2i 4 A NEW and UNIVERSAL SOLUTION 



CA 



confequently, fince t^f- — ?, we get 



__ fin AM cof AM 

 6 - fin AN cof AN : 

 therefore, becavife AM — ;/, it is rnanifeft that AN n <„ 



Thus, then, the refolution of the equation fin (n — ; vj 

 r= e fin n X cof n, coincides with that cafe of the general pro- 

 blem, " De inclinationibus" of the ancients, in which the two given 

 lines are fuppofed to be ftraight lines, interfering one another 



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



by the conftru&ion above, will be no other than the arch t, the 

 firft term in the feries that we have already difcuiTed : and, in like 



manner, if we take CF fucceflively equal to - x ^ ~~ *") x CA • 



£ fin (// — t) 



i (n — *•' ) i — *■") 



7 X fin(«-.^) X ; « X fin (« — »") X CA ' andfoon:. 

 we may find, by the fame conftrudtion, the other terms #\ *■",. 

 9T t &c. of that feries. 



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

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

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

 by means of the known and elementary problem, " De incline 

 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 conftrudtion 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 conducl his calculations, and who 

 will not be fatisfied with a fpeculation of the mind. 



10. The problem, " De inclinationibus" when the two lines 

 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. 



