[ 5°4 ] 
elliplis or lingle hyperbola, will be aiTigned by the 
36th proportion of the forelaid book of Pappus : 
and when oppolite fe&ions, by the 30th proportion, 
or be reduced to the preceding cafes thus. 
Since K G will be to G B as C A to A B, and 
I G to G A as B D to A B, the rectangle under 
K. G I will be to that under A G B, in the given 
ratio of the reftangle under AC, B D to the fq uare 
of A B. Therefore when the ratio of the redtan- 
gle under K E I to that under A I D is given, the 
redlangle under AID alfo bearing a given ratio to 
that under A G B, the ratio of the redtangle under 
K E I to that under A G B will be given, and in 
the lad; place the ratio of the redangle^under G E F 
to that under A G B will be given, this redtangle 
under G E F being the excefs of that under KGI 
above that under K E I [c]. And thereby the fedli- 
ons will be determined, as before. 
AND thus may the locus of the point fought be 
afiigned in all the cafes of this ancient problem, 
which Sir llaac Newton has diftindtly explained. 
T he other cafes, he has alluded to, may be treated, 
as follows. 
When three of the given lines fhall be parallel, 
as A C, B D, and H I, the fourth line being A B, 
FjV an ^ K E L M being parallel to A B, the 
0 9 ratio of the redtangle under K E L to the 
redfangle under E G and E M fhall be givenj; 
that is, three points A, B, and H being given in 
the line AB, with the line GE infilling on AB in 
a given angle, that the redtangle under AGB fhall be 
to that under G H and G E in a given ratio : then 
p] By Prop. 193, Lib. 7. Papp. 
take 
