[ 497 ] 
HT H E defcribing a conic fedtion through the 
angles of a quadrilateral with two parallel fides 
is fo ready a means of affigning loci for the folution 
of folid problems, that it cannot be doubted, but 
this gave rife to the general problem concerning 
three and four lines mentioned by Apollonius, and 
defcribed by Pappus ; and it may be learnt from 
Sirlfaac Newton, who has confidered the problem, 
how ealily the mod: extenfive form of it is reducible 
to the cafe, which I have fuppofed to give rife to 
it. 
Sir Ifaac Newton refers the general problem to 
this : Any quadrilateral A B C D being propofea, 
to find the locus of the point P, whereby PRQJaeing 
drawn parallel to A C and S P T parallel to A B, 
the ratio of the redtangle contained un- 
der QP, PR to that under S P, PT pi.,, 
fhall be given ; and this by purfuing 
the fteps, whereby he proves, that the point P will 
in every quadrilateral be in a conic fediion, may be 
readily reduced to the cafe of a quadrilateral with 
two fides parallel, after this manner. Draw B t 
and DN parallel to AC, then find the point M 
in N D, that the redtangle under M D N be to 
that under ANB in the ratio given, and draw C r 
M d. 
Here Rr will be to A Q, or SP, as M D to 
AN, and B /, or QJ*, to T/ as ND to NB 
whence the redtangle under R r , QJP will be to 
that under SP, T t as that under MDN to that 
under ANB, that is, in the ratio given of the rec- 
tangle under R P Q to that under S P T. Therefore, 
by taking the fum of the antecedents and of the 
confequents. 
