[ 5° 8 ] 
the iquare of B C will be compounded or the ratio- 
of the Iquare of MN to the l'quare of NA, and of 
the ratio of the redtangle under B A C to the fourth 
part of the fquare of BC; and thus the line A M 
will be divided in N in a given ratio, and the point 
N, one vertex of the diameter, to which B C is or- 
dinately applied, will be given. 
If A N be equal to N M, the point N will be 
tlie only vertex of this diameter, and the fedtion will 
be a parabola. 
Otherwile by taking the point O in A M extend- 
ed, fo that the ratio of A O to O M be the fame 
with that of AN to N M, the point O will be the 
other vertex of the diameter. 
And here if the ratio of A N to NM be that of 
a greater to a lefs, the point O will fall beyond M 
from A within the angle B AC, the conic fedtion 
being an ellipfis. 
But if the ratio of AN to NM be that of a 
lefs to a greater, the point O will fall on the other 
fide of A, and the fedtion will be an hyperbola. 
F . And in this cafe if the oppofite fedtion be 
drawn, that alfo will be the locus of the 
point D within the angle vertical to the angle 
B AC. 
In the laft place, if D be in either of the collate- 
ral angles, A M drawn as before will contain a Se- 
condary diameter in oppofite fedtions, one of which 
fhall touch BA in B, and the other C A in 
g ' I4 ‘ C. Then if one of thefe fedions pafs thro’ 
D, the fedtions will be given. For here P A (^be- 
ing drawn through A parallel to B C, the given ra- 
tio of the redtangle under CL, B K to the fquare 
of 
