lOO 



Let the diftance between the foci be divided in extreme and mean 

 Fig. 2. ratio, about the dividing point, ("nearer the higher focus) let a circle 

 be defcribed with a radius, equal the femiaxis major ; from the upper 

 focus draw a line interfering the circle and making with the axis 

 major an angle = the mean anomaly ; and join the point of interfec- 

 tion, and the lower focus by a line, which will cut the ellipfe in the 

 place of the planet, according to Mercator.* 



Now if M be the point of divifion, let MH= (3— ^5X«)="^- 

 Let DSU-a, and M D H=.v 

 then I. s, x—Ties, m 

 andj, H : J, SDH : : SD : SH 

 J, DSH:.f, HMD::MD:SD 



Therefore s, mxs, a : s, m — axst m — x : : i : ie 



or 2. J, m — axs, m — x—2es,7nxs,a 



Let the fucceffive fluxions of thefe equations be taken per Sal/urn. 



"When a=Jii, x=o, e—o and e is confhant we have 



1. x=nes, m — a—2es, tn 



2. x—o —as, m-{-2axcs, 7nzz2smx2aecs, m 



3. &c. —a\a ■\-'i^axcs,vi + 3^a'x, m= 



=^2,yas., mxcs, m — 2ea s, m 



Hence, 



• Montucla, Hid. Math. Tom. 2, p. 340, has ftated this hypothefis very imperfeflly 

 and inaccurately : his words are, " Nicoles Mercator y fit dans Ie fuite, une correc- 

 " tion. II partagea la diftance entre les foyers de I'ellipfe in moyenne et extreme 

 " raifon, de forte que Ie point de feflion tombat au dela du centre a I'egard du foyer 

 " occupe par la planete centrale, et ce fut ce foinl gu'il prit pour centre du movemiut 

 " moyin." 



