M. DE Zaci^'s ^"yirofiomicjl Obfcr^jJlioni^' 14;; 



'iinus totus =1 is to cofine m :: as cotangent /^ is to the tan- 



.gent of an angle which I put = ?/, and ()o'^ - k '^^ n vviii 



.give -an angle which I put = q. Laftly, the analog}' 



cof. 71 : cof. q.y.fin.k'. will give the cofine of an angle ij/, 



which is the" required motion upon the orbit, or the angle 



'Conipreherided between the two radius veclors m and ^, Let 



therefore ECPMND he the apparent parabolic path of a comet-; 



S the fuK's center ; M and Ntwo places of the comet, the angle 



INISN equal to its motion in longitude, ort'ie Comprehendeti 



angle 4^; P the perihelion-; it is required to tind the two aiio* 



malies ¥M, FN, that is, PSM and PSN, the perihelial dif- 



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



.perihelion P to M andN. i 



■n r ^ • - ' ' 



Keiolution>. 



SM-m 



NSB = .V 



In the right-angled tri^ingleSMRand SNV\Ve hav€ 

 MR = OS:=;;7fm. (4;:fi::.v) NV = QS = ^fin..r; 

 therefore OP^: |/) -m (fin. ij/ =t: x^ and PQ^ 

 Ip^^tim. X ; but by the nature of the parabola 

 MSB= (i^^x)\ nvehaveSMrrAP-f POandSN=AP^-PQ;thatls 

 Parameter — /> | '/;/ ^ |/» — f't (fin. J^^t.v) ^u = -zp^^f^ fo. x 



m + rn (fin-, i^rt -v) ~\p 

 /;/(!+ fin. ijy^tx) = {p 



I and I 4- fin. (4'=fc'V)— -^— 



iP 



fATtzjx, im, X. 



^ (l rp fill, x) =lf 



2fn 



1 =±: fin. X — ^; by 



2> - 



._i^ 



+ ■ 



_ P . 



2> 



reduc^ 



putting hito a fum i + fin. (t|^ — a) + irtrfin. x— ^ . , 

 tion made 2 it fin. .v + fin-. (ij/dtA-) = /"'-^-^J/) ; but by trigono- 

 metrical fonnulae we have fm. (ij;=t:.v) = fin. ^ cof. A;=tfin. ^: 

 <:of. vj/. Subftituting this expreffion in its place we obtain^, 



•2 =tfin. x+ fin.;!/ cof..v-j=fin. x cof.v!/ = (^^) p- By the fame 



femulsci'we have cof." iv= i -fin.* ;t and cof. x~\/ 1 — fin.^^P 



U 2 Sub'- 



