• 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 

 tilways very nearly equal to the eccentricity e ; and that, there- 

 fore, we may aflTame e zz s, &x. lead for a firft approximation to 

 the value of v. 



For it is clear, from the mofl elenientary principles, that the 



maximum value of fin d X cof c is eqvial to - : therefore the arch 

 n — V, determined by the equation fin (« — v) — i fin c 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 s =: i, the arch 



« — V can never be greater than -. 



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

 fines ; a propofition 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 7. Therefore, the fraction will always 



be very fiearly et[ual to unity ; and, confequeAtly, the value of e 



determined by the formula <? — a X i — "■' ^K\\\ in all cafes 



n — V ' ' 



whatfoever, differ but little from s. 



AssDMiNG, therefore, ^ = e, if we denote by r the value of 

 •J correfponding to this value of f, in the equation fin [n — v) 

 rz I? fin V X cof V ; we may confider ir as a firft Approximation 

 to half the arch of eccentric anomaly. 



5. Having thus f6und one approximate value of v, it is cafy 

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

 already invefti gated. 



VoL.V.— P. II. Dd For, 



