LUNAR DISTANCES.] 



NAVIGATION-NAUTICAL ASTRONOMY. 



1099 



absolute time : the mean-time clock at Greenwich might 

 mark this instant in the local reckoning of that place 

 as 3h. 20m. : an observer to the east of Greenwich might 

 find the time at his place to be 3h. 22m. : the difference 

 in the local reckoning would thus give a difference of 2m. 

 in the time, although both observers saw the flash at the 

 same absolute instant ; and we should conclude that the 

 easterly place of observation had half a degree, or 30 

 nautical miles of east longitude. It is the same in re- 

 ference to any other phenomenon : let only the time at 

 Greenwich be noted which marks the instant of its occur- 

 rence, and also the time at any other locality which marks 

 the same instant : the difference of the times thus noted 

 will be the longitude, in time, of that other locality from 

 Greenwich.* 



The heavenly bodies furnish signals in abundance 

 analogous to the rocket-signal here imagined, and astro- 

 nomers have only to make their selections suitably to the 

 circumstances of the observers : what are called tlie 

 lunar okservationt are the best adapted to determine the 

 longitude at tea. Certain stars, lying in or very near to 

 the moon's path, are chosen, and the distance of the 

 moon from each of these, as also from the sun at noon 

 (Greenwich time), and at every interval of three hours, is 

 predicted and recorded in the Kautical Almanac for every 

 day in the year. An observer at sea measures one of 

 these distances, and refers to the Nautical Almanac for 

 the Greenwich time when the same phenomenon hap- 

 pened ; he finds, most likely, that A is distance is inter- 

 mediate between a certain pair of the three-hourly dis- 

 tances ; he knows, therefore, that the Greenwich time is 

 intermediate between the recorded hours ; and, just as in 

 tlio case of any other varying quantity given in the 

 Almanac for regular intervals of time he calculates the 

 intermediate time corresponding to the intermediate dis- 

 tance, by proportion. The motion of the moon is found 

 to be sufficiently uniform, for a period of three hours, to 

 justify an intermediate position being inferred in this way, 

 and it is sufficiently rapid to render her change of place 

 sensible even in two or three seconds of time ; a trifling 

 correction for variable motion will, however, be noticed 

 hereafter. 



The lunar observations thus enable the mariner to 

 discover the time at Greenwich independently of the 

 chronometer, and thence to determine the error of his 

 time-keeper ; and, therefore, by taking the difference 

 between this error and that last recorded, to apply the 

 proper correction to the mean daily rate. The lunar 

 observations alone will not enable the navigator to find 

 his longitude ; but they enable him to find what o'clock 

 it is at Greenwich at the instant the lunar distance was 

 taken, and thus to correct his chronometer. 



Knowing, in this way, the time at Greenwich, we 

 can very accurately compute the declination of the sun 

 at the time of observation, or its right ascension, if the 

 distance be that between the moon and a star, and 

 thence, by means of the latitude of the place, supposed 

 to be already known, we can find, as explained at page 

 1094, et Kq., the time at that place, and thence the 

 longitude. 



It must be remarked that the obientd distance be- 

 tween two celestial objects is not the true distance, any 

 more than their observed altitudes are the true alti- 

 tudes ; these latter the true altitudes it is necessary 

 to find, in order, from the observed, to compute the true 

 distance ; for it is the true distance, as measured from 

 the centre of the earth, that is predicted in the Nautical 

 Almanac : in other words, it is the observed distance 

 cleared from the effects of parallax and refraction. Such 

 being the case, the reader will perceive that the deduc- 

 tion of the Greenwich time, from an observed distance, 

 is not wholly independent of all local information to be 

 afforded by the ship's account : the estimated time at 

 the ship, and the estimated longitude, are evidently 

 useful for the purpose of enabling us to get the semi- 

 diameter and horizontal parallax of the moon with greater 



The electric telegraph has been largely used on oimilar principles 

 throughout Europe during thr> ]a*t few years, for accurately determining 

 the longitude* of Tarious stations. ED. 



precision ; for these quantities, especially the latter, 

 sensibly vary from noon to midnight, and they are 

 elements which necessarily enter the calculations for 

 reducing the moon's observed altitude to the true alti- 

 tude. 



We shall now investigate formulae for computing the 

 true distance of the moon's centre from that of the 

 sun, or from a fixed star, by means of the apparent 

 and true altitudes of the two objects, and their apparent 

 distance that is, the observed distance corrected for 

 semi-diameter. 



INVESTIGATION: OP FORMULA FOB CLEARING THE 

 LUNAB DISTANCE. In the annexed diagram (Fig. 31), 

 let Z represent the zenith of the place of observation, and 



Fig. 31. 



ZM, ZS the two 

 verticals on which 

 the objects are ob- 

 served. Letm, sbe 

 the observed places 

 of the moon and 

 sun, or of the moon 

 and a fixed star, 

 and let M, S be their 

 true places. As the 

 moon is depressed 

 by parallax more than it is elevated by refraction, M 

 will be above m ; but the sun, on the contrary, being 

 more elevated by refraction than depressed by parallax, 

 S will be below i. 



The corrections for dip, index error, and semi-diameter 

 being applied, observation gives the apparent zenith 

 distances, Zm, Zs, and the apparent distance m s of the 

 objects themselves ; and the proper corrections for paral- 

 lax and refraction being applied to the apparent alti- 

 tudes, we get the true zenith distances Z M, Z S, and 

 the object of the present investigation is to determine 

 the true distance M S by computation. 



Let d stand for the apparent distance 



D . . . true distance 

 /', "' . . , apparent al titudes 

 A, A' . . . true altitudes 



By Spherical Trigonometry (page 658), the triangle 

 M X S gives for cos. Z, the following expression, 

 namely : 



cos. D sin. A sin. A' 



cc. Z = - - ;, 



cos. A cos. A 



and the triangle mZs gives in like manner 



cos. d sin. a sin. a' 

 cos. Z = - 



cos. a cos. a 



Henoe, for the determination of D we have tho equation 



cos. D sin. A sin. A _ cos, d sin, a sin, a,' 



cos. A cos. A' cos. a cos. a' 



from which we have for cos. D the expression 



^ cos. d sin. a sin. o' 



oo.D= , cos. A cos. A -I- sin. Asm. A. 



cos. a cos. a 



But since cos. (a -\- a") = cos. a cos. a' sin. a sin. a', 

 this is the same as . 



_. cos. d 4- cos. (a + a") cos. o cos. a' 



cos. D = i 3- ' , 



cos. a cos. a 



cos. A cos. A' + sin. A sin. A' 



( cos. d + cos. (a -f o) , 

 = ( cosT a cos. a' -iJcos.Acos.A'+sin.Asm.A'. 



If the 1 within the braces be suppressed, we may 

 restore it by adding cos. A cos. A at the end, as is ob- 

 vious. Hence 



_ cos. d -f cos. (a -f a') , . , 



cos. D >= cos. A cos. A cos. (A+ A ) 



cos. a cos. a 



The numerator of the fraction in this expression being 



