li'L l>E Zacu's Aft ronomicjl Qi)ft;rv£tt ions, 14.7 



lliius totus = I is to coliiie .-7; :: as cotangent k is to the tan- 

 gent of an angle which 1 put = ;7, and 90° - i -p /Z will 

 give an angle which I put = q, Luflly, the analogy 

 cof. « : cof. q :: fin. k ; w^ili giv^c the cofme of an angle if/, 

 wliich is the requircu motion upon the orbit, or the angle 

 compreheaded between the two radius vectors w and ^, l.t't 

 therefore ECPMND be the apparent parabolic path of a comet ; 

 S the lull's center; ]\'I and N two places of the comet, the angle 

 MSN equal to its motion in longitude, or thie comprehended 

 ■angle if/ ; P the perihelion ; it is required to find the two ano* 

 malies PAI, PN, that is, PSM and PSN, the perihelial dif- 

 tance SP, and the time the comet employed to come from its 

 perihelion P to M and N. 

 Refolution. 



:SM = 7n 



•SNrr^ 



MSN - ^ 

 NSB := .V 



MSB=(iI;r±^^) 



Parameter =p 



In the right-angled triangleSMRand SNVwe have 

 MR = OS";;/fin. (4.=t:.v) N V =: QS=:^ fm. .r; 

 therefore OP— Ip — m (fin. 4^ — -^0 ^^^^ PQ^ 

 \p^lJi.{\Vi.x\ but by the natureofthe parabola 

 w^ehaveSMz:.AP^POaiidSN:=AP + PQ;thatis 

 V2':=,\p — m (tin. ipzt.v) 1^,= ipz:izf4, fin. .v , 



m + m (fin. 4^=tzx^ ~ip f^ — H' ^^^^' ^~i P 

 /;/ ( I + fim ij;±.v) rr {p ^.t (l rp fm. x') =lp 



Slid I -f fin. (iLdr.v")— -^ I =i= fill. X := ~\hY 



putting Into a fum i + fin. (4/ — a*) + i t±rfin. .v t=: — +3^ ; reduc- 

 tion made 2^fin. x ^(\i\. (iP'^t^v) = ^-^-7-^) p ; but by trigono- 

 metrical formulae we have (in. (ij;— .v)=:lui. 4^ cof. ATrtrfin. x 

 ^of. ip. Subftituting this exprefTion in its place we obtain, 



2 ^ fin. x-\- fin.ip cof..Vrt:fm . x cof.vl' = ( ^~ — ^ j />. By the fame 

 formulas we have cof.- x—i^ fm.^ a; and cof. x — s/\ -lin. 



U 2 



X\, 



Sub-' 



