Of KEPLER'S PROBLEM. 237 



Then, to find a value of x, from the equation, x$ -J- a x — b : 

 fince b is fmall in comparifon of a, it is manifefl that x mull 

 be a very fmall fraction ; and, confequently, x> inconfiderable in 



refpect of a x : therefore x = - — .05 nearly : and, having cor- 



reeled this value by the common method, we fhall find, 



* = fin <p — .0547 rr fin 3 8', and fo <p rr ij6°52 f . 

 As the firft approximation which we are now computing, cannot 

 be exact, even to the neareft minute, it would be fruitlefs to pufii 

 the calculation to a great degree of accuracy. For the fame 

 reafon, I here ufe the proportion in the general rule, becaufe it 



requires but one operation for finding $, viz. cof : 



cof : : cof <Z> : cof -J/. 



2 



Hence ^ is found =3° 1', and therefore p — <p — -^ = 

 1 73 51'; this is the firft approximation to the eccentric ano- 

 maly reckoned from the aphelion, and is too fmall. 



2dly, To compute the fecond approximation, we have, 



r m — P 



w 2 fin 2 58' , 1 , 



e =r g X z= g X r^s> » whence - aa 1.06878 ; 



1 , x „ arc 2° c8 e % ' 



1 1 — .06878 =. a; — — — .0038749 — k Therefore, 



#3 _j_ a x — 3 . but we know, that a near value of x is .0547 t 

 and, having corrected this value, we fhall find, * z=. fin <p =. 

 .0540430 ; therefore, fin <p == :054043c = fin 3 5' 52", and <p = 



176 54 8". 



As 4> is here a fmall angle, we muft ufe the method of Art. 12. 



to find it with the requifite exactnefs : this gives tan A =3 



X ^-^ — X f ec 45 °, and fin 2: — tan - X fin 45 ° j there- 



cof 1 - 



2 



fore log. tan A = 8.8689481, and ^ = 2° 59' 32'', 



Vol. V.— P. II. H h wherefore 



