154 ME WALLACE'S Formula! for finding the Logarithms 

 Let us now put 



z = A log cos ;r, q = tan x sec # sin (a- + A a 1 ) A log sin a 1 , 



and instead of formula (C') we have this : 



To resolve this equation, we must form a series of approxi- 

 mations, 



z' = q, log tf' = log q + 2 log 2"' = log q + z", &c. 



and the quantities z', z", z", &c. will quickly approximate to the 

 correct value of z. 



4. To make the mode of proceeding perfectly clear, I shall 

 now give some examples. 



(1.) To find the log. sine of an angle whose cosine 

 9.9450803019 by VLACQ'S Arithmetica Logarithmica, which con- 

 tains a table of log. sines, &c. to every minute of the quadrant, 

 and to ten places of figures : 



Given cosihe, log cos (x + Aa-) 9'9450803019 



Next greater in Tab. log cos #(28 12') - g-9451254712 



A log cos x - -0000451693 



Log. 



cot x 10-2706770 



cosecx - 10-3255515 



cos(a- + Aa-) 9-9450803 



A log cos x - 5-6548434 



=4-1961522 y =-0001571.... First approx. 

 y = 1959951 y = -0001570345 = A log sin a- 



9-6744484704 = log sin x 



The sine required, 9'67 46055049 = log sin (x + 



The sine required is that of 28 12' 40", and its correct value, 

 as given in VLACQ'S Trigonometria Artificialis, is 9'6746055050, 

 with which our approximation agrees almost exactly. 



