I. 



GENERAL THEOREMS, CHIEFLY PORISMS, IN THE 

 HIGHER GEOMETRY.* 



THE following are a few propositions that have occurred to 

 me in the course of a considerable degree of attention which I 

 have happened to bestow on that interesting, though difficult 

 branch of speculative mathematics, the higher geometry. 

 They are all in some degree connected; the greater part 

 refer to the conic hyperbola, as related to a variety of other 

 curves. Almost the whole are of that kind called porisms, 

 whose nature and origin is now well known; and, if that 

 mathematician to whom we owe the first distinct and popular 

 account of this formerly mysterious, but most interesting 

 subject,! should chance to peruse these pages, he will find in 

 them additional proofs of the accuracy which characterizes 

 his inquiry into the discovery of this singularly-beautiful 

 species of proposition. 



Though each of the truths which I have here enunciated is 

 of a very general and extensive nature, yet they are all dis- 

 covered by the application of certain principles or properties 

 still more general ; and are thus only cases of propositions 

 still more extensive. Into a detail of these I cannot at 

 present enter : they compose a system of general methods, by 

 which the discovery of propositions is effected with certainty 

 and ease; and they are, very probabl}% in the doctrine of 

 curve lines, what the ancients appear to have prized so nmch 

 in plain geometry ; though unfortunately all that remains to 



* From Phil. Trans., 1798, part ii. 



t See Mr. Playfair's Paper in vol. iii. of the 'Edinburgh Transactions.' 



