118 



BowdifcJi^s Hemarlcson Doctor Stewart^ s Formula^ 



t 



3_ 



apogee to apogee to be represented h^ 300°. (1 f w}% we should 



4 



have hy Prop. 9. of the same work, h 



preceding equation would become 



2 



A irt 



2 



2 



n 



+ 5, so that the 



m 



2 



u 







0, Now, when 



^. 



4 



w is extremely small, the motion of the mocn from apogee to 



apogee is by formula (7) nearly equal to 36o° (1 4 ^ m 



, which 



beins: put equal to the assumed value 380° (t -f nY , gi^es 



r> 



n 



~ nf nearly ; and then the left hand side of the preceding 



expression would become 



2 



5, which is negative, whereas 



it was found to be positive for the value 



m 



0,0748013 corres- 



ponding to the present situation of the moon, consequently there 

 must be a value of m less than 0,074oOI3, that would satisfy the 



proposed equation 



2 



111 



9 



n 



5 



; and a slight attention to the 



subject will show that no great change in the present value of m 

 would be necessary to produce this effect, and thus AiU upon the 

 case which would make the computed distance of the sun from 

 the earth infinite. Less values of m would be impossihle iu this 

 method of calculation,' because the versed sine vs would be nega- 

 tlve. It was thought unnecessary to make this part of the calcu- 

 lation with any greater degree of precision, on account of the terms 

 of the order m' neglected in formula (7). upon which the value of 

 n depends. 



I shall clo^e these observations with the remark, that the mistake 

 here noticed in D >ctor Stewart's tracts may be attributed wholly 

 to the use of the geometrical method of investiiration to which that 



eminent geometrician was so very much attached — the failure of so 

 distinguished a mathematician shows how extremely difficult it Is 



