96 LATITUDE AND LONGITUDE. 



the Pole, ZMK, ZNKany two vertical circles in which 

 we have observed the object at the points M and N, P 

 NT, PMS, portions of two circles of declination extend- 

 ing to the equator at the points S and T. 



After the necessary corrections hereafter alluded to, 

 let us imagine we have found by observation the arcs 

 MZ and NZ, being the complements of two different 

 observed altitudes AM and BN ; then are also the arcs 

 MP and NP complements of the declinations, SMTN, 

 which declination is ascertained by the tables calculated. 

 Now the angle NPM is found by turning the time be- 

 tween the observations into degrees, and parts of a 

 degree, at the rate of 15 degrees an hour for the sun, 

 and 15 degrees, 2 minutes and 28 seconds for a star, 

 this consequently gives its measure. 



This understood, conceive the arc of a great circle 

 passing through the points M and N, we have then a 

 spherical triangle MPN ; two sides of which, MP, NP 

 are known and the contained angle ; we can therefore 

 by spherical trigonometry determine the angle PMN, 

 and the remaining side MN ; then in the spherical tri- 

 angle MZN we have the three sides known ; this, by 

 another process in spherical trigonometry, enables us 

 to determine the angle ZMN, take away from the whole 

 angle ZMN, the angle PMN, we have then the remain- 

 ing angle ZMP. 



But in the spherical triangle ZMP, the two sides Z 

 M, PM are known, one being the complement of the de- 

 clination, the other the complement of the altitude ; we 

 have consequently the two sides and the contained 

 angle : this enables us to calculate the remaining side 

 ZP, which it is evident is the complement of PH, the 

 elevation of the pole which is equal to the latitude. 



In general those methods by double altitudes, al- 

 though very beautiful in theory, are not so well in 

 practice, being liable to chances of errror, they are there- 

 fore seldom resorted to except in cases of necessity. 



To be concluded in our ttej/. 



