338 ON THE APPLICATION OF ANALYSIS TO THE DISCOVERY 



cient geometry, the views which are here opened may proba- 

 bly be regarded with surprise, not unmixed with regret, that 

 the increasing perfection of the language of symbols should 

 gradually cause it to usurp the hitherto exclusive domains of 

 the higher geometry. But whatever may be the discoveries to 

 which the geometrical interpretation of language, at once the 

 most comprehensive and condensed which human ingenuity 

 has devised, shall give birth, the restorer of the Porisms of 

 Euclid, and the author of the General Theorems, will retain 

 an undiminished reputation, and their works continue to be 

 studied, by all those who wish to acquire a correct taste for the 

 geometry of the ancients. Those propositions which have re- 

 ceived the appellation of Local Theorems and Porisms, may, 

 in one point of view, be considered as differing from theorems 

 and problems, by having something more general, or indeter- 

 minate, in their nature ; by affirming that some property is pos- 

 sessed not merely by one individual, but by every one, of 

 some class or species. It is on this circumstance that the alge- 

 braic investigation of porisms is founded : the arbitrary con- 

 stants of an equation of two variables, are made to submit to 

 certain conditions, which shall leave the variables themselves 

 still indeterminate. By generalising this process, we take, in- 

 stead of arbitrary constants, unknown functions of the variables 

 themselves ; and instead of the resulting algebraic equations, 

 we find functional equations which determine the form of the 

 functions we have assumed. 



This process, which will be better understood by the subse- 

 quent inquiries, leads us at once to the highest pitch of gene- 

 rality, and puts us in possession of innumerable porisms, and 

 local theorems, each comprehending whole classes of curves. 

 By limiting and determining the form of the arbitrary func- 

 tions involved in the solution, we gradually restrict the extent 



of 



