EXPLANATION AND USE OF THE TABLES. 



and the given log. we annex two ciphers, and divide the result by 



o 



of the difference between the next less and next greater log. ; and the 

 quotient is the seconds to be added to the degrees and minutes before 

 taken out. 



Ex. Required the degrees, minutes, and seconds corresponding to the 

 log. sin. 9.6 41 357. 



The sin. 25". 58'. is 9.641324 which is the log. next less than the given 

 one. The difference of these two logs, is 33, which by adding two ciphers 



becomes 3300, and this divided by of 260, or by 433, gives 8 nearly for 



the number of seconds ; .*. required arc is 25. 58'. 8". 



When the arc is small, a particular process is necessary as follows : 



To find the log. sine of a small arc less than 3. 



Add 4.685575 to the common log. of the arc reduced to seconds ; from 

 the sum subtract one-third of the log. secant less radius of the arc, and 

 the remainder will be the required log. sine. 



To find the log. tangent of a small arc. 



Add together the common log. of the arc, reduced to seconds, % of the 

 log. secant less radius of the arc, and 4. 68.") 575 ; and the sum will be the 

 required tangent. We have hence the following rules for performing 

 the reverse operations : 



To find a small arc whose log. sine is given. 



To % of the log. secant of the arc in the Table, whose log. sine most 

 nearly corresponds with the given log. sine, add the given log. sine, and 

 5.314125, and the sum will be the common log. of the seconds in the re- 

 quired arc. 



To find a small arc when its log. tangent is given. 



To the log. tangent add 5.314425, and from the sum subtract % of the 

 log. secant of the arc in the Table, whose tangent most nearly agrees 

 with the given tangent ; arid the remainder will be the log. of the se- 

 conds in the required arc. 



Ex. 1. Required the log. sine of 1. 28'. 13". or the log. cosine of 880, 



lo. 28'. 13" = 5293" log. 3.723702 



Constant No 4.685575 



8.409277 

 % log. secant lo. 28' sub. .000047 



lo. 28'. 13". log. sine 8.409230 



Ex. 2. Required the arc to the log. sine 7.963214. 



^ log. sec. 00. 32' 000006 



7.963214 

 Constant No 5.314425 



1895" log. 3.277645 



Whence the required arc is 31'. 35" 



Hence the arc to log. cosine 7.963214 is 89. 28'. 25". 



FINIS. 



