Of KEPLER'S PROBLEM. %o 7 



Let us now confider *r, the firft term in the feries of ap- 

 proximate arches ; this arch is the value of v in the equation 

 fin (n — v) — e X fin v X cof y, when g is fubflituted for e ; but, 



fince we have e — g X ^— -, it is evident, that e is lefs than e, 



« — v 



and confequently v will be greater than r. 



Again, take sr', the fecond term in the feries of approximate 



arches. This arch is the value of v in the equation 



Hn (« — v) zz. e X fin v X cof v, when i is fubflituted for e : Now, 



' __ v, fi n (» — «") i fin (« — ,"> , . 



e — e X ' and f = sX — ; and iince v has been 



n — a- // — f 



fhewn to be greater than t, it is evident that n — t will be 



greater than n — v : but the greater arch has to its fine the 



c ( \ 



greater ratio ; therefore the fraction - —, will be lefs than 



n — 3* 



the fraction ' confequently & will be lefs than e ; and 



n — v 



therefore t will be greater than v. 



• r • r fin (// — v) 

 And, in general, if, in the formula e rr g X — , we fub- 



ftitute a greater arch for v, we fhall have a value of e greater 

 than its true value ; but, if we fubflitute a lefs arch for i>, we 

 fhall have a value of e lefs than its true value : but we have de- 

 monflrated, that, in the equation fin (« — v) — e fin v X cof v, 

 the greater e is, the lefs will the arch v be : from which confider- 

 ations the truth of what we have afTerted above is evident, viz. 

 that the arches t, t , k", &c. continued indefinitely, are alter- 

 nately too fmall and too great. 



7. Let us now compare together the alternate terms in the 

 feries of approximate arches ; and it will not be difficult to per- 

 ceive, that the firft, third, fifth, &c. terms, which have been 

 fhewn to be all lefs than half the arch of eccentric anomaly, 

 continually increafe ; but that the fecond, fourth, fixth, &c. 

 terms, which have been fhewn to be all greaterthan half the 

 arch of eccentric anomaly, continually decreafe. 



D d 2 Foa 



