C 515 ] 
AFC will bear to that under E F B a lefs ratio than in 
any other fituation of the point F between E and B. 
Moreover CN is to G as the reCtangle under 
AFC to the fum of the reCtangles under AFC and 
EFBj therefore FN being equal to A F, when the 
problem is limited to this fingle folution, the rectangle 
under AFC fhall be to the rectangles under AFC 
and E F B together as the fum of AF and F C to 
Gj which is equal to the fum of A E and C B ; 
whence by diviiion the ratio of the reCtangle under 
AFC to that under E F B, when the problem is 
limited to this fingle folution, will be that of the 
fum of AF and C F to the excefs of F B above E F. 
Thus direCtly do thefe lemmas correfpond with A- 
pollonius’s firft mode of folution, and lead to the ge- 
neral principle of applying to a given line a reCtangle 
exceeding or deficient by a fquare, which ./hall be equal 
to a fpace given. This being a fimple cafe of the 28th 
and 29th proportions of the 6th book of Euclid's ele- 
ments, admits of a compendious folution. Such a one 
is exhibited by Snellius in his treatife on thefe problems 
(in Apollon. Batav.) and Des Cartes has exhibited 
another more contracted in it’s terms, but not there- 
fore more ufeful. It may alio be performed thus. 
If upon a given line A B any triangle A C B be 
ereCted at pleafure j then if the legs CA, CB, 
whether equal or unequal, be continued to Fig. 1 7. 
D and E, that the reCtangles under CAD 
and C B E be each equal to the given fpace, 
and a circle be defcribed through C, D, E cut- 
ting A B extended in F and G, the reCtangle un- 
der B F A and B G A will each be equal to F . 0 
the fpace given. Alfb if in the legs CA, ' IS,I °’ 
CB the reCtangles under CAD and CBE be each 
taken 
