[ 211 ] 



XXX. A short Method of Jinding the Latitude at Sea by 

 Double Altitudes and the Time between. Bu James Burns, 

 B.A. 



'T'HIS jDroblem, from its great utility to seamen, has engaged 

 ■*■ the attention of several eminent mathematicians : but 

 among the many attempts that have been made, not one short 

 direct and acairate method has been yet given. The solution 

 given by Douwes was the only one generally practised, for a 

 long time ; but there are many objections to this method, which 

 the scientific navigator will easily comprehend. In the first 

 place, its being a method oi false position^ and depending 

 chiefly on the latitude by account, renders it in most cases in- 

 accurate; and moreover, to ensure any thing like correctness, 

 the computed latitude should be nearer the true latitude than 

 that by account. Secondly, its necessary limitations, with re- 

 spect to the time, must often render it impracticable. — Another 

 method was given a few years ago by Dr. Brinkley, in the Nau- 

 tical Almanac ; but it has been found so tedious, that very 

 few seamen, I believe, have ever practised it. Besides, it is 

 liable to the principal objections that are made to Douwes's 

 solution ; for the latitude by account is retained in it, and two 

 or three repetitions and corrections of the calculation are ne- 

 cessary, before any conclusion to be depended on can be had: 

 and after all, a considerable degree of error might be involved in 

 the result. In the following investigation, — which is direct, — 

 the consideration of the latitude by account is therefore omitted. 

 All that is necessary to be known is, the time, the interval be- 

 tween the observations, and the altitudes ; all of which, from 

 the improved state of our chronometers and other instru- 

 ments, may be known with the greatest exactness. 

 Let A = greater altitude, 



a — less altitude, 



8 = O's declination, 



X = the latitude, 



T = the timey;om or to noon corresponding to A, 



i = the interval. 

 By elementary spherical trigonometry, we have, 

 sin. A =cos. (A ± 8) — vers. t. cos. 8 . cos. A. 



= cos. A. cos. 8 + sin.X . sin. 8 — [(1 — cos . t) . cos. 8 . 

 cos. X. ] . 



= COS. X . cos. 8 + sin. A . shi. 8 — (cos.8 . cos. X — cos. 8. 



cos. A. COS. t). 



= COS. A. COS. 5 + sin. A . sin. 8 — cos. 8. cos. A -|- cos. 8. 



COS. A . cos. T. 

 = 4^ sin. A . sin. 8 + cos. 8 . cos. A . cos. t. 



D d 2 By 



