214 



A NEW and UNIVERSAL SOLUTION 



confbquently, fince ^^ = ^■> we get 



fin AM 



cof AM 

 cof AN 



'^ - fin AN 

 therefore, becaufe AM = w, it is manifeft that AN = u 



Thus, then, the refohition of the equation fin (« — v^ 

 - e fin« X cof V, cohicides with that cafe of the general pro- 

 blem, " Dc inclinatiomhus" of the ancients, in which the two given 

 lines are fuppofed to be ftraight lines, interfeding one another 



at right angles. If we take CF = - x CA, the arch AN, found 



by the conftrudion above, will be no other than the arch r, the 

 firfl term in the feries that we have already difcufTed : and, in like 



manner, if we take CF fucceflively equal to i- x ^^ / ^ _' ^-. X CA ;; 



-x ,^""""\ xCA; - X J:"~ ""I. X CA ; andfoon:. 

 £ lin (« — T ) ' £ nn (« — n- ) 



we may find, by the fame conftrudion, the other terms t', t",. 

 tt'", &c. of that feries. 



If, therefore, we are to reft fatisfied with a geometrical con-- 

 ftrudlion, we may confider Kepler's problem as already re- 

 folved. For, it is manifeft. from what has been proved, that,, 

 by means of the known and elementary problem, " De inclina-. 

 tionibus" we may, in all cafes,, approximate to the arch of eccen- 

 tric anomaly as nearly as may be required. It muft be con- 

 fefled, however, that a conftrudion of this kind, let it be ever 

 fo ingenious or elegant, is of no ufe to the aftronomer, who 

 feeks for a rule by which to condudl his calculations, and who 

 will not be fatisfied with a fpeculation of the mind. 



lo. The problem, " De iiiclinationihus," when the two hnes 

 given by pofition are fuppofed to be ftraight lines, is, in gene^ 

 ral, a folid problem. The geometrical conftrudion cannot be 

 effeded, unlefs by the help of the conic fedions ; and the folu- 

 tion, by the modern algebra, leads to an equation of the fourth 



power. 



