of a particular distance becomes also an instantaneous phenomenon; and as such may be employed to determine the times, at the instant, at any two places where it may have been observed: or if a series of such distances have been calculated for given times, and for any one place, as Greenwich, and a distance have been observed at another place; an identity of distance will serve to determine the time of the observation according to the reckoning of time at the former place. PROB. X. 360. To find the longitude of a station or ship by means of the observed angular distance between the moon and the sun or a star. The latitude is supposed to be well known; and, in order that the polar distance of the sun may be found for the purpose of ascertaining the correct time at the place or ship, the longitude, as well as the hour of the day, must be known approximatively. In general, in making a lunar observation, one person observes the angular distance between the moon and sun, or the moon and a star, and at the same time two other observers take the altitudes of the celestial bodies. If the moon and sun be observed, the distance between either the eastern or western limb of the latter, and the enlightened limb of the former is taken; also the altitude of the upper or lower limb of the sun, and that of the enlightened limb of the moon. Now, after applying the corrections for the index m M p errors of the instruments, and the dip of the horizon if at sea, or taking the halves of the altitudes if an artificial horizon have been used; let the angular semidiameter of the sun, and the augmented semi-diameter of the moon be applied, by addition or subtraction, so as to reduce the observed altitudes and distance to the altitudes and distance of the centres. Then the two zenith distances and the direct distance of the centres will form in the heavens a triangle, as z SM, of which all the three sides will be affected by parallax and refraction. With the three sides thus found compute the angle SZM by the rules of spherical trigonometry: then, imagining ss to be the star's refraction, or the difference, in a vertical direction, between the sun's parallax and refraction, and Mm to be the difference, also in a vertical direction, between the moon's parallax and refraction (the former being added to zs because the sun's refraction is greater than his parallax, and the latter being subtracted from Zм for a contrary reason), the points s and m become the corrected places of the sun, or star, and the moon, or the places which the luminaries would appear to occupy in the heavens if seen from the centre of the earth and were not affected by atmospherical refraction. Therefore, in the triangle z sm, having two sides and the included angle, the remaining side sm, which is the true distance of the luminaries from each other, may be computed. This is the direct, and the most accurate method of finding the true distance between the centres of the sun and moon, or between a star and the centre of the moon; but the computation is laborious, and it requires great attention to the logarithms in determining the angle at z. It would be easier, and sufficiently correct, to proceed in the following manner. By means of the three sides ZM, Zs and MS, compute (art. 66.) the two angles ZMS and ZSM; and, ss being equal to the difference first mentioned above, let fall st perpendicularly on MS produced if necessary: then in the triangle sst considered as plane we should have st=ss cos. ZSM or Ss cos. sst. If the angle ZSM were obtuse, st would fall on the other side of s. Next, Mm being equal to the difference between the moon's refraction and her parallax in altitude, imagine about s or t as a pole a portion mp of a small circle to be described, so that sm may be considered as equal to tp; then the value of Mp may be found by a formula similar to that which was used in determining the latitude of a place by an altitude of the pole star (art. 352.): thus, Mm and Mp being expressed in seconds of a degree, and using only two terms (see art. 410.), Mp = Mm cos. ZMS-1 (Mm)2 sin.2 ZMS cotan. Ms sin. 1". The value of Mp must, if the angle at м be acute, be subtracted from Mt, that is from MS+st in order to have the value of pt or its presumed equivalent ms, which is the correct distance between the luminaries. If the angle at M be obtuse, the value of Mp must be added to Mt. The process of finding the angles at s and м would be facilitated by the use of spherical traverse tables corresponding to those which are given in Raper's "Navigation:" the formation of such tables has been indicated in art. 337. Having thus obtained the correct distance sm, there must be found in the Nautical Almanac the time at Greenwich when the distance between the sun, or star, and the moon is equal to that which has been found; then that time being compared with the mean time, at the station or ship, deduced from the sun's horary angle SPZ at the instant of the observation; the difference between them will be the required difference of longitude in time. And the place or ship will be westward or eastward of Greenwich according as the mean time is earlier or later in the day than that at Greenwich. Since the computed distance sm will seldom be exactly found in the Nautical Almanac, a proportion must be made by means of the difference between the distances for three hours; and in this operation the proportional or logistic logarithms ("Requisite Tables," tab. XV.), as they are called, are generally used. Ex. At Sandhurst, April 13. 1840, at 10 ho. 4' 48" by the watch, the following observations were made: Double altitude of ß, Geminorum, by reflexion Distance between the star and the moon's nearest limb = 81 1 20 = 81 14 50 = 60 42 10 mean time of =log. augmentation of moon's Double alt. star 1.20412 . 1.01488 = 10".3 81 proximative Augmentation 40 19 26.2 App. alt. moon's centre 49 29 52 Apparent zenith distance of the moon's centre (=z м) 49 40 44.1 Observed distance of the star from the moon's nearest limb=60 42 10 Computation of the angle мzs in the spherical triangle мzs (P repre Corrections for parallax and refraction in the altitudes or zenith dis tances. log. cos. moon's app. alt. (40° 19′ 16) 9.882200 =log. moon's parallax in altitude 3.525563 3.407763=2557".2 Refraction = 68 .5 Difference between the moon's parallax and re- True zenith distance of the moon (= zm) Apparent zenith distance of the star = = 49 29 52 True zenith distance of the star (zs) = 49 31 In the spherical triangle mzs. - log. cos. mzs (82° 57′ 26′′) 9.0885272 -log. cotan. zs (49 31) 0 9.8790057 9.8123965 9.9312431 By the Nautical Almanac, the true distances of the star from the moon's 61 13 7 True dist. by the observation=60 14 17 = 1 34 51 Proportional log. 1° 34′ 51′′ = 2782 Difference 36 1 36 1 = 6988 361. The proportional or logistic logarithms here employed are merely the common logarithms of the fractions arising from the division of the number of seconds in 3 hours, or 3 degrees, by the four terms of a proportion none of which exceeds that number of seconds; and they are convenient on many occasions similar to that which occurs in the above proposition. In order to explain their nature, let R represent any number, and let A: B :: C: D be any proportion; then R R R R we shall have Now if R = 10800 (the number of seconds in three hours) and none of the terms A, B, C, D R R exceed that number, the common logarithms of &c. will A'B' be those which are called the proportional logarithms of A, B, &c., and if three such logarithms be taken from a table, the proportional logarithm of the fourth term may be found exactly as when common logarithms are employed in working a proportion. If one of the terms, as A or B, be equal to 10800, its proportional logarithm will be zero; and the operation will then consist in merely adding together, or in subtracting one from another, the two proportional logarithms which have been taken from the tables; the sum or difference will be the proportional logarithm of the required term. Similar logarithms are formed by making R equal to 3600 (the number of seconds in one hour, or one degree), and they are used in a similar manner. 362. Besides a method which was proposed by Borda, Delambre has given in his " Astronomie" three formulæ for finding the correct lunar distance by logarithms, two formulæ requiring only the addition of natural cosines, two requiring natural versed sines, and two requiring both natural and logarithmic sines: he has also given a considerable number of formulæ by which the difference between the true and apparent distances may be computed directly.* The method of determining longitudes by lunar distances appears to have been first used by La Caille in his voyage to the Cape of Good Hope; and that astronomer gave certain formula by which the calculations might be abridged: the method did not, however, become general till the year 1767, when Dr. Maskeline first published the British Nautical Almanac, in which the distances of the moon from the sun and from certain stars are inserted, by computation, for every three hours according to the time at Greenwich. Previously to the publication of that work the longitude had been generally determined by means of the moon's horary angle. This method consisted in observing the altitude of the moon's enlightened limb and reducing it as usual to the altitude of the centre; then, knowing the latitude of the station and finding the moon's declination in the ephemeris for the time of observation (an approximate estimate of the longitude of *The practical seaman, in determining the longitude of his ship by lunar observations, will find the labour considerably diminished by the use of Mrs. Taylor's "Lunar Tables," third edition. |