• Of KEPLER'S PROBLEM. 205 



in other words, unlefs the problem, of which we are treating, 

 were already refolved. But it is eafy to demonflrate, that e is 

 "always very nearly equal to the eccentricity s ; and that, there- 

 fore, we may afTume e — 2, at lead for a firft approximation to 

 the value of v. 



For it is clear, from the mod elementary principles, that the 



■maximum value of fin 1) X cof v is equal to - : therefore the arch 



11 — v, determined by the equation fin [n — v) \zzz s fin v X cof v, 



when greateft of all, can never exceed -. It is alfo evident, 



from the nature of Kepler's 'problem, that s can never be 

 greater than unity ; becaufe the point D is fuppofed to be al- 

 ways taken in the diameter, and never without the circle. 

 Therefore, even in the extreme cafe, when sz 1, the arch 



« — v can never be greater than -. 



But fmall arches of a circle are very nearly equal to their 

 fines ; a proportion that we may extend, without great error, 

 to all arches not exceeding 30 °. Now, we have fhewn, that 



the length of the arch n -— - v can never exceed -, and there- 

 fore, that arch will always be lefs than the arch of 30 °, the fine 



of which is \. Therefore, the fraction will always 



be very nearly efcjual to unity ; and, confequently, the value of e 



determined by the formula e == 1 X m ^ n v > w |]] ' m a u ca f eS 



whatfoever, differ but little from e. 



Assuming, therefore, e — s, if we denote by tt the value of 

 v correfponding to this value of e 9 in the equation fin (n — v) 

 zr e fin v X cof v ; we may confider v as a firft approximation 

 to half the arch of eccentric anomaly. 



5. Having thus fbund one approximate value of v, it is eafy 

 to find as many others as we pleafe, by means of the formulas 

 already invefligated. 



Vol.V.-P.II. Dd For, 



