[S® 3 ] 
Moreover D K being to A C as K M to C M, the 
ratio of D K to A C, that is, the ratio of the rec- 
tangle under JBKD to the fquare of AC, will belefs 
than the ratio of KL to C L, or the ratio of 
the redangle under C K L to that under A B, CL; 
therefore, by permutation and inverdon, the ratio 
of the redangle under C K L to the redangle under 
B K D, that is, the given ratio of the redangle 
under N E P to that under A N C, equal to that 
under G E is greater than the ratio of that un- 
der A B, CL to the fquare of A C. And hence 
the oppoiite fedions pading through the angles of 
the quadrilateral A B C L, whofe fides A B, C L 
are parallel, will be given as before. 
When the given ratio of the fquare of O M to 
the redangle under L O K {hall be that of ^ 
the redangle under K ML to the fquare 
of half K L, whereby the given ratio of the 
redangle under FEG to that under A G B {hall be 
that of the redangle under AC, BD to the fquare 
of A B, the points T and V {hall unite in one, bi- 
feding K L, and the points O and R fhall alfo 
unite in one, dividing the line KLM harmonical- 
ly ; and then the Incus of the point E will be each 
of the diagonals of the quadrilateral. 
In the laft place, if the diagonals AD, B C of 
the quadrilateral were drawn, cutting G E p ifr g 
in I and K, and the ratio of the redtangle 
under K E I to that under AID were given, and 
not that of the redangle under G E F to that un- 
der A G B ; then the interfedion of thefe diago- 
nals, as L, will be in the line drawn from M bi- 
feding A C, and B D, and the point L will fall 
within the quadrilateral, whereby the locus, when an 
T 1 1 2 ellipds 
