[ 5°9 3 
of B C will be the fame with that of the given red- 
angle under BAC to the fquare of AP: therefore 
AP is given, and thence the fedions. For let Pv S 
be the fecondary diameter, to which BC is ordi- 
nately applied, and T the center of the oppofite 
fedions. Then the fquare of BM will be to the red- 
angle under AMT as the fquare of the tranfverfe di- 
ameter conjugate to the fecondary diameter R S to the 
fquare of this fecondary diameter ; and if a line were 
drawn from M to P, this would touch the hyper- 
bola BP in P Qg*], and the fquare of AP will be 
to the redangle under MAT in the fame ratio ; 
therefore the given ratio of the fquare of M B to the 
fquare of A P will be that of the redangle under 
AMT to the redangle under M A T, or the ratio 
of M T to AT; confequently the ratio of M T 
to A T is given, and thence the point T. But alfo 
the diameter R S is given in magnitude, the fquare 
of R T or of S T being equal to the redangle un- 
der M T A ; whence in the laffc place the tranfverfe 
diameter conjugate to this is alfo given ; for the fquare 
of this diameter is to the fquare of RT as the given 
fquare of B M to the redangle under AMT now 
alfo given. 
But a more fimple cafe may alfo be propofed in 
three lines, when the ratio of the redangle Fiff>154 
under EDF fhould be equal to the redan- 
gle under a given line, and that drawn from D to 
B C in a given angle. 
This line will bear, both to B E and F C, a given 
ratio, and the redangle under EDF will be in a given 
jjr] Apoll. conic. L. II. prop. 40. 
Vol. LIII. Uuu 
ratio 
