[ 5 « ] 
and dedud from both the redangles that under E I 
or H, and DF, whereby will be left the redangle 
under IDF equal to that under H and FG, both 
whofe fides are given. Draw therefore I K parallel 
to A E, and the redangle under IDF will be equal 
to this given redangle, the given lines KI, AF being 
the afymptotes to the hyperbola pafling through D. 
Coroll. If L M be drawn through B parallel to 
E F, L B fhall be equal to F G, and B M equal to 
E I or H, whereby the hyperbola oppofite to that 
pafting through D will pafs through B. 
SCHOLIUM. 
The proportions of Pappus, which have been refer- 
red to in pag. 500, 501, 504, 1. 2. are given by him, 
among others, for Lemmas fubfervient to the loft 
treatife of Apollonius De J'ediione determinata , and the 
four here cited refped and comprehend all the cafes 
of the problem, where three points are given in any 
line, and a fourth is required fuch, that the redangle 
under the fegments of the propofed line intercepted 
between the point fought, and two of the given 
points, fhall bear a given ratio to the fquare of the 
fegment terminated by the third point. 
The cafes indeed of the problem, from the diver- 
fity of fituation in the points given to the point fought 
and to one another, are in number fix. The given 
extreme of the fegment to conftitute the fquare may 
either be without the other two given points, or be- 
tween them. And when it is without, the point 
fought may be required to be taken without them all, 
either on the fide oppolite to the given extreme of the 
fegment to conftitute the fquare, which will be one 
cale,or it may be required to fall on the fame fide, 
U u u 2 which 
