74 



tracted from the cosine of a, gives the cosine of d. The 

 cosine of an arch greater than 90° being negative, occasions 

 here a distinction of cases. If we use versed instead of 

 -co-sines, as Dr. Mackay has done, and put ^ ^^ h ron i~ ^ 

 wo shall have v. sin cl = v. sin a -f N (v. sin D — v. sin A) 

 where no distinction of cases occur. But there are still 

 inconveniences in this method so improved. An extensive 

 tabic, with a double argument, is required for the value of 

 log. N (Tab. 9 of the requisite tables) and also logarithmic 

 tables to six or seven places requiring proportional parts. 

 But even with these inconveniences the method, on account 

 of its plainness and conciseness, is valuable, and scarcely 

 yields to any one that has been given, 



4. Instead of performing the multiplication by aid of log- 

 arithms, it may be done by the assistance of natural sines, 

 and the conclusion reduced to versed sines. Thus, let 

 2 cos M = N, Then cos d=cos a — 2 cos M (cos A — cos D) 

 = cos a — cos (A + M) — cos (A — M) + cos (D + M) + cos 

 (D— M) 



Or reducing this equation to versed sines by substituting 

 for cosine, 



V. sin d = V. sin a — v. sin (A + M) — v. sin (A — M) + v. sin 

 (D + M) — v.sin(D — M) 



This is one of Mr. Mendoza's formulas ; but he prefers a 

 similar one in which the sum of the altitudes occur instead of 

 the difference ; the former being somewhat more easily com- 

 puted than the latter. The inconvenience of this method of 

 multiplying, arises from the formation of the arguments. 



The 



