VOL. LXV.] PHILOSOPHICAL TRANSACTIONS. -, QQt 



astronomy, where spherical trigonometry can only be of use, are generally of 

 such a nature that we know nearly, or at least within a few degrees, what the 

 required side or angle is ; there is nothing therefore wanted. but to find how much 

 this quantity, or first approximation, differs from the true value of the side or 

 angle. Thus, in calculating the right ascension of any point of the ecliptic, 

 whose longitude and declination are known, instead of finding the right 

 ascension immediately, it will be more convenient to seek for the difference 

 between the longitude and right ascension immediately, and as it never 

 exceeds 2-^-°, 4 or 5 places of figures will always be sufficient to determine it 

 within a second. And in other similar cases, rules might be made agreeable to 

 the exigency of each particular case, which would be better than the application 

 of the general method of solution. Some examples of which shall be shown in 

 the following paper : the design of which is to point out a method of solving 

 several of the most useful questions in spherical trigonometry, in a manner 

 somewhat similar to that used in approximating to the roots of algebraic equa- 

 tions. This method is founded on the following 



Lemma. — If the radius be supposed equal to unity, the sine of the sum of 2 

 arcs, a and (3, is equal to sin. « + cos. a x sin. |3 — sin. a x vers. sin. (3. And 

 its cosine = cos. a. — sin. a X sin. j3 — cos. « X vers. sin. (3. For let the arc « 

 be RA, fig. 7, pi. 13, and the arcj3 be ab, their sines xa, bd, respectively; 

 then nb, being drawn perpendicular to the radius cr, will be the sine of « + (3. 

 Draw Dp and An parallel to cr. Then, by similar triangles, CA : ca :: bd : b^, 

 and CA : ao :: ad : np. Therefore, b6 (= Aa + b/) — pn) = Aa + 

 ca_x_BD _ Aox A D , ^^^^ jg^ gj^g (a + (3) = sin. « + cos. « X sin. (3 — sin. a X 



vers. sin. j3. In the same manner, drawing nq parallel to xa, we may prove cb 



, , \ AO X BD Ca X AD / 1 rt\ 



(= ca — bq — aq) = CA --^ — — , or cos. (a -|- j3) = cos. x — sm. 



A X sin. (3 — cos. a. X vers. p. 



In what follows, for brevity sake, the arc is expressed by a Greek letter ; its 

 sine by the capital character ; and the cosine by the small italic character of the 

 same letter. In this notation, the 1 theorems will stand thus, sin. (a -f |3) 

 = A -|- OB — A X vs. (3, and cos. (a + (3) = a — ab — a X vs. (3. 



Corol. I. — Since the tangent is equal to the sine divided by the cosine, we 



u 11 u .. / 1 n\ A + as — A X V3./3 a , b , a ^^ . , 



shall have tang. (^ + (3) = ^_,^_,^ —^ = a + ^ + ? X ^«- f^ "e^'-'X- 



Corol. 1. — If we change the sign of p, we shall have sin. (« — ' (3) = a — as 

 — A X vs. (3. Cos. (a — P) = a + ab — a X vs. |3. And tang. (« — (3) 



A B 



o a* ■ a 



7- T^+—^ VS. p. 



By the help of these theorems, knowing nearly what any quantity in a sphe- 

 rical triangle is, we may find its correction, thus : if we have to find the cosine 



4 T 2 



