and Levelling, t§-e. 341 



trusted those made on the 21st of August on the pole-star, 

 which I continued to observe from about 10 o'clock in the 

 evening to 1 o'clock next morning. During this period I 

 completed eight series of double observations, reversing the 

 circle each time, or sixteen single observations, comprehend- 

 ing forty-eight readings of the verniers, accompanied by the 

 times of observations, and the indications of the level. 



The circle, indeed, is rather small for examining so delicate 

 a question, being only six inches in diameter, having three 

 verniers, each shewing 10", and a level whose divisions each 

 indicate 2". I shall, however, claim no more for it than what 

 competent judges may be inclined to grant ; and they may 

 put such confidence in my final result as they conscientiously 

 believe it deserves. I may add, however, that I believe it to 

 be in contemplation by Colonel Colby to examine this point 

 carefully at no distant period, by means of the Ordnance 

 zenith sector, which will be set up somewhere in this vicinity 

 expressly with that view. 



For ordinary purposes the tables to determine the latitude 

 by the pole-star at the end of the Nautical Almanac are very 

 convenient when no great precision is wanted, and the inter- 

 polation made by even proportion. I used these chiefly as a 

 check, however, and made my computations by the following 

 formulae, easily investigated, and in the application of which 

 I used proportional logarithms to five places of decimals, 

 which, for this purpose, give results in general sufficiently 

 accurate. 



Let t be the sidereal time after the upper transit of the 

 pole-star, and p its polar distance. 



1. Log secant t + prop, log p = prop, log u. Now if x be 

 the latitude of the foot of the perpendicular arc from the star 

 upon the meridian, and z the true zenith distance, 



2. Log sin X = log sec p + log cos u + log cos z. Now 

 making I the true latitude, 



3. /=X±« 



where it must be observed that in the 1st and 4th quadrant 

 after transit the sign of u is — , while in the 2d and 3d it 

 is + . 



