Mr . Brougham’s general Theorems , &c. 379 
they are, very probably, in the doctrine of curve lines, what the 
ancients appear to have prized so much. in plain geometry ; al- 
though, unfortunately, all that remains to us of that treasure, is 
the knowledge of its high value. Neither have I added the de- 
monstrations, which are all purely geometrical, granting the 
methods of tangents and quadratures : I have given an example, 
in the abridged synthesis, of what I consider as one of the most 
intricate. It is unnecessary to apologize any farther for the 
conciseness of this tract. Let it be remembered, that were 
each proposition followed by its analysis and composition, and 
the corollaries, scholia, limitations, and problems, immediately 
suggested by it, without any trouble on the reader's part, the 
whole would form a large volume, in the style of the ancient 
geometers ; containing the investigation of a series of con- 
nected truths, in one branch of the mathematics, all arising 
from varying the combinations of certain data enumerated in 
a general enunciation.* 
As a collection of curious general 'truths, of a nature, so far 
as I know, hitherto quite unknown, I am persuaded this Paper, 
with all its defects, may not be unacceptable to those who feel 
pleasure in contemplating the varied and beautiful relations 
between abstract quantities, the wonderful and extensive ana- 
logies which every step of our progress in the higher parts of 
geometry opens to our view. 
prop. 1. porism. (Tab. XXI. fig. 1.) 
% 
A conic hyperbola being given, a point may be found, such, 
that every straight line drawn from it to the curve, shall cut, in 
* See the celebrated account of ancient geometrical works, in the eleventh book of 
Pappus Alexan drxnus. 
3C2 
