certainty for Greenwich time *; but a telescope magnifying at least forty times is required for their observation; and those of different powers are found to give such different results as to the moment of immersion or emersion, that the method is not susceptible of the accuracy it would appear to promise, and is moreover almost impracticable at sea. In determining the longitude by this method, the local time must be found by observations of one or more fixed stars, unless it is known from a chronometer whose error and rate has been previously ascertained. The eclipses of the sun and moon also enable us to determine the longitude; the former with considerable accuracy; but their rare occurrence renders them of little or no practical benefit, and the results obtained by the eclipses of the moon are generally unsatisfactory, owing to the indistinct outline of the shadow of the earth's border. The three methods upon which the most dependence can be placed, are-Ist, by a “lunar observation,” which, as before stated, possesses the great advantage of being easily taken at sea; 2ndly, by the meridional transits of the moon, compared with those of certain stars previously agreed on, which are given in the Nautical Almanac under the head of “ Moon Culminating Stars ;” and 3rdly, by occultations of the fixed stars by the moon.—The two latter methods are the most accurate of any, but the first of them requires the use of a transit instrument, and the latter a good telescope; both involve also long and intricate calculations, which will be found fully detailed in the works of Dr. Pearson, and in chapter 37 of Woodhouse's Astronomy. The methods given in the following pages considerably shorten the labour of the more accurate computations, and are the same as those in Mr. Riddle's “ Navigation.” 1 * The time occupied by light in travelling from the sun to the earth is also ascertained by means of the eclipses of Jupiter's satellites. The difference of distance the light has to travel from Jupiter to the earth, on the occasion of an eclipse of one of the satellites, happening when they are in opposition or in conjunction, is evidently the major axis of the earth's orbit. This difference has been ascertained to be 16m 268.4, which gives 8m 138.2 for the time occupied by light in passing from the sun to the earth. The distance of the sun from the earth was determined by means of the transit of Venus over the sun's disc. Method lst.—By a Lunar Observation. The observations for this method of ascertaining the longitude of any place can be taken by one individual; but as there are three elements required as data, which, if not obtained simultaneously, must be reduced to what they would have been if taken at the same moment of time, it is better, if possible, to have that number of observers. The lunar distance, which is of the first importance, is measured by bringing the enlightened edge of the moon and the star, or the edge of the moon and either limb of the sun, in perfect contact. The other observations required are, the altitudes of the moon, and that of the other object, whether it be the sun, a fixed star, or a planet *; and as these are only taken for the purpose of correcting the angular distance, by clearing it from the effects of parallax and refraction, they do not require the same accuracy, or an equal degree of dexterity in observing. When the observations are made consecutively by one person, the two altitudes are first taken (noticing of course the times); then the lunar distance repeated any number of times, from whence a mean of the times and distance is deduced; and afterwards the altitudes again in reverse order, which altitudes are to be reduced to the same time as that of the mean of the lunar distances. It being of great moment to simplify and render easy the solution of this problem, which is of the most vital importance at sea, a number of celebrated practical astronomers have turned their attention to the subject, and tables for “clearing the lunar distance” are to be found in all works on Nautical Astronomy, by the use of which the operation is undoubtedly very much shortenedt; but as none of these methods show the steps by which this object is attained, the example given below is worked out by spherical trigonometry, and the process will be rendered perfectly easy and intelligible by the following description : * These altitudes, if not observed, can be calculated when the latitude is known; by which method more accurate results are obtained. + Dr. Pearson enumerates no less than twenty-four astronomers who have published different methods of facilitating the “Clearing the Lunar Distance.” M E H In the accompanying figure Z represents the zenith, P the pole, M the observed place of the moon, and S that of the sun or star. The data given are M S, the measured angular distance; and ZM and ZS the two zenith distances (or co-altitudes) from whence the angle MZS is found, the value of which is evidently not affected by refraction or parallax, which, acting in vertical lines, cause the true place of the moon to be elevated above its apparent place (the parallax, from her vicinity to the earth, being a greater quantity than the correction for refraction), and that of the sun or star, to be depressed below its apparent place. Let M' and S represent the corrected places of these bodies, and we have then Z M' and ZS--the zenith distances corrected for refraction and parallax—and the angle Z' before found, to find the true lunar distance M'S in the triangle Z M'S'. The apparent time represented by the angle ZPS may be found in the triangle ZP S, having SS, PS, and ZP the co-latitude, if the exact error of the chronometer at the moment is not already known; and ! this time, compared with the Greenwich time at which the lunar distance is found from the Nautical Almanac to be the same, gives the difference of longitude east or west of the meridian of that place. The example below will show all the steps of the operation. On May 4, 1838, at 10b 41m 458.8 by chronometer, the following observations were taken in latitude 51° 23:40 north, to find the longitude; the chronometer having been previously ascertained the same evening to be 3m 34s too fast. Double altitude--) 74° 42' 35", taken with a sextant; index error-22". Altitude Spica Virginis 28° 15' 50"-with alt. and az. inst. ; index error-28“. Distance D-* 31° 25' 55"-with repeating circle. 1st-Then in the triangle Z MS we have the three sides to find the angle MZS. (a) M S = 31 11 1.2 (6) ZS = 61 44 38 ar. comp. sin 0.0551028 (c) ZM = 52 53 47.3 ar. comp. sin 0.0982439 Then to correct the zenith distances for refraction and parallax : = Then in the triangle Z M'S', we have zs 61 46 23 to find M'S the corrected lunar distance. and angle Z= 35 46 50 Formula, tan á cos 2 x tan Z M' a'' = ZS a' cos a' cos M'S = cosine Z M' x cos a |