8 GENERAL THEOREMS, CHIEFLY PORISMS, 



us of that treasure is the knowledge of its high value. I 

 have not added the demonstrations, which are all purely 

 geometrical, granting the methods of tangents and quadra- 

 tures : I have given an example in the abridged synthesis of 

 what I consider as one of the most intricate. It is un- 

 necessary to apologise any further 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 connected 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 unknown, I am persuaded that 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 ex- 

 tensive analogies which every step of our progress in the 

 higher parts of geometry opens to our view. 



PROP. 1. Porism. Fig. 10. A conic hyperbola being 

 given, a point may be found, such, that every straight line 

 drawn from it to the curve, shall cut, 

 p 10 \M i* 1 a given ratio, that part of a straight 

 **" line passing through a given point which 

 is intercepted between a point in the 

 curve not given, but which may be found, 

 and the ordinate to the point where the 

 first-mentioned line meets the curve. Let 

 x be the point to be found, N A the line passing through the 

 given point N, and M any point whatever in the curve ; join 

 x M, and draw the ordinate M p ; then AC is to c p in a given 

 ratio. 



* See the celebrated account of ancient geometrical works, in the 

 seventh book of Pappus. 



