160 MR WALLACE'S Formula for finding the Logarithms 



.0000000001. Hence we may infer, that with a table of logar- 

 ithmic sines and tangents to every minute, and to ten decimal 

 places, such as that in VLACQ'S Arithmetica Logarithmica, the 

 log sine and cosine may be found from any log tangent correct 

 to at least 10 decimals between a> = 2 and x 88. Beyond 

 these limits the accuracy will diminish, but still they will be true 

 to seven figures, when ae 10', and when x 89 50'. With 

 VLACQ'S Trigonometria Artificialis, which gives the sines and 

 tangents to every ten seconds and to ten figures, the formulae 

 will be accurate throughout the whole table. 



10. The formulas (D), (E) when properly adapted to calcula- 

 tion, will stand thus, 



. sin (x + A .r) 



A log sin x =. sin x cos x cot (x + Aa? ) ^ '- A log tan x ; 



sin x 



V, :,: - : ; J :!..,, r; . : . ..,-. (E') fi ,' .::':" 



. COS (# 4- A #) 



A log cos x sin # cos x tan (# + A x) ' * A log tan x ; 



COS 3C 



And in logarithms 



(D") 



log (A log sin x) = log (sin x cos a- cot (x + A a?) A log tan x} + A log sin #, 

 log (A log cos x) log {sin x cos a? tan (x + A a?) -A log tan x} ( Ale 



11. The mode of deducing A log sin x and A log cos x from 

 these formulae, is exactly the same as we have already employed 

 in art. 3. That is, putting 



y = A log sin a?, z = A log cos a; , 



r 



tan (x + A x) ' 



= sin * cos * tan (a- + A*), 



