ZXPLANATIOiT AND USE OTF THE TABLES. 



No. answering to 2.993789 is 985.8. If the number be required to a great- 

 er No. of places than four, find the difference between the given and the 

 next less log. To this annex on the right hand as many ciphers as there 

 are figures required above four. Divide the whole by the difference be- 

 tween the next less and next greater log., and the quotient annexed to 

 the four figures formerly found will be the natural number required. 

 Thus required the No. to 6 places answering to the log. 4.C87956. The 

 nearest less log. than this is 687886 corresponding to which is the No. 4874. 

 The difference between 687956 and 68788(5 is 70, to this annex 2 ciphers 

 and it becomes 7000, which being divided by 89, the difference between 

 the next less and next greater log. gives 79, .'. the number required ia 

 48747.9. 



TABLE II. 



1. To find the logarithmic sine, cosine, $c. answering to any given de- 

 gree or minute. 



Find the given degrees at the top of the page, if less than 45, and the 

 minutes in the left hand column ; opposite to which, and under the word 

 sine, cosine, &c. is the number required. But if the given degrees be 

 greater than 45 and less than 90, find them at tli(* bottom, and the re- 

 quired sine, cosine, &c. will be found above the word sine, cosine, &c. 

 opposite to the given number of minutes in the right hand column. If 

 the given arc exceed 90, find the sine, cosine, &c. of its supplement. 

 Thus the log. sine of 23. 28' is 9.600118; and the cotangent of 55". 57' is 

 9.829805. If the No. of minutes be odd, and .'. not contained in the 

 Table, proceed as directed for the odd numbers, Table I. 



To find the logarithmic sine, tangent, $c. of an arc expressed in de- 

 grees, minutes, and seconds. 



Find the sine, tangent, &c. corresponding to the given degree and 

 minute, and also that answering to .the next greater minute; multiply 

 the difference between them by the given number of seconds, and divide 

 the product by 60 ; then the quotient added to the sine, tangent, &c. of 

 the given degree and minute, or subtracted from the cosine, cotangent, 

 &c. will give the quantity required nearly. 

 Ex. Required the log. sine of 23. 27' 40". 



Log. sin. 23<> 27' 9.599827 



23 28 9.600118 



Difference 291 



which multiplied by 40, and divided by 60, gives 194, and this added to 

 9.599827 gives the required logarithm 9.600021. 



2. To find the degrees and minutes answering to any given logarithmic 

 tine, tangent, Sfc. 



Find the nearest log. to that given in the proper column : if the title 

 be at the top of the column, you have the number of degrees at the top 

 of the page, and the minutes in the column on the left hand; but should 

 the title be at the bottom of the column, you have the degrees at the bot- 

 tom of the page, and the minutes in the column on the right hand. If 

 the given log. seems to belong to the odd minutes, proceed as directed 

 Art. 2. Table I. Thus log. sin. 9.457584 answers to 16. 40'. Log. tan. 

 10.535401 answers t 73. 45'. But if the seconds in the arc are also re- 

 quired, we seek in the proper column for the logarithm which is next 

 less than the given one, when the logs, in the column are increasing; 

 but next greater, when they are decreasing, and take the degrees and 

 minutes corresponding to that logarithm for the degrees and minutes in 

 the required arc. Then to the difference between the logarithm so found 



