150 MR WALLACE'S Formula for finding the Logarithms 



1. Let A# be the finite increment of any angle #, (the letter 

 A being employed as the characteristic of a finite difference) and 

 let m = . 4342945, the modulus of the common system of lo- 

 gariths. By TAYLOR'S theorem, 



1 (A a/) 2 sin a? (A a/) 3 -. 

 logcos(;r+A;r) = logcosa; m{ tan x-*x+ -^^ $ + -^^ ^- + &e. }, 



1 (A a-) 2 , cos a? (A rf 

 log sin (a:+Atf) = log sin ^ + m { cot a? A a; ^g^ ^ + -^^ ^ -- Stc, }, 



Employing now the notation of the theory of differences, and 

 expressing, 



log cos (a- + A x) log cos a? by A log cos x, 

 log sin (x + A x) log sin a? by A log sin x ; 



we get these two formula, 



1 (Aa?)* cosir (A a?) 3 

 ? i- f 2. + s l n rj-^ L -&c. }. (2), 



Again, by TAYLOR'S theorem, we have 



(A*)* + &c. 



cos* x cos* 



/-. \t n 

 - - A ar + ^^ (A *) - 



1 t cos x 



\x T- ** j " gi 



and hence 



m-A f (A a;)* 







1 (A a?)* sin a? (Aa?) 3 



-^-g- *** 



COS^CA^ 



,-- c. 



But by the calculus of sines, 



sin (2 a?+ A,r) 



sin (2 x + A a.') 

 cot (x + Aar) + cot a? - s i n ^ 9 i n (a; + A a^) "' 



Therefore, 



