156 MR WALLACE'S Formula for finding the Logarithms 



Given sine, log sin (x + A#) = 9'99993516626353 

 Next less, log sin x (89) = 9'99993384980922 



The approx. value of cosine, 8-2374908820 

 Its true value is, - 8-28749095... 



Error of Approx. -00000007 



From this example it appears, that, supposing the sine of an 

 angle about 89, or the cosine of an angle about 1 to be known 

 with sufficient accuracy, the formula may be trusted to give the 

 cosine or sine true to seven decimal places. Towards the middle 

 of the quadrant its accuracy is much greater. 



In these examples, the logarithmic differences to be found 

 have been expressed by seven or more significant figures : it was 

 therefore necessary in the calculation to take out the logarithms 

 to at least as many places ; but when the given sine or cosine 

 consists of only seven figures, besides the index, the logarithms 

 need not be carried so far ; as in this example. 



(4.) To find the sine corresponding to log cos (x + *x) = 9 9409872 

 In BUTTON'S Tables, next greater, log cos #(60 49/)= 9-9410461 



cot * 10-25298 



cosec* 10-31193 



cos(,r + A.r) - 9-94098 



log (_ A log cos x) 5-7701 2 



A log cos x 0000589 



logp 



y 



1-27601 i/ = -0001888... First approx. 

 -27582 t/' = -0001885 = A log sin* 

 9-6880688 = log sin a; 



Sine required, 9'6882573 



