162 MR WALLACE'S Formula for finding the Logarithms 



A log sin x = cos* x A log tan x m (A xf + &c. 

 A log cos x = sin 2 x A log tan x + m (A .r) z + &c. 



Hence it appears, that our two formulae F are only approxima- 

 tions of the first order ; but that the error, as far as it depends 

 on the square of A x, is independent of the angle #, or is a 

 constant quantity for a given value of A x. 



If we make the increment A x an angle of one minute, we 

 have m (A #)* = .0000000367. Thus they appear to be correct 

 in the first seven decimal places ; we may therefore safely em- 

 ploy them with HUTTON'S or SHERWIN'S Tables, reserving the 

 others for Tables of greater extent. 



14. Let it be required to find the logarithmic sine and co- 

 sine corresponding to the tangent 10*0763404 10548. 



The given tan (x + A x) 10-076340410548 

 The next less (Bmccs's Trig. Brit.) tan x (50) 10-076186469801 



A log tan x = -000153940747 

 To find log sin (,r + A x) 

 sin x 9-8842539666 



cos x 9-8080674968 



cot (x + A x) 9-9236595895 

 A log tan x 4-1873535895 



logr =5-8033346424 z' = -000063583 First Ap. 



log r + .?'=: -8033982254 z" = 0000635914 Second Ap. 



logr + .s" ~ -8033982338 %'" - -000063591378 = A log sin x 



sin^r = 9.884253966554 







sin (x + A x) = 9.884317557932 



To find log cos (,r + A x) 



sin x 9-8842539666 



COS.T 9-8080674968 



tan (x + A x) 10^0763404105 



A log tan x 4-1873535895 



log* = 5-9560154634 tf = -000090368 First Ap 



log* * = -9559250954 z" = -0000903494 Second Ap. 



logs 2" = -9559251140 z"' = -000090349367 = A log cos j- 



COS.T = 9-808067496752 



cos (x + A x) = 9-8079771 47385 



