L 5° 1 J _ 
R I will lie given, the conic ledtion being here u 
parabola, tills diameter having thus but one ver- 
tex. 
Hitherto the point E, when the line E F G falls 
between A C and B D, is without the quadrilw 
teral', and within the lines A B, CD, when E F G 
is without the quadrilateral. 
But when E is within the lines AC, B D in the 
hr ft cafe, and without in the lecond, the locus or 
the point E will be oppolite fedtions, each palling 
through two angles of the quadrilateral. 
When one fedtion paftes through A and C, and 
the other through B and D ; then if the diameter 
M I be drawn, as before, and to K L be applied a 
redtangle deficient by a fquare, to which ^ 
the given redtangle under KML (hall be in 1& ' 
the given ratio of the fquare of M O to the redtan- 
gle under K O L, or of the fquare of M R to the 
redtangle under K R L, the points T and V, con- 
fti tuting the redtangles under KT L andunder K VL, 
being thus found, M O will be to O T, and M R 
to R V, in this given ratio (by prop* 30. L.7. Papp.) 
O and T being the vertexes of the diameter M I. 
But the redtangles under K T L, K V L cannot 
be aftigned, as here required, unlefs the ratio given 
for that of the fquare of O M to the redtangle under 
KO L, or that of the fquare of R M to the rec- 
tangle under K R L, be not lefs than that of the 
redtangle under K M L to the fquare of half K L ; 
that is, when the ratio of the fquare of O N to the 
redtangle under ANB, and that of the fquare of 
R Q_to the redtangle under A Q B, or that of the 
given ratio of the redtangle under F E G to that 
under A G B is not lefs than that of the redtangle 
Vol. LIII. Ttt under 
