[ 5 * 6 ] 
taken equal to the fpace given, and a circle in 
like manner be defcribed through C, D, E, cutting 
A B in F and G, the redangles under A F B and 
AGB will each be equal to the given fpace. 
Here it is evident, that the fpace given muft not 
exceed the fquare of half A B, when equal, the cir- 
cle will touch A B in it’s middle point. 
POSTSCRIPT. 
A S this application to a given line of a redangle ex- 
ceeding or deficient by a fquare, or the more 
general problem treated of in the fixth book of the 
elements, of applying a fpace to a line fo as to exceed 
or be deficient by a parallelogram given in fpecies, 
is the mod obvious refult, to which the analyfis of 
plane problems, not too fimple to require this con- 
ifrudion, leads ; fo the defcriptions of the conic fec- 
tions here treated of, ftand in the like head in regard 
to the higher order of problems ftyled folid from the 
ufe of the conic fedions deemed necefiary for their 
genuine folution. And thefe are the only modes of 
folution, the modern algebra, which grounds its ope- 
rations on one or two elementary propofitions only, 
naturally leads to. But as the form of analyfis a- 
mongft the antients, by expatiating through a larger 
field, often was found to arrive at conclufions much 
more concife and elegant, than could offer themfelves 
in a more confined track ; the antient fages in geome- 
try, that the folid order of problems might not want 
this advantage, fought out that copious and judicious 
colledion of properties attending the conic fedions, 
which 
