ItrSAR DISTANCES.] 



NAVIGATION NAUTICAL ASTRONOMY. 



1101 



the results, and the unhesitating confidence that may 

 consequently be always placed in the conclusions of the 

 direct methods, by Spherical Trigonometry, produce a 

 degree of satisfaction which more than compensates for 

 the trouble of a few additional references to those com- 

 mon tables, with which eTery calculator may reasonably 

 be expected to be abundantly familiar. 



It will, no doubt, occur to the reader, that instead of 

 comp. cos., secant may be used ; observing to reject the 

 index 10 from each secant. 



2. The apparent distance of the moon's centre from 

 the star Regulus was 63" 35' 14'; the apparent altitude o:~ 

 the moon's centre 24 29' 44'; the apparent altitude 01 

 the star 45 9' 12"; the true altitude of the moon's centre 

 25 17" 45 ; and the true altitude of the star 45 8 15 . 

 Required the true distance. 



d 63 35' 14" 



o 24 29 44 comp. cos. -0409017 

 a' 45 9 12 comp. cos. '1510S03 



2)133 14 10 



sum 66 37 5 



sum-/. </ 3 1 51 



A 25 17 45 



A' 45 8 15 



A + A' 70 20 



cos. 9-59863r>9 

 cos. 9-9993921 

 cos. 9-9562230 

 cos. 9-8484402 



2)39-5953332 



19-7970666 

 cos. 9 -9122099+ , 



sin. 9 8854507 



being 

 added 



cos. 9 -8063479+ > 



sin. 9 -7185578= their sum. 



i (A + A 1 ) 35 13 

 50 11 23 



. 



i D 31 32 17i 

 .'. D = 03 4 35 



ON MAKING THE OBSERVATIONS. In taking a lunar 

 distance at sea, it is desirable that there should be three 

 observers ; one of these measures the distance between 

 the limbs of the sun and moon, or between the limb of 

 the moon and the selected star, and the other two take 

 the altitudes of the objects, at the instant the distance is 

 obtained. A single distance, and a single corresponding 

 pair of altitudes, are not in general considered as suf- 

 ficient ; but a set of distances, and a set of corresponding 

 altitudes, are usually taken, allowing as little time as 

 possible to intervene between each observation : the mean 

 of the distance, and the corresponding means of the alti- 

 tudes, are then employed in the calculation. 



If, however, there be but one observer, it will be 

 necessary that he have the aid of an assistant, to note 

 by the watch intervals of time, the observer proceeding 

 as follows. 1, Let the altitude of the sun or star be 

 taken ; 2, then the altitude of the moon ; 3, a set of 

 distances ; 4, another altitude of the moon ; 5, another 

 altitude of the sun or star ; the times of each observation 

 being noted. Then let the means of the distances and 

 tiinuK of observing them be taken ; and reduce the alti- 

 tudes to the mean time thus found by proportion. These 

 are the directions given by Norie ; but Lieut. Raper, an 

 experienced practical navigator, recommends as follows : 

 When the observer is alope, he will first observe the 

 altitude of the object farthest from the meridian, then 

 that of the other object, and then the distance, con- 

 cluding with the altitudes in the reverse order ; the 

 reason of this order is, that the outer object preserves 

 uniformity in its change of altitude for a longer time 

 than the other, and consequently its altitude may be 

 reduced, by simple proportion, to an intermediate time, 

 with less error than the altitude of the other object ; we 

 may add, however, that it is desirable that the outer 

 object should not be the moon. 



When the ship has much motion, the observer fixes 

 himself in a corner, or lies on his back on the deck, to 

 take the distance, in order to remove, as much as pos- 

 sible, the sense of bodily effort and inconvenience which 



disturbs the eye and the attention. Three or five dis- 

 tances, at least, should be taken ; less precision is neces- 

 sary in taking the altitudes than in observing the 

 distance, so that one altitude of each object, taken 

 with ordinary care, will in general suffice, when the 

 time at the ship is not to be deduced as well as the true 

 distance. 



OTHER METHODS OF CI.EARING THE DISTANCE. As 

 the problem of deducing the true lunar distance from 

 the observed distance is one of such note and importance 

 in Nautical Astronomy, we shall present the reader with 

 some other methods, all, like that of Borda just given, 

 conducted by the rigorous principles of Spherical Trigo- 

 nometry, and requiring only the common logarithmic 

 and trigonometrical tables ; that is, only the logarithms 

 of numbers, and the natural and logarithmic sines and 

 cosines. 



The first of these additional methods which we shall 

 offer is that of Delambre, the formula for which occurs 

 in the investigation of the method given at page 1100 ; 

 namely cos. D 



^2 cos. J {(a + a!) + d} cos. | {(a + a) v> d} cos. A cos. A 



cos. a cos. a' 

 cos. (A -f A r ). 



The logarithm of the first expression in this formula is 

 calculated as in the method of Borda ; then the natural 

 number answering to that logarithm is taken out of the 

 table, and the natural cosine of (A + A.") subtracted 

 from it ; the remainder is the natural cosine of the true 

 distance. As this is a sort of mixed method, requiring 

 reference to both logarithmic and natural cosines, atten- 

 tion must be paid to the change of radius in passing from 

 one to the other : the expression which it is here pro 

 posed to find the common logarithm of, will supply four 

 additive logarithms and two subtractive ones ; hence two 

 tens, or 20, must be suppressed in the result (page 1038) ; 

 but if the arithmetical complements of log. cos. a, log. 

 cos. a', be added, then four tens, or 40, must be sup- 

 pressed ; and the result will then be, the ordinary loga- 

 rithm of the expression ; and from the number answering 

 thereto, the nat. cos. (A + A') is to be subtracted, aa 

 the formula indicates. 



Taking the example at page 1100, in which 



d = 83 57' 33", a = 27 34' 5", ' = 48 27' 32" 

 A = 28 2V 48", A' = 48 20' 4!)" 



Tho operation by the above method of Delambre will 

 stand arranged as follows : 



d 83 57' 33" 

 o 27 34 5 

 a' 48 27 32 



2)159 59 10 



sum 79 59 35 

 Js"um</>d3 57 58 

 A 28 20 48 

 A' 48 26 49 



comp. cos. -0523390 

 comp. cos. -1783835 

 log. 2 -301030Q 



cos. 9-2399G8G 

 cos. 9-9989587 

 cos. 9-9445275 

 cos. 9-8217187 



log. -3442921 = 1-5309200 (40 sup- 

 A + A' 70 47 37 nat. cos. -2284595 pressed). 



D 83 20 54 nat. cos. -1158320 

 Other fc rmulfo may be investigated as follows : 



Referring to the two expressions for cos. Z at paga 

 1099, we have 



cos. Z 



1 + cos. Z 



cos. D + cos. A cos. A' sin. A sin. A* 

 cos. A cos. A' 



_ cos. D + cos. (A + A. 

 cos. A cos. A' 



cos. d-\- cos. a cos. a' sin. a sin. a 

 c JM. a cos. a 



