[ 5*4 ] 
gle under AFC together with that under EFB 
will be equal to the redangle under G and D F. 
Here if it were propofed to find the point F, that the 
ratio of the redangle under AFC to that under EFB 
fhould be given.the ratio of the rectangle under AFC 
to that under DF and the given line G would be given. 
But this analyfis may be carried on to a compleat 
folution of the problem thus. If C N be taken to 
G in the given ratio of the redangle under AFC to 
that under DF and G, | j__j j | 
the point N will be given, a C D E F B N 
and the redangle under j \ 
A F, C N will be to G 
that under A F, G in this ratio of CN to G ; confe- 
quently the excefs of the redangle under A F, C N 
above that under AFC, that is, the redangle under 
A F N, will be to the excefs of the redangle under 
A F and G above that under D F and G, or the given 
redangle under A D, G, in the fame given ratio, and 
in the lad: place the redangle under AFN will equal 
the given redangle under A D and C N. 
Here I have chofen this proportion in particular, 
becaufe the cafe of the problem, to which it is fubfer- 
vient, is fubjed to a determination, when F N fhall 
be equal to A F. And then the redangle under 
AFN being equal to that under AD and CN, as CN to 
FN fo is AF to AD, and by divifion as CF to FN fo DF 
to AD; therefore when AF is equal to FN, CF will 
be to A F as FD to A D : confequently CD to F D 
as FD to AD, and the fquare of DF equal to the 
redangle under ADC, when the problem admits of 
a fingle folution only, wherein the rectangle under 
AFC 
