224 



Mr. R. Templeton on a Method of 



In applying this formula, it is only necessary to replace what- 

 ever be the index proper to the given number by the temporary 



index 3, and to add in the constant log ^ ( = '3622157). 



Next let us take the case of such logarithmic sines as are not 

 given to seconds of arch in Hutton's Tables, that is, of 2° and 

 upwards. Here we have the following rule*: — To the constant 

 4*3233591 add the cotangent of the arch to the nearest tabular 

 minute, also the logarithm of the given number of seconds; 

 from the sum subtract half the number (in Hutton's Tables) 

 answering to it ; the remainder will be the logarithm of the dif- 

 ferential quantity answering to the given seconds of arch, to 

 be added to the tabular sine. 



To test the accuracy of the rule, let it be required to find 

 the sine of 2° 1 / from the sines of 2° 0' and 2° 2'; it will be 

 a pretty fair trial, as the first differences are very large and the 

 second nearly equal to 300. 



const. . 4-3233591 

 cotan 2° 0' 1-4569192 



const. . 4-3233591 

 cotan 2° 2' 1-4497317 



log 60" . 



1-7781513 





log 



60" . 



1-7781513 





7-5584266 



- 18088-. 



1 Mn 



' 2 X 







7-5512421 

 + 17791. 





7-5566178 









7-5530212 



sin 2° . 



•0036026 

 8-5428192 





sin 



2° 2' 



-•0035729 

 8-5499948 



sin 2° 1' 



8-5464218 





sin 



2°1' 



8-5464219 



Mn 



The correct number lies between these, the second being some- 

 what in excess. This method can be used until the differences 

 become nearly constant, when its utility ceases. 



The formula places within our reach great facilities for taking 

 out sines from Tables, which, if tangents be omitted, may be made 

 very compact, that is, at 10" intervals in the earlier portions of 

 the Table. The sines so computed are perfectly true to nine 

 places, in the tenth place with a slight error, which, however, 

 being constant over a great extent of Table, can be easily provided 

 for. The following Table of sines to each second of arch com- 

 puted as above will give an idea of its trustworthiness at certain 

 points of the canon — its exactitude in other parts may be inferred. 



* Since b log sin x = M cot x 



