the Double AUittide Problem. 169 



cosine of the declination, and the sum will be the sine of arc 

 ^rst. 



2. Add together the secant of ax'c first, and the sine of the 

 declination, and the sum will be the cosine of arc second; which 

 will be acute when the latitude and declination are of the same 

 denomination, but obtuse when they are of different ones. 



3. Add together the cosect. of arc first, the cosine of half the 

 sum of the altitudes, and tlie sine of half their difference ; and 

 the sum will be the sine of arc third, 



4. Add together the secant of arc first, the sine of half the 

 sum of the altitudes, the cosine of half their difference, and the 

 secant of arc third, and the sum will be the cosine o^ arc fourth. 



5. When the zenith and elevated pole are on the same side 

 of the great circle passing through the places of the sun at 

 the times of observation, the dlff'ere?ice of arcs second and 

 fourth wUl be arc Jifth, otherwise their sum will be arcjifth. 



6. Add together the cosiyies of arcs third and fifth, and their 

 sum will be the sine of the latitude. 



Note. When the declination and latitude are nearly equal, 

 and of the same name, it may sometimes be doubtful whether 

 the sum or difference of arcs second and fourth ought to be 

 taken for arc fifth. But the computation is soon made on both 

 suppositions ; for cosine of arc filth is the last logarithm whicli 

 is taken from the tables, and the other parts of the calculation 

 are therefore not affected by the change. One of the results 

 must certainly be the required latitude, and the latitude by 

 account will generally be sufficient to determine which of them 

 ought to be taken. But when the sum of arcs second and 

 fourth is equal to 90°, or greater, it can only be their difference 

 which is arc fifth. 



Foiin of Calculation. 



sin H.E.T. ° ' 



8111 Dec cos 



Vol.60. No. 293. Si'it. 1822. Y Ex- 



