to the same point of the heavens; and consequently, the equinoctial points have a retrograde motion: this motion, though very small in a year or two, would in the compass of a few centuries amount to something considerable. It therefore did not escape the notice of the early astronomers, although their observations were very inaccurate in comparison with those of the moderns it is called the precession of the equinoxes. Hipparchus was, we believe, the first person who noticed this motion; and, by comparing his own observations with some that were made about 155 years previous to his, he judged the motion to be nearly 1° in 100 years. The observations of Albategnius in the year 878, compared with those of 1738, give 51" 9" for the annual precession. M. de la Caille's observations, compared with those given by Flamsteed, determine the precession in an 100 years, or the secular precession, to be 1° 23′ 45′′, and the annual precession 50. Thus is this motion determined to as great accuracy as may reasonably be expected; some of its effects will naturally fall under our consideration in the next chapter. CHAPTER IV. On determining the relative Situations of the fixed Stars; with Remarks on their Appearances, the Conftellations, &c. ART. 58. THOSE stars which when seen either by the naked eye, or by means of telescopes, keep constantly in the same situation with respect to one another, are called fired stars, as has been already mentioned (Art. 27.). If we contemplate any number of such stars, which to our view form a triangle, a trapezium, or any other figure, regular or irregular; since those stars have continued in the same relative situation ever since we have observed them, and have appeared to astronomers in the same situation with respect to each other as long back as the records of authentic history will carry us, namely, for some thousands of years; we may fairly conclude from analogy, that they have appeared very nearly as they now do ever since the creation of the earth, and will probably continue to appear so as long as the earth shall endure. Some few particular changes must be taken as exceptions from this general inference. 59. The fixed stars appear to us of different magnitudes this may arise either from their different sizes, or from their unequal distances from us; or in some instances both these.causes may combine to make the apparent difference. However, be this as it will, astronomers have from their apparent magnitudes distinguished the stars into six orders or classes: the first contains those of the largest apparent size, the second those which appear next in bigness, and D so on, till we come to stars of the sixth magnitude, among which last class are all those that can just be seen without telescopes: those which are only seen with telescopes are called telescopic stars. Now although this distribution of the stars is generally received, it is not to be imagined that all the stars of each class are exactly of the same apparent size; for there is almost an innumerable difference either in respect of apparent magnitude, colour, or brilliancy; and in some of these respects many of the stars appear to undergo changes, so that the same star may be reckoned by some astronomers in the first class, while others estimate it as belonging to the second or third. But the general divisions may be looked upon as usually adhered to, since the deviations from them are but few. 60. But, leaving what else is proper to remark respecting their magnitudes and changes to a subsequent article (Art. 683. &c.), we will now proceed to shew the methods of determining the places of any of them, with respect to the most important circles of the sphere: for the accurate determination of the places of the fixed stars will be of considerable utility in enabling us to ascertain the places of the planets, &c. by means of their apparent distances from any of those stars whose situations are known. 61. The declination of any star may be easily found by observing the following rule: Take the meridian altitude of the star, at any place where the latitude is known; the complement of this is the zenith distance, and is called north or south, as the star is north of south at the time of observation. Then, 1. When the latitude of the place and zenith distance of the star are of different kinds, namely, one north and the other south, their difference will be the declination; and it is of the same kind with the latitude, when that is the greatest of the two, otherwise it is of the contrary kind. the latitude and the zenith distance are of the same kind, i. e. both north, or both south, their sum is the declination; and it is of the same kind with the latitude. To prove the truth of this rule, turn to fig. 7, Pl. I. where Z is the zenith of the place, EQ the equinoctial, and EZ the latitude. 1. Let r represent the place of a star on the meridian, and Zr the zenith distance, the latitude being greater: then, Er (the declination) will be equal to EZ-Zr (the zenith distance): again, let c be the place of a star in the meridian, when the zenith distance exceeds the latitude; then E c (the declination) Zc (the zenith distance) — E Z (the latitude). And it is manifest, that in the former instance Z and r are on the same side of the equinoctial; and that in the latter case Z and c are on contrary sides. 2dly. Let y be the place of a star on the meridian, having its zenith distance Zy of the same kind with E Z the latí tude of the place: then Ey (the declination) = EZ +Zy; and the declination is of the same kind as the latitude, because Z and y are on the same side of the equinoctial. Q. E. D. For an example, suppose that in north latitude 52° 15′, the meridian altitude of a star is 51° 28′ on the south; then 38° 32′ the zenith distance, being taken from 52° 15′ the latitude, leaves 13° 43′ for the declination of the star north. 62. Having, by means like the above, found the declination of a star, it becomes requisite, in the next place, to know the right ascension (Art. 44.), as its situation with regard to the equator will then be known. Now the right ascension being estimated from the point where the equator and ecliptic intersect each other in the spring, a point which is marked out by nothing that comes under the cognizance of our senses; some phænomenon, therefore, must be chosen, whose right ascension is either given, or may be readily known, at any time that the right ascensions of other objects may be discovered by comparison with it. For this purpose nothing appears so proper as the sun; because its motion is the most simple, and its right ascension quickly found. 63. For if, in fig. 9, Pl. I. we have given QS the declination of the sun (which may be easily taken every day at noon by observation), and the angle SEQ the obliquity of the ecliptic-i. e. one leg of a right-angled spherical triangle, and its opposite angle, to find the adjacent leg EQ, the right ascension-it may be done by this proportion; as the tangent of the obliquity of the ecliptic: the tangent of the declination radius the sine of the right ascension reckoned from the nearer equinoctial point. : : For example, suppose on the 13th of February, the sun's south declination is found to be 13° 24', and the obliquity of the ecliptic is 23° 28'; we shall thus find the sun's right ascension: As tangt. 23° 28' 9.6376106 To tangt. 13° 24' 9°3770030 So is radius To sine 33° 16′ 58′′ 9*7393924 Here 33° 16′ 58′′ is the sun's distance from T; but as the declination is at that time decreasing, and the sun approaching Y, this must be taken from 360°, and the remainder 326° 43′ 2′′ is the right ascension. In a similar manner may the sun's right ascension be calculated for every day at noon, and arranged in tables for use for any intermediate time between one day at noon and the following, the right ascension may be determined by proportion. The longitude ES of the sun, when required, may be readily found by the rules to ascertain the hypothenuse of the same triangle. 64. The apparent diurnal motion of the heavenly bodies being uniform, and performed in circles parallel to the equator, the interval of the times in which two stars pass over any meridian must bear the |