[ 499 ] 
diameter of the ffedtion, to which AC and BD 
are lines ordinately applied, will he in the line 
M I ; and if NP, QS are tangents to the p. 
fedtion, and parallel to A C and BD, the " lt; ’ 
points O, R, in which they interfect M I, will be 
the points of their contadi, and the vertexes of that 
diameter. But the fquare of N O is to the rectangle 
under A N B, and the fquare of Q^R to the rect- 
angle under A QJ3, as the rectangle under E G H 
or F E G to that under AGB, therefore in a given 
ratio ; but the ratio of N M to NO, the fame as 
that of QJVI to QJG is alfo given ; whence the 
ratio of the fquare of N M to the rediangle under 
A N B, or of the fquare of O M to the redtangle 
under K O L, is given, as likewife the ratio of the 
fquare of R M to the redtangle under K R L. 
Now in the ellipfis the fquare of M O, the di- 
ffance of the remoter vertex of the diame- 
ter O R from M, is greater than the redt- = ’ J 
angle under KOL; that is, the ratio given of the 
rectangle under FEG to that under AGB muft be 
greater than the ratio of the fquare of half the differ- 
ence between A C and B D to the fquare of A B = , 
But in the hyperbola the fquare of M O is lefs than 
the rediangle under KOL ; whereby the ratio of 
the redtangle under FEG to that under p . 
AGB fhall be lefs than that of the fquare 
of half the difference between A C and B D to the 
fquare of A B [«]. In 
[a] As the fquare of O M fhall be greater or lefs than the rect- 
angle under KOL, the fquare of NM will be refpedtively greater 
or lefs than the redtangle under ANB ; therefore the ratio of the 
fquare of N O to the redtangle under ANB, that is, of the 
redtangle under FEG to that under AGB, will be accordingly 
greater 
