[ 525 ] 
equality N S will be to N K as A G to A Q, that is, 
as K R to Q^K ; and in the lad place N S to K S as 
K R to QJl, that is, the rectangle under S K R equal 
to the given redangle under N S, QJl • whence the 
point K, the pofition of K D, and thence the point 
D will be given. 
But it D K be not ordinately applied to L M, let 
D O be oidinately applied to L M. Then here the 
rectangle under A QJl, equal to the fquare of QM, 
will be equal to that under O QG, and 
G t0 A Q_as QJl to O Q^whence by 2 ^* 
compofition AG to AQjis OR to OQ^ But BN be- “ 
ing now alio taken equal to I, and N S to A B as 
I to A Q, A B will be here in like manner to 
K N as AG to I, and NS to K N as A G to A Qj 
therefore NS will be to KN as OR to OQ, and' 
by converfion NS to K S as OR to Q^R. But 
N S and QJl being both given in magnitude, if S P 
be taken to N S as QR to P R, the point P will be 
given, and alfo by equality S P will be to K S as O R 
to PR; whence if R V be drawn parallel to D O, 
and S T to K D, both R V and ST will be given 
in pofition, one pafiing through the given point R, 
parallel <o the ordinates applied to the axis LM, and 
the other through the point S alfo given, and parallel 
to K D or C B : alfo DTV being drawn parallel to 
M L, D T will be equal to K S and D V equal to 
O R, therefore as S P to D T fo D V to P R, and the 
redfangle under SPR equal to that under TDV, 
confequently the point D in an hyperbola paffing thro’ 
P, and having for afymptotes the lines ST, RV given 
in pofition. 
Yyy 
In 
