388 Mr. Brougham’s general Theorems 
s , ■ } 
remains the area MPN, equal to x AP x MP -- — q — x PT; 
q — p q — P 
which was before demonstrated to be, together with APM, 
equal to . ACD. Wherefore MPN, together with APM, 
that is, the area AMN is equal to . ACD; consequently, 
AMN : ACD :: M : M -f- N; and ( dividendo ) AMN : NMDC 
: : M : N. An area has therefore been found, which the hy- 
perbola IC' always cuts in a given ratio. 
Wherefore, a conic hyperbola being given, &c. Q. E. D. 
Scholium. This proposition points out, in a very striking 
manner, the connection between all parabolas and hyperbolas, 
and their common connection with the conic hyperbola. The 
demonstration which I have given is much abridged ; and, to 
avoid circumlocution, algebraic symbols, and even ideas, have 
been introduced : but, by attending to the several steps, any 
one will easily perceive that it may be translated into geome- 
trical language, and conducted upon purely geometrical prin- 
ciples, if any numbers be substituted for m , n, p , and q ; or 
if these letters be made representatives of lines, and if concise- 
ness be less rigidly studied. 
PROP. XIV. THEOREM. 
A common logarithmic being given, if from a given point, 
as origin, a parabola (or hyperbola) of any order whatever be 
described, cutting, in a given ratio, a given area of the loga- 
rithmic, the point where this curve meets the logarithmic is 
always situated in a conic hyperbola, which may be found. 
Scholium. This proposition is, properly speaking, neither a 
porism, a theorem, nor a problem. It is not a theorem, be- 
cause something is left to be found, or, as Pappus expresses 
