212 Mr. Burns's Method ofjinding the Latitude at Sea. 



By a similar investigation we should find, 



sin. a = + sin. A. sin. 8 + cos. 8. cos. \. cos. (t + »)- 

 •.' by subtraction, 

 sin. A — sin. a = cos. 8 . cos. X . cos. t — cos. D . cos. X . 

 COS. (t + »)• 

 = COS. X. {cos. 8. cos. T — cos. 8 . COS. (t + i)] . 

 •.• COS. X = 



sin. A - sin. a _ 2 cos, j {A + a), sin. ^ ( A - n) _ 



COI. J. COS. T — COS. S. COS. (t -|- i) COS. S . JcOS. T — COS. (t + /; j 



2 COS. HA + a) sin. H A - ") _ _coM: (A +«K sin.^ A-a) _ /j) 

 ~r. /2t + /\ . , . ~ sin.(T+^-)-sin.i..cos.?. 



2 sin. I \ . sin. J / . cos. d. 



Such is the foinnula of calculation, when the observations 

 are made on the sa7)ie side of noon ; and the practical rule may 

 be expressed as follows : Add together the log. cos. of half 

 the sum, and the log. sin. of half the difference of the two al- 

 titudes; from the sum of these two logarithms, increased by 20 

 in the index, subtract the log. cos. of the declination, the log. 

 sin. of the time, and half interval reduced to degrees, and log. 

 sin. of half the interval ; — the remainder will be the log. cos. of 

 the true latitude. 



When the observations are made on different sides of noon, 

 the investigation will proceed similarly, but will be rather more 

 simple. 



Let T = the greater i\me from or to noon, 

 T = the less time. 

 We shall then have, 



sin. A = + sin. x. sin. I + cos. X . cos. 8 . cos. t. 



sin. a = + sin. X . sin. 8 + cos. X . cos. 8 . cos. T. 

 ••• sin. A — sin. a = cos. X . cos. 8 . cos. t — cos. X . cos. 8 . cos. T. 



= cos. X . (cos. 8 . cos. T — cos. 8 . cos. T). 

 '.• cos. X = 



sin. A — sin. a sin. A — sin. a 



COS. S . cos . T— cos. J . COS. T cos. J (cos. T — cos. T) 



2 cos ^(A + g). sin. ^(A -a) _ (cos.^(A + a). sin. §(A- a) . . 

 2sin. ^T-t-r). sin. -KT-Tjcos. S ~ sin. J(T-|-r) bin.i(T-r)cos.r " ^ 



From one or the other of the expressions (1) or (2), ac- 

 cording to the case, the latitude may be directly found, with- 

 out repetition or correction ; and the computation is as short 

 as could be desired, and that by the common tables of loga- 

 rithms. We will now give an example or two, from which 

 the inaccuracy of any indirect method, and the consequent 

 danger of depending on it, will appear very clearly. Let us 

 take one of the examples given in the Nautical Alnjanac, 

 wrought by the foregoing formula (1) ilqixaaab .ES'jnsTjn; 



Alt. 



